diff --git a/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/phase1.log b/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/phase1.log
new file mode 100644
index 00000000..c4030aac
--- /dev/null
+++ b/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/phase1.log
@@ -0,0 +1,9 @@
+Phase 1 Acquire for score_dynamics_giorgini: https://youtu.be/P75iVMmbqQk
+Artifacts: C:\projects\manual_slop\conductor\tracks\video_analysis_score_dynamics_giorgini_20260621\artifacts
+Step 1: extract_transcript (yt-dlp VTT directly)
+  OK: wrote C:\projects\manual_slop\conductor\tracks\video_analysis_score_dynamics_giorgini_20260621\artifacts\transcript.json (2998 segments)
+Step 2: download_video
+{
+  "status": "error",
+  "error": "download_video: YtdlpError: ERROR: unable to download video data: HTTP Error 403: Forbidden\n"
+}
diff --git a/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/transcript.json b/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/transcript.json
new file mode 100644
index 00000000..20d83131
--- /dev/null
+++ b/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/transcript.json
@@ -0,0 +1,20940 @@
+{
+  "video_id": "P75iVMmbqQk",
+  "segments": [
+    {
+      "start": 0.0,
+      "duration": 0.0,
+      "text": "for<00:00:00.560><c> inviting</c><00:00:01.080><c> me</c><00:00:01.280><c> at</c><00:00:01.440><c> your</c>"
+    },
+    {
+      "start": 2.39,
+      "duration": 0.0,
+      "text": "for inviting me at your"
+    },
+    {
+      "start": 2.4,
+      "duration": 0.0,
+      "text": "for inviting me at your group<00:00:02.600><c> meeting.</c><00:00:03.280><c> Um</c>"
+    },
+    {
+      "start": 3.95,
+      "duration": 0.0,
+      "text": "group meeting. Um"
+    },
+    {
+      "start": 3.96,
+      "duration": 0.0,
+      "text": "group meeting. Um So,<00:00:04.160><c> in</c><00:00:04.400><c> today's</c><00:00:04.720><c> presentation,</c><00:00:06.160><c> uh</c><00:00:06.240><c> I</c><00:00:06.520><c> will</c>"
+    },
+    {
+      "start": 7.39,
+      "duration": 0.0,
+      "text": "So, in today's presentation, uh I will"
+    },
+    {
+      "start": 7.4,
+      "duration": 0.0,
+      "text": "So, in today's presentation, uh I will talk<00:00:07.640><c> about</c><00:00:08.600><c> building</c><00:00:09.720><c> mathematical</c><00:00:10.360><c> models</c>"
+    },
+    {
+      "start": 11.55,
+      "duration": 0.0,
+      "text": "talk about building mathematical models"
+    },
+    {
+      "start": 11.56,
+      "duration": 0.0,
+      "text": "talk about building mathematical models from<00:00:12.600><c> high</c><00:00:12.880><c> high-dimensional</c><00:00:13.960><c> partially</c>"
+    },
+    {
+      "start": 14.35,
+      "duration": 0.0,
+      "text": "from high high-dimensional partially"
+    },
+    {
+      "start": 14.36,
+      "duration": 0.0,
+      "text": "from high high-dimensional partially observed"
+    },
+    {
+      "start": 15.79,
+      "duration": 0.0,
+      "text": "observed"
+    },
+    {
+      "start": 15.8,
+      "duration": 0.0,
+      "text": "observed dynamical<00:00:16.360><c> systems.</c>"
+    },
+    {
+      "start": 17.59,
+      "duration": 0.0,
+      "text": "dynamical systems."
+    },
+    {
+      "start": 17.6,
+      "duration": 0.0,
+      "text": "dynamical systems. So,<00:00:17.680><c> I</c><00:00:17.720><c> think</c><00:00:18.040><c> this</c><00:00:18.280><c> is</c><00:00:18.480><c> a</c><00:00:18.840><c> central</c><00:00:19.240><c> topic</c><00:00:20.040><c> in</c>"
+    },
+    {
+      "start": 20.31,
+      "duration": 0.0,
+      "text": "So, I think this is a central topic in"
+    },
+    {
+      "start": 20.32,
+      "duration": 0.0,
+      "text": "So, I think this is a central topic in many<00:00:20.560><c> scientific</c><00:00:21.040><c> fields</c><00:00:21.760><c> ranging</c><00:00:22.320><c> from</c>"
+    },
+    {
+      "start": 22.95,
+      "duration": 0.0,
+      "text": "many scientific fields ranging from"
+    },
+    {
+      "start": 22.96,
+      "duration": 0.0,
+      "text": "many scientific fields ranging from geophysical<00:00:23.560><c> fluid</c><00:00:24.000><c> dynamics,</c><00:00:24.640><c> which</c><00:00:24.800><c> is</c><00:00:24.880><c> the</c>"
+    },
+    {
+      "start": 24.99,
+      "duration": 0.0,
+      "text": "geophysical fluid dynamics, which is the"
+    },
+    {
+      "start": 25.0,
+      "duration": 0.0,
+      "text": "geophysical fluid dynamics, which is the field<00:00:25.280><c> that</c><00:00:25.480><c> is</c><00:00:25.640><c> the</c><00:00:25.760><c> closest</c><00:00:26.480><c> uh</c>"
+    },
+    {
+      "start": 27.23,
+      "duration": 0.0,
+      "text": "field that is the closest uh"
+    },
+    {
+      "start": 27.24,
+      "duration": 0.0,
+      "text": "field that is the closest uh to<00:00:27.520><c> my</c><00:00:27.720><c> research,</c><00:00:28.440><c> but</c><00:00:28.680><c> also</c><00:00:29.720><c> other</c><00:00:30.080><c> physical</c>"
+    },
+    {
+      "start": 30.43,
+      "duration": 0.0,
+      "text": "to my research, but also other physical"
+    },
+    {
+      "start": 30.44,
+      "duration": 0.0,
+      "text": "to my research, but also other physical systems<00:00:31.200><c> like</c><00:00:31.560><c> molecular</c><00:00:32.160><c> systems</c><00:00:33.000><c> or</c>"
+    },
+    {
+      "start": 33.31,
+      "duration": 0.0,
+      "text": "systems like molecular systems or"
+    },
+    {
+      "start": 33.32,
+      "duration": 0.0,
+      "text": "systems like molecular systems or neuroscience.<00:00:34.360><c> So,</c><00:00:34.680><c> so</c><00:00:34.800><c> every</c><00:00:35.000><c> times</c><00:00:35.520><c> we</c><00:00:35.680><c> have</c>"
+    },
+    {
+      "start": 36.39,
+      "duration": 0.0,
+      "text": "neuroscience. So, so every times we have"
+    },
+    {
+      "start": 36.4,
+      "duration": 0.0,
+      "text": "neuroscience. So, so every times we have access<00:00:37.000><c> to</c><00:00:37.160><c> data,</c><00:00:37.680><c> so</c><00:00:37.800><c> we</c><00:00:37.920><c> have</c><00:00:38.200><c> observations</c>"
+    },
+    {
+      "start": 39.87,
+      "duration": 0.0,
+      "text": "access to data, so we have observations"
+    },
+    {
+      "start": 39.88,
+      "duration": 0.0,
+      "text": "access to data, so we have observations of<00:00:40.440><c> the</c><00:00:40.560><c> underlying</c><00:00:41.040><c> system,</c><00:00:41.920><c> and</c><00:00:42.120><c> we</c><00:00:42.240><c> want</c><00:00:42.800><c> to</c>"
+    },
+    {
+      "start": 42.91,
+      "duration": 0.0,
+      "text": "of the underlying system, and we want to"
+    },
+    {
+      "start": 42.92,
+      "duration": 0.0,
+      "text": "of the underlying system, and we want to build<00:00:43.640><c> a</c><00:00:43.720><c> mathematical</c><00:00:44.360><c> model</c><00:00:44.920><c> that</c><00:00:45.080><c> is</c><00:00:45.280><c> able</c>"
+    },
+    {
+      "start": 45.91,
+      "duration": 0.0,
+      "text": "build a mathematical model that is able"
+    },
+    {
+      "start": 45.92,
+      "duration": 0.0,
+      "text": "build a mathematical model that is able to<00:00:46.040><c> reproduce</c><00:00:47.120><c> and</c><00:00:47.280><c> capture</c><00:00:48.240><c> the</c><00:00:48.360><c> main</c>"
+    },
+    {
+      "start": 48.83,
+      "duration": 0.0,
+      "text": "to reproduce and capture the main"
+    },
+    {
+      "start": 48.84,
+      "duration": 0.0,
+      "text": "to reproduce and capture the main causality<00:00:49.400><c> relationships</c><00:00:50.280><c> between</c><00:00:50.600><c> the</c>"
+    },
+    {
+      "start": 50.71,
+      "duration": 0.0,
+      "text": "causality relationships between the"
+    },
+    {
+      "start": 50.72,
+      "duration": 0.0,
+      "text": "causality relationships between the variable<00:00:51.160><c> of</c><00:00:51.280><c> the</c><00:00:51.360><c> system,</c><00:00:52.560><c> and</c><00:00:52.760><c> also</c>"
+    },
+    {
+      "start": 53.07,
+      "duration": 0.0,
+      "text": "variable of the system, and also"
+    },
+    {
+      "start": 53.08,
+      "duration": 0.0,
+      "text": "variable of the system, and also hopefully<00:00:53.680><c> to</c><00:00:53.840><c> perform</c><00:00:54.760><c> predictions</c><00:00:55.760><c> and</c>"
+    },
+    {
+      "start": 55.95,
+      "duration": 0.0,
+      "text": "hopefully to perform predictions and"
+    },
+    {
+      "start": 55.96,
+      "duration": 0.0,
+      "text": "hopefully to perform predictions and that<00:00:56.280><c> uncertainty</c><00:00:57.600><c> quantification.</c>"
+    },
+    {
+      "start": 59.79,
+      "duration": 0.0,
+      "text": "that uncertainty quantification."
+    },
+    {
+      "start": 59.8,
+      "duration": 0.0,
+      "text": "that uncertainty quantification. So,<00:00:59.920><c> of</c><00:00:59.960><c> course</c><00:01:00.280><c> for</c><00:01:00.480><c> this</c><00:01:01.040><c> specific</c><00:01:01.520><c> class</c><00:01:01.800><c> of</c>"
+    },
+    {
+      "start": 61.91,
+      "duration": 0.0,
+      "text": "So, of course for this specific class of"
+    },
+    {
+      "start": 61.92,
+      "duration": 0.0,
+      "text": "So, of course for this specific class of system"
+    },
+    {
+      "start": 63.23,
+      "duration": 0.0,
+      "text": "system"
+    },
+    {
+      "start": 63.24,
+      "duration": 0.0,
+      "text": "system that<00:01:03.480><c> are</c><00:01:03.680><c> extremely</c><00:01:04.120><c> high-dimensional</c><00:01:05.160><c> and</c>"
+    },
+    {
+      "start": 65.35,
+      "duration": 0.0,
+      "text": "that are extremely high-dimensional and"
+    },
+    {
+      "start": 65.36,
+      "duration": 0.0,
+      "text": "that are extremely high-dimensional and also<00:01:05.720><c> and</c><00:01:06.000><c> also</c><00:01:06.280><c> multiscale</c><00:01:07.080><c> and</c><00:01:07.280><c> partially</c>"
+    },
+    {
+      "start": 67.67,
+      "duration": 0.0,
+      "text": "also and also multiscale and partially"
+    },
+    {
+      "start": 67.68,
+      "duration": 0.0,
+      "text": "also and also multiscale and partially observed,<00:01:08.600><c> uh</c>"
+    },
+    {
+      "start": 69.55,
+      "duration": 0.0,
+      "text": "observed, uh"
+    },
+    {
+      "start": 69.56,
+      "duration": 0.0,
+      "text": "observed, uh it<00:01:09.920><c> is</c><00:01:10.040><c> quite</c><00:01:10.280><c> meaningless</c><00:01:11.360><c> to</c><00:01:11.880><c> try</c><00:01:12.480><c> to</c>"
+    },
+    {
+      "start": 72.63,
+      "duration": 0.0,
+      "text": "it is quite meaningless to try to"
+    },
+    {
+      "start": 72.64,
+      "duration": 0.0,
+      "text": "it is quite meaningless to try to perform<00:01:13.960><c> trajectory</c><00:01:14.680><c> shadowing.</c><00:01:15.440><c> So,</c><00:01:15.640><c> it's</c>"
+    },
+    {
+      "start": 75.75,
+      "duration": 0.0,
+      "text": "perform trajectory shadowing. So, it's"
+    },
+    {
+      "start": 75.76,
+      "duration": 0.0,
+      "text": "perform trajectory shadowing. So, it's quite<00:01:15.920><c> meaningless</c><00:01:16.680><c> to</c><00:01:17.160><c> try</c><00:01:17.440><c> to</c><00:01:17.560><c> build</c><00:01:17.760><c> a</c>"
+    },
+    {
+      "start": 77.83,
+      "duration": 0.0,
+      "text": "quite meaningless to try to build a"
+    },
+    {
+      "start": 77.84,
+      "duration": 0.0,
+      "text": "quite meaningless to try to build a model<00:01:18.240><c> that</c><00:01:18.440><c> is</c><00:01:18.640><c> able</c><00:01:19.000><c> to</c><00:01:19.160><c> precisely</c>"
+    },
+    {
+      "start": 80.91,
+      "duration": 0.0,
+      "text": "model that is able to precisely"
+    },
+    {
+      "start": 80.92,
+      "duration": 0.0,
+      "text": "model that is able to precisely predict<00:01:21.560><c> the</c><00:01:21.720><c> trajectory</c><00:01:22.360><c> of</c><00:01:22.480><c> the</c><00:01:22.560><c> system.</c>"
+    },
+    {
+      "start": 83.87,
+      "duration": 0.0,
+      "text": "predict the trajectory of the system."
+    },
+    {
+      "start": 83.88,
+      "duration": 0.0,
+      "text": "predict the trajectory of the system. Instead,<00:01:24.320><c> what</c><00:01:24.520><c> we</c><00:01:24.640><c> are</c><00:01:24.800><c> usually</c><00:01:25.720><c> So,</c><00:01:26.120><c> what</c>"
+    },
+    {
+      "start": 86.43,
+      "duration": 0.0,
+      "text": "Instead, what we are usually So, what"
+    },
+    {
+      "start": 86.44,
+      "duration": 0.0,
+      "text": "Instead, what we are usually So, what our<00:01:26.680><c> goal</c><00:01:27.240><c> becomes</c><00:01:27.760><c> usually</c><00:01:28.720><c> is</c><00:01:28.960><c> to</c>"
+    },
+    {
+      "start": 90.35,
+      "duration": 0.0,
+      "text": "our goal becomes usually is to"
+    },
+    {
+      "start": 90.36,
+      "duration": 0.0,
+      "text": "our goal becomes usually is to build<00:01:30.680><c> a</c><00:01:30.720><c> mathematical</c><00:01:31.280><c> model</c><00:01:31.840><c> that</c><00:01:32.080><c> is</c><00:01:32.280><c> able</c>"
+    },
+    {
+      "start": 93.07,
+      "duration": 0.0,
+      "text": "build a mathematical model that is able"
+    },
+    {
+      "start": 93.08,
+      "duration": 0.0,
+      "text": "build a mathematical model that is able to<00:01:33.200><c> reproduce</c><00:01:34.400><c> some</c><00:01:34.960><c> key</c><00:01:35.760><c> statistical</c><00:01:36.840><c> and</c>"
+    },
+    {
+      "start": 97.07,
+      "duration": 0.0,
+      "text": "to reproduce some key statistical and"
+    },
+    {
+      "start": 97.08,
+      "duration": 0.0,
+      "text": "to reproduce some key statistical and dynamical<00:01:37.880><c> observable</c><00:01:39.040><c> of</c><00:01:39.560><c> the</c>"
+    },
+    {
+      "start": 100.31,
+      "duration": 0.0,
+      "text": "dynamical observable of the"
+    },
+    {
+      "start": 100.32,
+      "duration": 0.0,
+      "text": "dynamical observable of the the<00:01:40.800><c> data</c><00:01:41.120><c> set</c><00:01:41.960><c> that</c><00:01:42.160><c> we</c><00:01:42.280><c> are</c><00:01:42.360><c> observing.</c>"
+    },
+    {
+      "start": 103.59,
+      "duration": 0.0,
+      "text": "the data set that we are observing."
+    },
+    {
+      "start": 103.6,
+      "duration": 0.0,
+      "text": "the data set that we are observing. Those<00:01:43.880><c> observables</c><00:01:44.760><c> can</c><00:01:45.000><c> be,</c><00:01:45.120><c> for</c><00:01:45.280><c> example,</c>"
+    },
+    {
+      "start": 106.19,
+      "duration": 0.0,
+      "text": "Those observables can be, for example,"
+    },
+    {
+      "start": 106.2,
+      "duration": 0.0,
+      "text": "Those observables can be, for example, some<00:01:46.400><c> moments</c><00:01:47.400><c> of</c><00:01:47.680><c> the</c><00:01:48.280><c> steady</c><00:01:48.520><c> state</c>"
+    },
+    {
+      "start": 108.87,
+      "duration": 0.0,
+      "text": "some moments of the steady state"
+    },
+    {
+      "start": 108.88,
+      "duration": 0.0,
+      "text": "some moments of the steady state distribution,<00:01:50.080><c> the</c><00:01:50.200><c> whole</c><00:01:50.480><c> steady</c><00:01:50.760><c> state</c>"
+    },
+    {
+      "start": 111.19,
+      "duration": 0.0,
+      "text": "distribution, the whole steady state"
+    },
+    {
+      "start": 111.2,
+      "duration": 0.0,
+      "text": "distribution, the whole steady state distribution,"
+    },
+    {
+      "start": 112.83,
+      "duration": 0.0,
+      "text": "distribution,"
+    },
+    {
+      "start": 112.84,
+      "duration": 0.0,
+      "text": "distribution, and<00:01:53.040><c> also</c><00:01:53.520><c> some</c>"
+    },
+    {
+      "start": 114.99,
+      "duration": 0.0,
+      "text": "and also some"
+    },
+    {
+      "start": 115.0,
+      "duration": 0.0,
+      "text": "and also some uh<00:01:55.080><c> state</c><00:01:55.600><c> or</c><00:01:55.920><c> temporal</c><00:01:56.600><c> correlations</c>"
+    },
+    {
+      "start": 117.75,
+      "duration": 0.0,
+      "text": "uh state or temporal correlations"
+    },
+    {
+      "start": 117.76,
+      "duration": 0.0,
+      "text": "uh state or temporal correlations between<00:01:58.320><c> the</c>"
+    },
+    {
+      "start": 119.31,
+      "duration": 0.0,
+      "text": "between the"
+    },
+    {
+      "start": 119.32,
+      "duration": 0.0,
+      "text": "between the observed<00:01:59.800><c> variables</c><00:02:00.400><c> of</c><00:02:00.480><c> the</c><00:02:00.560><c> system.</c>"
+    },
+    {
+      "start": 122.35,
+      "duration": 0.0,
+      "text": "observed variables of the system."
+    },
+    {
+      "start": 122.36,
+      "duration": 0.0,
+      "text": "observed variables of the system. So,<00:02:02.440><c> this</c><00:02:02.640><c> is</c><00:02:03.280><c> the</c><00:02:03.440><c> goal</c><00:02:04.120><c> of</c><00:02:04.800><c> this</c><00:02:05.200><c> talk.</c><00:02:05.680><c> So,</c>"
+    },
+    {
+      "start": 126.51,
+      "duration": 0.0,
+      "text": "So, this is the goal of this talk. So,"
+    },
+    {
+      "start": 126.52,
+      "duration": 0.0,
+      "text": "So, this is the goal of this talk. So, develop<00:02:06.960><c> a</c><00:02:07.000><c> mathematical</c><00:02:07.560><c> model</c><00:02:07.920><c> that</c><00:02:08.119><c> is</c>"
+    },
+    {
+      "start": 128.27,
+      "duration": 0.0,
+      "text": "develop a mathematical model that is"
+    },
+    {
+      "start": 128.28,
+      "duration": 0.0,
+      "text": "develop a mathematical model that is able<00:02:08.840><c> uh</c>"
+    },
+    {
+      "start": 130.15,
+      "duration": 0.0,
+      "text": "able uh"
+    },
+    {
+      "start": 130.16,
+      "duration": 0.0,
+      "text": "able uh uh<00:02:10.880><c> to</c><00:02:11.000><c> reproduce</c><00:02:11.640><c> this</c><00:02:11.880><c> target</c><00:02:13.000><c> statistical</c>"
+    },
+    {
+      "start": 133.67,
+      "duration": 0.0,
+      "text": "uh to reproduce this target statistical"
+    },
+    {
+      "start": 133.68,
+      "duration": 0.0,
+      "text": "uh to reproduce this target statistical and<00:02:13.920><c> dynamical</c><00:02:14.400><c> observables.</c>"
+    },
+    {
+      "start": 135.91,
+      "duration": 0.0,
+      "text": "and dynamical observables."
+    },
+    {
+      "start": 135.92,
+      "duration": 0.0,
+      "text": "and dynamical observables. So,<00:02:16.080><c> because</c><00:02:16.560><c> of</c><00:02:16.960><c> this</c><00:02:17.800><c> timescale</c><00:02:18.400><c> separation</c>"
+    },
+    {
+      "start": 139.79,
+      "duration": 0.0,
+      "text": "So, because of this timescale separation"
+    },
+    {
+      "start": 139.8,
+      "duration": 0.0,
+      "text": "So, because of this timescale separation of<00:02:20.520><c> the</c><00:02:20.640><c> data,</c><00:02:21.160><c> so</c><00:02:21.320><c> the</c><00:02:21.440><c> fact</c><00:02:21.760><c> that</c><00:02:21.880><c> the</c><00:02:21.960><c> data</c>"
+    },
+    {
+      "start": 142.27,
+      "duration": 0.0,
+      "text": "of the data, so the fact that the data"
+    },
+    {
+      "start": 142.28,
+      "duration": 0.0,
+      "text": "of the data, so the fact that the data can<00:02:22.480><c> be</c><00:02:23.280><c> multiscale,</c><00:02:24.760><c> I</c><00:02:24.880><c> will</c><00:02:25.080><c> use</c>"
+    },
+    {
+      "start": 146.19,
+      "duration": 0.0,
+      "text": "can be multiscale, I will use"
+    },
+    {
+      "start": 146.2,
+      "duration": 0.0,
+      "text": "can be multiscale, I will use a<00:02:26.240><c> stochastic</c><00:02:26.840><c> model</c><00:02:27.560><c> where</c><00:02:27.880><c> we</c><00:02:28.040><c> have</c><00:02:28.840><c> a</c><00:02:28.920><c> first</c>"
+    },
+    {
+      "start": 149.19,
+      "duration": 0.0,
+      "text": "a stochastic model where we have a first"
+    },
+    {
+      "start": 149.2,
+      "duration": 0.0,
+      "text": "a stochastic model where we have a first component,<00:02:30.080><c> the</c><00:02:30.240><c> drift</c><00:02:30.560><c> term</c><00:02:31.040><c> f</c><00:02:31.160><c> of</c><00:02:31.360><c> x,</c><00:02:32.080><c> which</c>"
+    },
+    {
+      "start": 153.15,
+      "duration": 0.0,
+      "text": "component, the drift term f of x, which"
+    },
+    {
+      "start": 153.16,
+      "duration": 0.0,
+      "text": "component, the drift term f of x, which uh<00:02:33.720><c> will</c><00:02:33.960><c> model</c><00:02:34.640><c> the</c><00:02:34.760><c> deterministic</c><00:02:35.640><c> and</c><00:02:36.240><c> slow</c>"
+    },
+    {
+      "start": 156.51,
+      "duration": 0.0,
+      "text": "uh will model the deterministic and slow"
+    },
+    {
+      "start": 156.52,
+      "duration": 0.0,
+      "text": "uh will model the deterministic and slow variable<00:02:37.000><c> component</c><00:02:37.720><c> of</c><00:02:37.880><c> my</c><00:02:38.360><c> data</c><00:02:38.640><c> set.</c>"
+    },
+    {
+      "start": 159.75,
+      "duration": 0.0,
+      "text": "variable component of my data set."
+    },
+    {
+      "start": 159.76,
+      "duration": 0.0,
+      "text": "variable component of my data set. And<00:02:39.920><c> I</c>"
+    },
+    {
+      "start": 160.51,
+      "duration": 0.0,
+      "text": "And I"
+    },
+    {
+      "start": 160.52,
+      "duration": 0.0,
+      "text": "And I And<00:02:40.720><c> I</c><00:02:40.800><c> also</c><00:02:41.080><c> have</c><00:02:41.320><c> a</c><00:02:41.400><c> stochastic</c><00:02:41.880><c> component</c>"
+    },
+    {
+      "start": 162.59,
+      "duration": 0.0,
+      "text": "And I also have a stochastic component"
+    },
+    {
+      "start": 162.6,
+      "duration": 0.0,
+      "text": "And I also have a stochastic component that<00:02:42.760><c> takes</c><00:02:43.000><c> into</c><00:02:43.200><c> account</c><00:02:44.280><c> the</c><00:02:44.440><c> faster</c>"
+    },
+    {
+      "start": 164.87,
+      "duration": 0.0,
+      "text": "that takes into account the faster"
+    },
+    {
+      "start": 164.88,
+      "duration": 0.0,
+      "text": "that takes into account the faster dynamics,"
+    },
+    {
+      "start": 166.15,
+      "duration": 0.0,
+      "text": "dynamics,"
+    },
+    {
+      "start": 166.16,
+      "duration": 0.0,
+      "text": "dynamics, the<00:02:46.280><c> unresolved</c><00:02:47.400><c> fast</c><00:02:47.800><c> fluctuations</c><00:02:48.720><c> of</c><00:02:48.840><c> the</c>"
+    },
+    {
+      "start": 168.95,
+      "duration": 0.0,
+      "text": "the unresolved fast fluctuations of the"
+    },
+    {
+      "start": 168.96,
+      "duration": 0.0,
+      "text": "the unresolved fast fluctuations of the data<00:02:49.240><c> set.</c>"
+    },
+    {
+      "start": 170.43,
+      "duration": 0.0,
+      "text": "data set."
+    },
+    {
+      "start": 170.44,
+      "duration": 0.0,
+      "text": "data set. So,<00:02:50.520><c> this</c><00:02:51.160><c> will</c><00:02:51.360><c> be</c><00:02:51.600><c> the</c><00:02:51.720><c> model</c>"
+    },
+    {
+      "start": 172.83,
+      "duration": 0.0,
+      "text": "So, this will be the model"
+    },
+    {
+      "start": 172.84,
+      "duration": 0.0,
+      "text": "So, this will be the model that<00:02:53.040><c> I</c><00:02:53.120><c> will</c><00:02:53.800><c> that</c><00:02:54.000><c> I</c><00:02:54.080><c> want</c><00:02:54.320><c> to</c><00:02:54.400><c> construct</c>"
+    },
+    {
+      "start": 175.19,
+      "duration": 0.0,
+      "text": "that I will that I want to construct"
+    },
+    {
+      "start": 175.2,
+      "duration": 0.0,
+      "text": "that I will that I want to construct from<00:02:55.600><c> the</c><00:02:55.920><c> the</c><00:02:56.000><c> observations</c>"
+    },
+    {
+      "start": 177.63,
+      "duration": 0.0,
+      "text": "from the the observations"
+    },
+    {
+      "start": 177.64,
+      "duration": 0.0,
+      "text": "from the the observations with<00:02:58.160><c> the</c>"
+    },
+    {
+      "start": 179.19,
+      "duration": 0.0,
+      "text": "with the"
+    },
+    {
+      "start": 179.2,
+      "duration": 0.0,
+      "text": "with the final<00:02:59.560><c> goal</c>"
+    },
+    {
+      "start": 180.55,
+      "duration": 0.0,
+      "text": "final goal"
+    },
+    {
+      "start": 180.56,
+      "duration": 0.0,
+      "text": "final goal of<00:03:01.120><c> being</c><00:03:01.480><c> able</c><00:03:02.400><c> to</c><00:03:02.520><c> reproduce</c>"
+    },
+    {
+      "start": 183.91,
+      "duration": 0.0,
+      "text": "of being able to reproduce"
+    },
+    {
+      "start": 183.92,
+      "duration": 0.0,
+      "text": "of being able to reproduce in"
+    },
+    {
+      "start": 185.99,
+      "duration": 0.0,
+      "text": "in"
+    },
+    {
+      "start": 186.0,
+      "duration": 0.0,
+      "text": "in an<00:03:06.680><c> efficient</c><00:03:07.120><c> way</c><00:03:08.040><c> this</c><00:03:08.200><c> target</c><00:03:08.840><c> statistical</c>"
+    },
+    {
+      "start": 189.47,
+      "duration": 0.0,
+      "text": "an efficient way this target statistical"
+    },
+    {
+      "start": 189.48,
+      "duration": 0.0,
+      "text": "an efficient way this target statistical and<00:03:09.600><c> dynamical</c><00:03:10.120><c> observables.</c>"
+    },
+    {
+      "start": 191.63,
+      "duration": 0.0,
+      "text": "and dynamical observables."
+    },
+    {
+      "start": 191.64,
+      "duration": 0.0,
+      "text": "and dynamical observables. Okay,<00:03:11.880><c> so</c><00:03:12.240><c> right</c><00:03:12.440><c> now</c><00:03:13.120><c> I</c><00:03:13.240><c> have</c><00:03:13.440><c> been</c><00:03:13.720><c> extremely</c>"
+    },
+    {
+      "start": 194.11,
+      "duration": 0.0,
+      "text": "Okay, so right now I have been extremely"
+    },
+    {
+      "start": 194.12,
+      "duration": 0.0,
+      "text": "Okay, so right now I have been extremely general.<00:03:14.840><c> So,</c><00:03:15.080><c> I</c><00:03:15.160><c> have</c>"
+    },
+    {
+      "start": 196.35,
+      "duration": 0.0,
+      "text": "general. So, I have"
+    },
+    {
+      "start": 196.36,
+      "duration": 0.0,
+      "text": "general. So, I have talked<00:03:16.600><c> about</c><00:03:17.080><c> a</c><00:03:17.160><c> general</c><00:03:17.520><c> model</c><00:03:18.160><c> to</c><00:03:18.280><c> solve</c>"
+    },
+    {
+      "start": 198.71,
+      "duration": 0.0,
+      "text": "talked about a general model to solve"
+    },
+    {
+      "start": 198.72,
+      "duration": 0.0,
+      "text": "talked about a general model to solve this<00:03:19.000><c> very</c><00:03:19.280><c> general</c><00:03:19.640><c> problem.</c>"
+    },
+    {
+      "start": 200.91,
+      "duration": 0.0,
+      "text": "this very general problem."
+    },
+    {
+      "start": 200.92,
+      "duration": 0.0,
+      "text": "this very general problem. In<00:03:21.200><c> the</c><00:03:21.320><c> next</c><00:03:21.800><c> two</c><00:03:21.960><c> slides,</c><00:03:22.800><c> I</c><00:03:22.960><c> want</c><00:03:23.280><c> to</c><00:03:23.400><c> give</c><00:03:23.640><c> a</c>"
+    },
+    {
+      "start": 203.67,
+      "duration": 0.0,
+      "text": "In the next two slides, I want to give a"
+    },
+    {
+      "start": 203.68,
+      "duration": 0.0,
+      "text": "In the next two slides, I want to give a bit<00:03:23.880><c> more</c><00:03:24.040><c> details</c><00:03:24.760><c> about</c><00:03:25.360><c> the</c><00:03:25.440><c> first</c><00:03:26.280><c> what</c>"
+    },
+    {
+      "start": 206.59,
+      "duration": 0.0,
+      "text": "bit more details about the first what"
+    },
+    {
+      "start": 206.6,
+      "duration": 0.0,
+      "text": "bit more details about the first what are<00:03:27.120><c> the</c><00:03:27.360><c> requirements</c><00:03:28.320><c> that</c><00:03:28.560><c> we</c><00:03:28.680><c> want</c><00:03:29.120><c> from</c>"
+    },
+    {
+      "start": 209.31,
+      "duration": 0.0,
+      "text": "are the requirements that we want from"
+    },
+    {
+      "start": 209.32,
+      "duration": 0.0,
+      "text": "are the requirements that we want from our<00:03:29.520><c> model,</c>"
+    },
+    {
+      "start": 210.59,
+      "duration": 0.0,
+      "text": "our model,"
+    },
+    {
+      "start": 210.6,
+      "duration": 0.0,
+      "text": "our model, and<00:03:30.760><c> then</c><00:03:31.000><c> also</c><00:03:31.440><c> what</c><00:03:31.640><c> are</c><00:03:31.760><c> the</c><00:03:31.880><c> assumptions</c>"
+    },
+    {
+      "start": 212.63,
+      "duration": 0.0,
+      "text": "and then also what are the assumptions"
+    },
+    {
+      "start": 212.64,
+      "duration": 0.0,
+      "text": "and then also what are the assumptions that<00:03:32.880><c> we're</c><00:03:33.000><c> doing</c><00:03:33.720><c> on</c><00:03:33.880><c> the</c><00:03:33.960><c> observed</c><00:03:34.400><c> data</c>"
+    },
+    {
+      "start": 214.63,
+      "duration": 0.0,
+      "text": "that we're doing on the observed data"
+    },
+    {
+      "start": 214.64,
+      "duration": 0.0,
+      "text": "that we're doing on the observed data set."
+    },
+    {
+      "start": 217.56,
+      "duration": 0.0,
+      "text": "So,<00:03:37.720><c> first</c>"
+    },
+    {
+      "start": 219.19,
+      "duration": 0.0,
+      "text": "So, first"
+    },
+    {
+      "start": 219.2,
+      "duration": 0.0,
+      "text": "So, first let's<00:03:39.480><c> see</c><00:03:39.840><c> what</c><00:03:40.160><c> are</c><00:03:40.840><c> the</c><00:03:40.960><c> requirement</c><00:03:42.080><c> that</c>"
+    },
+    {
+      "start": 222.27,
+      "duration": 0.0,
+      "text": "let's see what are the requirement that"
+    },
+    {
+      "start": 222.28,
+      "duration": 0.0,
+      "text": "let's see what are the requirement that we<00:03:42.440><c> want</c>"
+    },
+    {
+      "start": 223.55,
+      "duration": 0.0,
+      "text": "we want"
+    },
+    {
+      "start": 223.56,
+      "duration": 0.0,
+      "text": "we want from<00:03:43.840><c> our</c><00:03:44.200><c> modeling</c><00:03:44.640><c> strategy.</c>"
+    },
+    {
+      "start": 225.87,
+      "duration": 0.0,
+      "text": "from our modeling strategy."
+    },
+    {
+      "start": 225.88,
+      "duration": 0.0,
+      "text": "from our modeling strategy. So,<00:03:45.960><c> as</c><00:03:46.120><c> I</c><00:03:46.200><c> said,</c><00:03:46.920><c> we</c><00:03:47.040><c> would</c><00:03:47.240><c> like</c><00:03:47.680><c> to</c><00:03:47.840><c> model</c>"
+    },
+    {
+      "start": 228.39,
+      "duration": 0.0,
+      "text": "So, as I said, we would like to model"
+    },
+    {
+      "start": 228.4,
+      "duration": 0.0,
+      "text": "So, as I said, we would like to model real<00:03:48.760><c> data</c><00:03:49.280><c> that</c><00:03:49.520><c> can</c><00:03:49.800><c> be</c><00:03:50.320><c> extremely</c>"
+    },
+    {
+      "start": 230.91,
+      "duration": 0.0,
+      "text": "real data that can be extremely"
+    },
+    {
+      "start": 230.92,
+      "duration": 0.0,
+      "text": "real data that can be extremely high-dimensional"
+    },
+    {
+      "start": 233.15,
+      "duration": 0.0,
+      "text": "high-dimensional"
+    },
+    {
+      "start": 233.16,
+      "duration": 0.0,
+      "text": "high-dimensional and<00:03:53.480><c> also</c><00:03:54.280><c> maybe</c><00:03:54.800><c> also</c><00:03:55.320><c> unevenly</c><00:03:55.920><c> sampled</c><00:03:57.000><c> or</c>"
+    },
+    {
+      "start": 237.43,
+      "duration": 0.0,
+      "text": "and also maybe also unevenly sampled or"
+    },
+    {
+      "start": 237.44,
+      "duration": 0.0,
+      "text": "and also maybe also unevenly sampled or have"
+    },
+    {
+      "start": 238.47,
+      "duration": 0.0,
+      "text": "have"
+    },
+    {
+      "start": 238.48,
+      "duration": 0.0,
+      "text": "have a<00:03:58.560><c> very</c><00:03:58.920><c> low</c><00:03:59.160><c> sampling</c><00:03:59.640><c> frequency.</c><00:04:00.520><c> So,</c><00:04:00.720><c> these</c>"
+    },
+    {
+      "start": 240.95,
+      "duration": 0.0,
+      "text": "a very low sampling frequency. So, these"
+    },
+    {
+      "start": 240.96,
+      "duration": 0.0,
+      "text": "a very low sampling frequency. So, these are<00:04:01.160><c> all</c>"
+    },
+    {
+      "start": 242.43,
+      "duration": 0.0,
+      "text": "are all"
+    },
+    {
+      "start": 242.44,
+      "duration": 0.0,
+      "text": "are all features<00:04:03.480><c> of</c><00:04:03.880><c> our</c><00:04:04.200><c> real</c><00:04:04.480><c> data</c><00:04:04.760><c> set</c><00:04:05.000><c> that</c><00:04:05.160><c> we</c>"
+    },
+    {
+      "start": 245.27,
+      "duration": 0.0,
+      "text": "features of our real data set that we"
+    },
+    {
+      "start": 245.28,
+      "duration": 0.0,
+      "text": "features of our real data set that we have<00:04:05.440><c> to</c><00:04:05.560><c> take</c><00:04:05.800><c> into</c><00:04:05.960><c> account</c><00:04:06.600><c> when</c><00:04:06.920><c> we</c><00:04:07.120><c> are</c>"
+    },
+    {
+      "start": 247.75,
+      "duration": 0.0,
+      "text": "have to take into account when we are"
+    },
+    {
+      "start": 247.76,
+      "duration": 0.0,
+      "text": "have to take into account when we are building<00:04:08.200><c> a</c><00:04:08.240><c> mathematical</c><00:04:08.840><c> model</c><00:04:09.280><c> for</c><00:04:09.440><c> it.</c>"
+    },
+    {
+      "start": 250.63,
+      "duration": 0.0,
+      "text": "building a mathematical model for it."
+    },
+    {
+      "start": 250.64,
+      "duration": 0.0,
+      "text": "building a mathematical model for it. So,<00:04:10.720><c> this</c><00:04:10.920><c> essentially</c><00:04:11.320><c> implies</c><00:04:12.040><c> first</c>"
+    },
+    {
+      "start": 253.67,
+      "duration": 0.0,
+      "text": "So, this essentially implies first"
+    },
+    {
+      "start": 253.68,
+      "duration": 0.0,
+      "text": "So, this essentially implies first that<00:04:14.200><c> since</c>"
+    },
+    {
+      "start": 255.31,
+      "duration": 0.0,
+      "text": "that since"
+    },
+    {
+      "start": 255.32,
+      "duration": 0.0,
+      "text": "that since we<00:04:15.480><c> want</c><00:04:15.920><c> our</c><00:04:16.200><c> modeling</c><00:04:16.640><c> strategy</c><00:04:17.160><c> to</c><00:04:17.320><c> scale</c>"
+    },
+    {
+      "start": 257.789,
+      "duration": 0.0,
+      "text": "we want our modeling strategy to scale"
+    },
+    {
+      "start": 257.799,
+      "duration": 0.0,
+      "text": "we want our modeling strategy to scale very<00:04:18.040><c> well</c><00:04:18.280><c> with</c><00:04:18.400><c> the</c><00:04:18.480><c> dimension,</c>"
+    },
+    {
+      "start": 259.91,
+      "duration": 0.0,
+      "text": "very well with the dimension,"
+    },
+    {
+      "start": 259.92,
+      "duration": 0.0,
+      "text": "very well with the dimension, and<00:04:20.400><c> since</c><00:04:20.840><c> also</c><00:04:21.239><c> the</c><00:04:21.400><c> model</c><00:04:22.079><c> that</c><00:04:22.280><c> we</c><00:04:22.360><c> want</c><00:04:22.600><c> to</c>"
+    },
+    {
+      "start": 262.67,
+      "duration": 0.0,
+      "text": "and since also the model that we want to"
+    },
+    {
+      "start": 262.68,
+      "duration": 0.0,
+      "text": "and since also the model that we want to develop<00:04:23.360><c> can</c><00:04:23.560><c> be</c><00:04:23.720><c> extremely</c>"
+    },
+    {
+      "start": 264.11,
+      "duration": 0.0,
+      "text": "develop can be extremely"
+    },
+    {
+      "start": 264.12,
+      "duration": 0.0,
+      "text": "develop can be extremely high-dimensional<00:04:25.120><c> and</c>"
+    },
+    {
+      "start": 265.91,
+      "duration": 0.0,
+      "text": "high-dimensional and"
+    },
+    {
+      "start": 265.92,
+      "duration": 0.0,
+      "text": "high-dimensional and computationally<00:04:26.680><c> expensive</c><00:04:27.400><c> to</c><00:04:27.520><c> integrate,</c>"
+    },
+    {
+      "start": 268.83,
+      "duration": 0.0,
+      "text": "computationally expensive to integrate,"
+    },
+    {
+      "start": 268.84,
+      "duration": 0.0,
+      "text": "computationally expensive to integrate, we<00:04:29.040><c> would</c><00:04:29.240><c> like</c><00:04:29.640><c> to</c><00:04:29.760><c> be</c><00:04:29.960><c> able</c><00:04:30.520><c> to</c><00:04:30.800><c> reconstruct</c>"
+    },
+    {
+      "start": 271.87,
+      "duration": 0.0,
+      "text": "we would like to be able to reconstruct"
+    },
+    {
+      "start": 271.88,
+      "duration": 0.0,
+      "text": "we would like to be able to reconstruct the<00:04:31.960><c> mathematical</c><00:04:32.560><c> model</c><00:04:33.040><c> from</c><00:04:33.280><c> data</c><00:04:34.000><c> using</c>"
+    },
+    {
+      "start": 274.79,
+      "duration": 0.0,
+      "text": "the mathematical model from data using"
+    },
+    {
+      "start": 274.8,
+      "duration": 0.0,
+      "text": "the mathematical model from data using as<00:04:35.000><c> few</c><00:04:35.240><c> model</c><00:04:35.560><c> integrations</c>"
+    },
+    {
+      "start": 276.79,
+      "duration": 0.0,
+      "text": "as few model integrations"
+    },
+    {
+      "start": 276.8,
+      "duration": 0.0,
+      "text": "as few model integrations as<00:04:36.960><c> possible.</c>"
+    },
+    {
+      "start": 278.19,
+      "duration": 0.0,
+      "text": "as possible."
+    },
+    {
+      "start": 278.2,
+      "duration": 0.0,
+      "text": "as possible. So,<00:04:38.400><c> a</c><00:04:38.480><c> naive</c><00:04:38.760><c> approach</c><00:04:39.560><c> to</c><00:04:39.720><c> build</c><00:04:40.040><c> a</c>"
+    },
+    {
+      "start": 280.07,
+      "duration": 0.0,
+      "text": "So, a naive approach to build a"
+    },
+    {
+      "start": 280.08,
+      "duration": 0.0,
+      "text": "So, a naive approach to build a mathematical<00:04:40.760><c> model</c><00:04:41.520><c> can</c><00:04:41.800><c> be</c><00:04:42.080><c> to</c><00:04:42.240><c> start</c><00:04:42.680><c> with</c>"
+    },
+    {
+      "start": 282.87,
+      "duration": 0.0,
+      "text": "mathematical model can be to start with"
+    },
+    {
+      "start": 282.88,
+      "duration": 0.0,
+      "text": "mathematical model can be to start with a<00:04:42.960><c> model</c><00:04:43.320><c> ansatz,</c><00:04:44.040><c> integrate</c><00:04:44.640><c> it</c><00:04:44.840><c> forward</c><00:04:45.600><c> for</c>"
+    },
+    {
+      "start": 285.75,
+      "duration": 0.0,
+      "text": "a model ansatz, integrate it forward for"
+    },
+    {
+      "start": 285.76,
+      "duration": 0.0,
+      "text": "a model ansatz, integrate it forward for a<00:04:45.840><c> given</c><00:04:46.160><c> time,</c>"
+    },
+    {
+      "start": 287.59,
+      "duration": 0.0,
+      "text": "a given time,"
+    },
+    {
+      "start": 287.6,
+      "duration": 0.0,
+      "text": "a given time, then<00:04:48.280><c> calculate</c>"
+    },
+    {
+      "start": 289.83,
+      "duration": 0.0,
+      "text": "then calculate"
+    },
+    {
+      "start": 289.84,
+      "duration": 0.0,
+      "text": "then calculate all<00:04:50.200><c> the</c><00:04:50.360><c> statistical</c><00:04:51.000><c> and</c><00:04:51.160><c> dynamical</c>"
+    },
+    {
+      "start": 291.63,
+      "duration": 0.0,
+      "text": "all the statistical and dynamical"
+    },
+    {
+      "start": 291.64,
+      "duration": 0.0,
+      "text": "all the statistical and dynamical observables,<00:04:52.520><c> compare</c><00:04:53.040><c> them</c><00:04:53.400><c> with</c><00:04:53.520><c> the</c>"
+    },
+    {
+      "start": 293.63,
+      "duration": 0.0,
+      "text": "observables, compare them with the"
+    },
+    {
+      "start": 293.64,
+      "duration": 0.0,
+      "text": "observables, compare them with the targets,"
+    },
+    {
+      "start": 294.99,
+      "duration": 0.0,
+      "text": "targets,"
+    },
+    {
+      "start": 295.0,
+      "duration": 0.0,
+      "text": "targets, evaluate<00:04:55.520><c> a</c><00:04:55.560><c> loss</c><00:04:55.800><c> function,</c><00:04:56.640><c> and</c><00:04:56.840><c> use</c><00:04:57.040><c> that</c>"
+    },
+    {
+      "start": 297.43,
+      "duration": 0.0,
+      "text": "evaluate a loss function, and use that"
+    },
+    {
+      "start": 297.44,
+      "duration": 0.0,
+      "text": "evaluate a loss function, and use that to<00:04:57.560><c> update</c><00:04:57.960><c> the</c><00:04:58.040><c> model.</c><00:04:58.880><c> But,</c><00:04:59.120><c> this</c><00:04:59.480><c> will</c>"
+    },
+    {
+      "start": 299.71,
+      "duration": 0.0,
+      "text": "to update the model. But, this will"
+    },
+    {
+      "start": 299.72,
+      "duration": 0.0,
+      "text": "to update the model. But, this will require<00:05:00.360><c> many</c><00:05:00.640><c> model</c><00:05:00.920><c> integrations</c><00:05:01.760><c> that</c><00:05:01.960><c> can</c>"
+    },
+    {
+      "start": 302.11,
+      "duration": 0.0,
+      "text": "require many model integrations that can"
+    },
+    {
+      "start": 302.12,
+      "duration": 0.0,
+      "text": "require many model integrations that can become<00:05:02.480><c> extremely</c><00:05:02.880><c> computationally</c>"
+    },
+    {
+      "start": 303.51,
+      "duration": 0.0,
+      "text": "become extremely computationally"
+    },
+    {
+      "start": 303.52,
+      "duration": 0.0,
+      "text": "become extremely computationally expensive.<00:05:04.760><c> So,</c><00:05:05.120><c> our</c><00:05:05.400><c> modeling</c><00:05:05.880><c> strategy</c>"
+    },
+    {
+      "start": 307.75,
+      "duration": 0.0,
+      "text": "expensive. So, our modeling strategy"
+    },
+    {
+      "start": 307.76,
+      "duration": 0.0,
+      "text": "expensive. So, our modeling strategy So,<00:05:08.560><c> one</c><00:05:08.760><c> of</c><00:05:08.880><c> the</c><00:05:08.960><c> main</c><00:05:09.200><c> goal</c><00:05:09.480><c> of</c><00:05:09.680><c> our</c><00:05:09.880><c> modeling</c>"
+    },
+    {
+      "start": 310.23,
+      "duration": 0.0,
+      "text": "So, one of the main goal of our modeling"
+    },
+    {
+      "start": 310.24,
+      "duration": 0.0,
+      "text": "So, one of the main goal of our modeling strategy<00:05:10.840><c> would</c><00:05:11.120><c> be</c><00:05:11.600><c> to</c><00:05:12.680><c> use</c><00:05:13.280><c> as</c><00:05:13.520><c> few</c><00:05:13.720><c> model</c>"
+    },
+    {
+      "start": 313.99,
+      "duration": 0.0,
+      "text": "strategy would be to use as few model"
+    },
+    {
+      "start": 314.0,
+      "duration": 0.0,
+      "text": "strategy would be to use as few model integrations<00:05:14.960><c> as</c><00:05:15.120><c> possible.</c>"
+    },
+    {
+      "start": 317.11,
+      "duration": 0.0,
+      "text": "integrations as possible."
+    },
+    {
+      "start": 317.12,
+      "duration": 0.0,
+      "text": "integrations as possible. Another<00:05:18.120><c> requirement</c><00:05:18.960><c> that</c><00:05:19.120><c> we</c><00:05:19.240><c> would</c><00:05:19.840><c> that</c>"
+    },
+    {
+      "start": 319.99,
+      "duration": 0.0,
+      "text": "Another requirement that we would that"
+    },
+    {
+      "start": 320.0,
+      "duration": 0.0,
+      "text": "Another requirement that we would that we<00:05:20.160><c> want</c>"
+    },
+    {
+      "start": 321.31,
+      "duration": 0.0,
+      "text": "we want"
+    },
+    {
+      "start": 321.32,
+      "duration": 0.0,
+      "text": "we want our<00:05:21.560><c> modeling</c><00:05:22.200><c> strategy</c><00:05:22.760><c> to</c><00:05:22.880><c> have</c><00:05:23.640><c> is</c><00:05:23.840><c> to</c>"
+    },
+    {
+      "start": 323.91,
+      "duration": 0.0,
+      "text": "our modeling strategy to have is to"
+    },
+    {
+      "start": 323.92,
+      "duration": 0.0,
+      "text": "our modeling strategy to have is to avoid<00:05:24.440><c> the</c><00:05:24.520><c> state</c><00:05:24.800><c> space</c><00:05:25.200><c> clustering.</c>"
+    },
+    {
+      "start": 326.63,
+      "duration": 0.0,
+      "text": "avoid the state space clustering."
+    },
+    {
+      "start": 326.64,
+      "duration": 0.0,
+      "text": "avoid the state space clustering. So,<00:05:26.720><c> we</c><00:05:26.840><c> would</c><00:05:27.000><c> like</c><00:05:28.000><c> to</c><00:05:28.200><c> avoid</c><00:05:29.240><c> clustering</c>"
+    },
+    {
+      "start": 329.83,
+      "duration": 0.0,
+      "text": "So, we would like to avoid clustering"
+    },
+    {
+      "start": 329.84,
+      "duration": 0.0,
+      "text": "So, we would like to avoid clustering the<00:05:30.000><c> state</c><00:05:30.360><c> space</c><00:05:31.160><c> to</c><00:05:31.280><c> estimate</c><00:05:31.920><c> the</c><00:05:32.040><c> average</c>"
+    },
+    {
+      "start": 332.43,
+      "duration": 0.0,
+      "text": "the state space to estimate the average"
+    },
+    {
+      "start": 332.44,
+      "duration": 0.0,
+      "text": "the state space to estimate the average velocity<00:05:32.880><c> field</c>"
+    },
+    {
+      "start": 334.27,
+      "duration": 0.0,
+      "text": "velocity field"
+    },
+    {
+      "start": 334.28,
+      "duration": 0.0,
+      "text": "velocity field because<00:05:34.680><c> essentially</c>"
+    },
+    {
+      "start": 335.95,
+      "duration": 0.0,
+      "text": "because essentially"
+    },
+    {
+      "start": 335.96,
+      "duration": 0.0,
+      "text": "because essentially So,<00:05:36.480><c> clustering</c><00:05:37.000><c> the</c><00:05:37.120><c> state</c><00:05:37.400><c> space,</c><00:05:38.240><c> even</c><00:05:38.520><c> if</c>"
+    },
+    {
+      "start": 338.79,
+      "duration": 0.0,
+      "text": "So, clustering the state space, even if"
+    },
+    {
+      "start": 338.8,
+      "duration": 0.0,
+      "text": "So, clustering the state space, even if we<00:05:38.920><c> are</c><00:05:39.040><c> using</c><00:05:39.440><c> an</c><00:05:39.560><c> extremely</c><00:05:40.400><c> efficient</c>"
+    },
+    {
+      "start": 340.95,
+      "duration": 0.0,
+      "text": "we are using an extremely efficient"
+    },
+    {
+      "start": 340.96,
+      "duration": 0.0,
+      "text": "we are using an extremely efficient clustering<00:05:41.480><c> algorithm</c><00:05:42.240><c> like,</c><00:05:42.600><c> I</c><00:05:42.640><c> don't</c><00:05:42.800><c> know,</c>"
+    },
+    {
+      "start": 343.31,
+      "duration": 0.0,
+      "text": "clustering algorithm like, I don't know,"
+    },
+    {
+      "start": 343.32,
+      "duration": 0.0,
+      "text": "clustering algorithm like, I don't know, bisecting<00:05:44.040><c> k-means</c><00:05:44.960><c> clustering</c><00:05:45.440><c> algorithm,</c>"
+    },
+    {
+      "start": 347.03,
+      "duration": 0.0,
+      "text": "bisecting k-means clustering algorithm,"
+    },
+    {
+      "start": 347.04,
+      "duration": 0.0,
+      "text": "bisecting k-means clustering algorithm, suffer<00:05:47.600><c> from</c><00:05:48.000><c> the</c><00:05:48.120><c> curse</c><00:05:48.400><c> of</c><00:05:48.520><c> dimensionality.</c>"
+    },
+    {
+      "start": 349.47,
+      "duration": 0.0,
+      "text": "suffer from the curse of dimensionality."
+    },
+    {
+      "start": 349.48,
+      "duration": 0.0,
+      "text": "suffer from the curse of dimensionality. So,<00:05:49.800><c> if</c><00:05:49.960><c> we</c><00:05:50.040><c> are</c><00:05:50.160><c> considering</c><00:05:50.800><c> a</c><00:05:50.840><c> data</c><00:05:51.160><c> set</c><00:05:52.120><c> or</c>"
+    },
+    {
+      "start": 352.31,
+      "duration": 0.0,
+      "text": "So, if we are considering a data set or"
+    },
+    {
+      "start": 352.32,
+      "duration": 0.0,
+      "text": "So, if we are considering a data set or a<00:05:52.360><c> system</c><00:05:53.040><c> with</c><00:05:53.320><c> dimension</c><00:05:53.840><c> larger</c><00:05:54.320><c> than</c><00:05:54.680><c> a</c>"
+    },
+    {
+      "start": 354.75,
+      "duration": 0.0,
+      "text": "a system with dimension larger than a"
+    },
+    {
+      "start": 354.76,
+      "duration": 0.0,
+      "text": "a system with dimension larger than a few<00:05:54.960><c> dozens,</c><00:05:55.920><c> clustering</c><00:05:56.440><c> becomes</c><00:05:56.880><c> extremely</c>"
+    },
+    {
+      "start": 357.31,
+      "duration": 0.0,
+      "text": "few dozens, clustering becomes extremely"
+    },
+    {
+      "start": 357.32,
+      "duration": 0.0,
+      "text": "few dozens, clustering becomes extremely computationally<00:05:57.920><c> expensive,</c><00:05:58.680><c> so</c><00:05:58.800><c> we</c><00:05:58.920><c> would</c>"
+    },
+    {
+      "start": 359.07,
+      "duration": 0.0,
+      "text": "computationally expensive, so we would"
+    },
+    {
+      "start": 359.08,
+      "duration": 0.0,
+      "text": "computationally expensive, so we would like<00:05:59.320><c> to</c><00:05:59.400><c> avoid</c><00:05:59.720><c> clustering.</c>"
+    },
+    {
+      "start": 361.19,
+      "duration": 0.0,
+      "text": "like to avoid clustering."
+    },
+    {
+      "start": 361.2,
+      "duration": 0.0,
+      "text": "like to avoid clustering. And<00:06:01.400><c> also</c><00:06:01.720><c> we</c><00:06:01.840><c> would</c><00:06:02.040><c> like</c><00:06:03.000><c> to</c><00:06:03.200><c> use</c><00:06:03.560><c> a</c><00:06:03.640><c> finite</c>"
+    },
+    {
+      "start": 363.95,
+      "duration": 0.0,
+      "text": "And also we would like to use a finite"
+    },
+    {
+      "start": 363.96,
+      "duration": 0.0,
+      "text": "And also we would like to use a finite difference<00:06:04.400><c> estimation</c><00:06:05.520><c> to</c><00:06:05.680><c> estimate</c><00:06:06.320><c> the</c>"
+    },
+    {
+      "start": 366.43,
+      "duration": 0.0,
+      "text": "difference estimation to estimate the"
+    },
+    {
+      "start": 366.44,
+      "duration": 0.0,
+      "text": "difference estimation to estimate the velocity<00:06:07.520><c> from</c><00:06:07.760><c> trajectories.</c><00:06:08.680><c> So,</c><00:06:09.000><c> we</c><00:06:09.120><c> are</c>"
+    },
+    {
+      "start": 369.27,
+      "duration": 0.0,
+      "text": "velocity from trajectories. So, we are"
+    },
+    {
+      "start": 369.28,
+      "duration": 0.0,
+      "text": "velocity from trajectories. So, we are assuming<00:06:10.280><c> that</c><00:06:10.600><c> our</c><00:06:11.480><c> realistic</c><00:06:12.080><c> data</c><00:06:12.320><c> set</c><00:06:12.920><c> can</c>"
+    },
+    {
+      "start": 373.19,
+      "duration": 0.0,
+      "text": "assuming that our realistic data set can"
+    },
+    {
+      "start": 373.2,
+      "duration": 0.0,
+      "text": "assuming that our realistic data set can be<00:06:13.840><c> can</c><00:06:14.000><c> have</c><00:06:14.280><c> a</c><00:06:14.360><c> low</c><00:06:14.560><c> frequen-</c><00:06:15.240><c> a</c><00:06:15.320><c> low</c>"
+    },
+    {
+      "start": 375.47,
+      "duration": 0.0,
+      "text": "be can have a low frequen- a low"
+    },
+    {
+      "start": 375.48,
+      "duration": 0.0,
+      "text": "be can have a low frequen- a low sampling<00:06:15.960><c> frequency,</c><00:06:17.120><c> which</c><00:06:17.280><c> essentially</c>"
+    },
+    {
+      "start": 377.79,
+      "duration": 0.0,
+      "text": "sampling frequency, which essentially"
+    },
+    {
+      "start": 377.8,
+      "duration": 0.0,
+      "text": "sampling frequency, which essentially means<00:06:18.200><c> that</c><00:06:18.440><c> we</c><00:06:18.560><c> cannot</c><00:06:19.320><c> trust</c><00:06:19.880><c> the</c><00:06:20.320><c> data</c><00:06:20.600><c> set</c>"
+    },
+    {
+      "start": 381.15,
+      "duration": 0.0,
+      "text": "means that we cannot trust the data set"
+    },
+    {
+      "start": 381.16,
+      "duration": 0.0,
+      "text": "means that we cannot trust the data set to<00:06:21.480><c> recover</c>"
+    },
+    {
+      "start": 383.19,
+      "duration": 0.0,
+      "text": "to recover"
+    },
+    {
+      "start": 383.2,
+      "duration": 0.0,
+      "text": "to recover the<00:06:23.920><c> velocity</c>"
+    },
+    {
+      "start": 385.55,
+      "duration": 0.0,
+      "text": "the velocity"
+    },
+    {
+      "start": 385.56,
+      "duration": 0.0,
+      "text": "the velocity of<00:06:25.800><c> our</c><00:06:26.080><c> dynamics</c><00:06:27.000><c> using</c><00:06:27.440><c> finite</c><00:06:27.720><c> difference.</c>"
+    },
+    {
+      "start": 388.79,
+      "duration": 0.0,
+      "text": "of our dynamics using finite difference."
+    },
+    {
+      "start": 388.8,
+      "duration": 0.0,
+      "text": "of our dynamics using finite difference. So,<00:06:28.960><c> these</c><00:06:29.200><c> are</c><00:06:29.360><c> the</c>"
+    },
+    {
+      "start": 390.79,
+      "duration": 0.0,
+      "text": "So, these are the"
+    },
+    {
+      "start": 390.8,
+      "duration": 0.0,
+      "text": "So, these are the three<00:06:31.120><c> main</c>"
+    },
+    {
+      "start": 392.47,
+      "duration": 0.0,
+      "text": "three main"
+    },
+    {
+      "start": 392.48,
+      "duration": 0.0,
+      "text": "three main um"
+    },
+    {
+      "start": 393.99,
+      "duration": 0.0,
+      "text": "um"
+    },
+    {
+      "start": 394.0,
+      "duration": 0.0,
+      "text": "um constraints<00:06:34.720><c> that</c><00:06:34.920><c> we</c><00:06:35.040><c> want</c><00:06:35.720><c> our</c><00:06:36.000><c> modeling</c>"
+    },
+    {
+      "start": 396.39,
+      "duration": 0.0,
+      "text": "constraints that we want our modeling"
+    },
+    {
+      "start": 396.4,
+      "duration": 0.0,
+      "text": "constraints that we want our modeling strategy<00:06:36.880><c> to</c><00:06:37.040><c> have</c><00:06:37.760><c> in</c><00:06:37.920><c> order</c><00:06:38.240><c> to</c><00:06:38.400><c> be</c><00:06:38.880><c> scalable</c>"
+    },
+    {
+      "start": 399.55,
+      "duration": 0.0,
+      "text": "strategy to have in order to be scalable"
+    },
+    {
+      "start": 399.56,
+      "duration": 0.0,
+      "text": "strategy to have in order to be scalable to<00:06:39.680><c> high-dimensional</c><00:06:40.320><c> system</c><00:06:41.320><c> and</c><00:06:41.520><c> also</c>"
+    },
+    {
+      "start": 403.55,
+      "duration": 0.0,
+      "text": "to high-dimensional system and also"
+    },
+    {
+      "start": 403.56,
+      "duration": 0.0,
+      "text": "to high-dimensional system and also being<00:06:43.800><c> used</c><00:06:44.360><c> to</c><00:06:44.600><c> realistic</c><00:06:45.160><c> settings.</c>"
+    },
+    {
+      "start": 406.91,
+      "duration": 0.0,
+      "text": "being used to realistic settings."
+    },
+    {
+      "start": 406.92,
+      "duration": 0.0,
+      "text": "being used to realistic settings. Then,<00:06:47.320><c> let's</c><00:06:47.600><c> see</c><00:06:47.800><c> what</c><00:06:48.120><c> are</c><00:06:48.400><c> instead</c><00:06:48.880><c> the</c>"
+    },
+    {
+      "start": 408.95,
+      "duration": 0.0,
+      "text": "Then, let's see what are instead the"
+    },
+    {
+      "start": 408.96,
+      "duration": 0.0,
+      "text": "Then, let's see what are instead the constraints"
+    },
+    {
+      "start": 410.31,
+      "duration": 0.0,
+      "text": "constraints"
+    },
+    {
+      "start": 410.32,
+      "duration": 0.0,
+      "text": "constraints that<00:06:50.880><c> we</c><00:06:51.080><c> want</c><00:06:51.440><c> to</c><00:06:51.520><c> impose</c><00:06:52.000><c> on</c><00:06:52.120><c> our</c><00:06:52.320><c> system.</c>"
+    },
+    {
+      "start": 412.83,
+      "duration": 0.0,
+      "text": "that we want to impose on our system."
+    },
+    {
+      "start": 412.84,
+      "duration": 0.0,
+      "text": "that we want to impose on our system. So,<00:06:53.080><c> on</c><00:06:53.280><c> which</c><00:06:53.480><c> kind</c><00:06:53.760><c> of</c><00:06:53.880><c> systems</c><00:06:55.040><c> this</c><00:06:55.720><c> So,</c>"
+    },
+    {
+      "start": 415.83,
+      "duration": 0.0,
+      "text": "So, on which kind of systems this So,"
+    },
+    {
+      "start": 415.84,
+      "duration": 0.0,
+      "text": "So, on which kind of systems this So, the<00:06:55.960><c> model</c><00:06:56.520><c> the</c><00:06:56.640><c> modeling</c><00:06:57.040><c> strategy</c><00:06:57.560><c> that</c><00:06:57.760><c> I</c>"
+    },
+    {
+      "start": 417.79,
+      "duration": 0.0,
+      "text": "the model the modeling strategy that I"
+    },
+    {
+      "start": 417.8,
+      "duration": 0.0,
+      "text": "the model the modeling strategy that I will<00:06:58.000><c> propose</c><00:06:58.800><c> in</c><00:06:58.920><c> this</c><00:06:59.080><c> talk</c><00:06:59.960><c> can</c><00:07:00.160><c> be</c>"
+    },
+    {
+      "start": 420.23,
+      "duration": 0.0,
+      "text": "will propose in this talk can be"
+    },
+    {
+      "start": 420.24,
+      "duration": 0.0,
+      "text": "will propose in this talk can be applied."
+    },
+    {
+      "start": 421.67,
+      "duration": 0.0,
+      "text": "applied."
+    },
+    {
+      "start": 421.68,
+      "duration": 0.0,
+      "text": "applied. So,<00:07:01.760><c> first</c><00:07:02.000><c> of</c><00:07:02.200><c> all,</c><00:07:02.960><c> I</c><00:07:03.200><c> will</c><00:07:03.400><c> assume</c><00:07:03.920><c> that</c><00:07:04.080><c> the</c>"
+    },
+    {
+      "start": 424.15,
+      "duration": 0.0,
+      "text": "So, first of all, I will assume that the"
+    },
+    {
+      "start": 424.16,
+      "duration": 0.0,
+      "text": "So, first of all, I will assume that the system<00:07:04.840><c> I'm</c><00:07:05.040><c> studying</c><00:07:05.880><c> has</c><00:07:06.640><c> statistical</c>"
+    },
+    {
+      "start": 427.27,
+      "duration": 0.0,
+      "text": "system I'm studying has statistical"
+    },
+    {
+      "start": 427.28,
+      "duration": 0.0,
+      "text": "system I'm studying has statistical stationarity.<00:07:08.120><c> So,</c><00:07:08.240><c> essentially</c><00:07:08.760><c> that</c><00:07:08.920><c> we</c>"
+    },
+    {
+      "start": 429.03,
+      "duration": 0.0,
+      "text": "stationarity. So, essentially that we"
+    },
+    {
+      "start": 429.04,
+      "duration": 0.0,
+      "text": "stationarity. So, essentially that we can<00:07:09.920><c> define</c><00:07:10.680><c> a</c><00:07:10.760><c> steady</c><00:07:11.040><c> state</c><00:07:11.400><c> distribution.</c>"
+    },
+    {
+      "start": 433.31,
+      "duration": 0.0,
+      "text": "can define a steady state distribution."
+    },
+    {
+      "start": 433.32,
+      "duration": 0.0,
+      "text": "can define a steady state distribution. So,<00:07:13.560><c> of</c><00:07:13.760><c> course</c><00:07:14.280><c> this</c><00:07:14.480><c> can</c><00:07:14.640><c> be</c><00:07:14.760><c> generalized</c>"
+    },
+    {
+      "start": 435.51,
+      "duration": 0.0,
+      "text": "So, of course this can be generalized"
+    },
+    {
+      "start": 435.52,
+      "duration": 0.0,
+      "text": "So, of course this can be generalized also<00:07:15.880><c> to</c><00:07:16.040><c> cyclostationary</c><00:07:17.280><c> data</c><00:07:18.200><c> by</c>"
+    },
+    {
+      "start": 438.63,
+      "duration": 0.0,
+      "text": "also to cyclostationary data by"
+    },
+    {
+      "start": 438.64,
+      "duration": 0.0,
+      "text": "also to cyclostationary data by augmenting<00:07:19.400><c> the</c><00:07:19.520><c> state</c><00:07:19.800><c> space</c><00:07:20.680><c> or,</c><00:07:21.080><c> for</c>"
+    },
+    {
+      "start": 441.23,
+      "duration": 0.0,
+      "text": "augmenting the state space or, for"
+    },
+    {
+      "start": 441.24,
+      "duration": 0.0,
+      "text": "augmenting the state space or, for example,<00:07:22.240><c> to</c>"
+    },
+    {
+      "start": 444.03,
+      "duration": 0.0,
+      "text": "example, to"
+    },
+    {
+      "start": 444.04,
+      "duration": 0.0,
+      "text": "example, to data<00:07:24.360><c> set</c><00:07:24.840><c> that</c><00:07:25.160><c> show</c><00:07:25.760><c> a</c><00:07:25.880><c> low</c><00:07:26.160><c> trends</c><00:07:27.400><c> that</c><00:07:27.600><c> can</c>"
+    },
+    {
+      "start": 447.83,
+      "duration": 0.0,
+      "text": "data set that show a low trends that can"
+    },
+    {
+      "start": 447.84,
+      "duration": 0.0,
+      "text": "data set that show a low trends that can be<00:07:28.040><c> detrended</c><00:07:29.160><c> and</c><00:07:29.400><c> then</c><00:07:29.600><c> we</c><00:07:29.720><c> can</c><00:07:30.000><c> get</c><00:07:30.320><c> a</c>"
+    },
+    {
+      "start": 450.39,
+      "duration": 0.0,
+      "text": "be detrended and then we can get a"
+    },
+    {
+      "start": 450.4,
+      "duration": 0.0,
+      "text": "be detrended and then we can get a stationary"
+    },
+    {
+      "start": 451.71,
+      "duration": 0.0,
+      "text": "stationary"
+    },
+    {
+      "start": 451.72,
+      "duration": 0.0,
+      "text": "stationary um"
+    },
+    {
+      "start": 453.03,
+      "duration": 0.0,
+      "text": "um"
+    },
+    {
+      "start": 453.04,
+      "duration": 0.0,
+      "text": "um data<00:07:33.400><c> set.</c>"
+    },
+    {
+      "start": 454.67,
+      "duration": 0.0,
+      "text": "data set."
+    },
+    {
+      "start": 454.68,
+      "duration": 0.0,
+      "text": "data set. I<00:07:35.360><c> will</c><00:07:35.560><c> also</c><00:07:35.840><c> assume</c><00:07:36.720><c> that</c><00:07:36.880><c> the</c><00:07:37.000><c> system</c><00:07:38.000><c> has</c>"
+    },
+    {
+      "start": 459.15,
+      "duration": 0.0,
+      "text": "I will also assume that the system has"
+    },
+    {
+      "start": 459.16,
+      "duration": 0.0,
+      "text": "I will also assume that the system has um<00:07:39.360><c> so,</c><00:07:39.600><c> is</c><00:07:39.760><c> ergodic</c><00:07:40.640><c> and</c><00:07:40.800><c> mixing,</c><00:07:41.600><c> which</c>"
+    },
+    {
+      "start": 461.75,
+      "duration": 0.0,
+      "text": "um so, is ergodic and mixing, which"
+    },
+    {
+      "start": 461.76,
+      "duration": 0.0,
+      "text": "um so, is ergodic and mixing, which essentially<00:07:42.200><c> means</c><00:07:42.960><c> that</c><00:07:43.800><c> time</c><00:07:44.080><c> averages</c>"
+    },
+    {
+      "start": 465.51,
+      "duration": 0.0,
+      "text": "essentially means that time averages"
+    },
+    {
+      "start": 465.52,
+      "duration": 0.0,
+      "text": "essentially means that time averages along"
+    },
+    {
+      "start": 466.79,
+      "duration": 0.0,
+      "text": "along"
+    },
+    {
+      "start": 466.8,
+      "duration": 0.0,
+      "text": "along long<00:07:47.120><c> trajectory</c>"
+    },
+    {
+      "start": 468.91,
+      "duration": 0.0,
+      "text": "long trajectory"
+    },
+    {
+      "start": 468.92,
+      "duration": 0.0,
+      "text": "long trajectory will<00:07:49.120><c> converge</c><00:07:49.760><c> to</c><00:07:49.880><c> ensemble</c><00:07:50.400><c> averages,</c><00:07:51.520><c> and</c>"
+    },
+    {
+      "start": 471.75,
+      "duration": 0.0,
+      "text": "will converge to ensemble averages, and"
+    },
+    {
+      "start": 471.76,
+      "duration": 0.0,
+      "text": "will converge to ensemble averages, and also<00:07:52.400><c> that</c>"
+    },
+    {
+      "start": 473.75,
+      "duration": 0.0,
+      "text": "also that"
+    },
+    {
+      "start": 473.76,
+      "duration": 0.0,
+      "text": "also that the<00:07:53.880><c> correlation</c><00:07:54.440><c> function</c><00:07:55.240><c> decays</c>"
+    },
+    {
+      "start": 476.07,
+      "duration": 0.0,
+      "text": "the correlation function decays"
+    },
+    {
+      "start": 476.08,
+      "duration": 0.0,
+      "text": "the correlation function decays sufficiently<00:07:56.640><c> fast.</c>"
+    },
+    {
+      "start": 478.19,
+      "duration": 0.0,
+      "text": "sufficiently fast."
+    },
+    {
+      "start": 478.2,
+      "duration": 0.0,
+      "text": "sufficiently fast. And<00:07:58.400><c> then</c><00:07:59.120><c> I'm</c><00:07:59.440><c> also</c><00:08:00.120><c> assuming</c><00:08:01.360><c> that</c><00:08:01.840><c> we</c><00:08:02.000><c> have</c>"
+    },
+    {
+      "start": 482.55,
+      "duration": 0.0,
+      "text": "And then I'm also assuming that we have"
+    },
+    {
+      "start": 482.56,
+      "duration": 0.0,
+      "text": "And then I'm also assuming that we have an<00:08:02.680><c> effective</c><00:08:03.560><c> timescale</c><00:08:04.120><c> separation.</c><00:08:05.040><c> So,</c>"
+    },
+    {
+      "start": 485.19,
+      "duration": 0.0,
+      "text": "an effective timescale separation. So,"
+    },
+    {
+      "start": 485.2,
+      "duration": 0.0,
+      "text": "an effective timescale separation. So, this<00:08:05.400><c> essentially</c><00:08:06.160><c> allow</c><00:08:06.560><c> us</c><00:08:07.200><c> to</c><00:08:07.840><c> model</c><00:08:08.480><c> the</c>"
+    },
+    {
+      "start": 488.59,
+      "duration": 0.0,
+      "text": "this essentially allow us to model the"
+    },
+    {
+      "start": 488.6,
+      "duration": 0.0,
+      "text": "this essentially allow us to model the data<00:08:08.880><c> set</c><00:08:09.200><c> using</c>"
+    },
+    {
+      "start": 490.75,
+      "duration": 0.0,
+      "text": "data set using"
+    },
+    {
+      "start": 490.76,
+      "duration": 0.0,
+      "text": "data set using the<00:08:10.920><c> Langevin</c><00:08:11.320><c> equation</c><00:08:12.000><c> that</c><00:08:12.240><c> I</c><00:08:12.280><c> showed</c><00:08:12.640><c> you</c>"
+    },
+    {
+      "start": 492.75,
+      "duration": 0.0,
+      "text": "the Langevin equation that I showed you"
+    },
+    {
+      "start": 492.76,
+      "duration": 0.0,
+      "text": "the Langevin equation that I showed you before.<00:08:13.400><c> So,</c><00:08:13.560><c> essentially</c><00:08:14.040><c> treat</c><00:08:14.480><c> the</c><00:08:14.600><c> fast</c>"
+    },
+    {
+      "start": 494.87,
+      "duration": 0.0,
+      "text": "before. So, essentially treat the fast"
+    },
+    {
+      "start": 494.88,
+      "duration": 0.0,
+      "text": "before. So, essentially treat the fast timescale<00:08:15.920><c> as</c><00:08:16.200><c> a</c><00:08:16.320><c> noise</c><00:08:16.720><c> process,</c><00:08:17.800><c> and</c><00:08:17.960><c> then</c>"
+    },
+    {
+      "start": 498.35,
+      "duration": 0.0,
+      "text": "timescale as a noise process, and then"
+    },
+    {
+      "start": 498.36,
+      "duration": 0.0,
+      "text": "timescale as a noise process, and then build<00:08:18.640><c> a</c><00:08:18.720><c> drift</c><00:08:19.480><c> term</c><00:08:20.120><c> for</c><00:08:20.280><c> the</c><00:08:20.640><c> slow</c>"
+    },
+    {
+      "start": 500.91,
+      "duration": 0.0,
+      "text": "build a drift term for the slow"
+    },
+    {
+      "start": 500.92,
+      "duration": 0.0,
+      "text": "build a drift term for the slow timescales."
+    },
+    {
+      "start": 502.55,
+      "duration": 0.0,
+      "text": "timescales."
+    },
+    {
+      "start": 502.56,
+      "duration": 0.0,
+      "text": "timescales. I<00:08:22.680><c> will</c><00:08:22.840><c> also</c><00:08:23.080><c> assume</c><00:08:23.600><c> to</c><00:08:23.760><c> have</c><00:08:24.320><c> enough</c>"
+    },
+    {
+      "start": 504.75,
+      "duration": 0.0,
+      "text": "I will also assume to have enough"
+    },
+    {
+      "start": 504.76,
+      "duration": 0.0,
+      "text": "I will also assume to have enough observables<00:08:25.800><c> of</c><00:08:26.000><c> the</c><00:08:26.120><c> slow</c><00:08:26.440><c> timescales,</c>"
+    },
+    {
+      "start": 507.95,
+      "duration": 0.0,
+      "text": "observables of the slow timescales,"
+    },
+    {
+      "start": 507.96,
+      "duration": 0.0,
+      "text": "observables of the slow timescales, and<00:08:28.680><c> so</c><00:08:29.320><c> in</c><00:08:29.440><c> such</c><00:08:29.640><c> a</c><00:08:29.680><c> way</c><00:08:30.200><c> I</c><00:08:30.360><c> can</c><00:08:30.600><c> get</c>"
+    },
+    {
+      "start": 511.95,
+      "duration": 0.0,
+      "text": "and so in such a way I can get"
+    },
+    {
+      "start": 511.96,
+      "duration": 0.0,
+      "text": "and so in such a way I can get a<00:08:32.120><c> full</c><00:08:32.400><c> recovery</c><00:08:33.599><c> of</c><00:08:34.120><c> the</c><00:08:34.280><c> slow</c><00:08:34.840><c> variables.</c>"
+    },
+    {
+      "start": 515.95,
+      "duration": 0.0,
+      "text": "a full recovery of the slow variables."
+    },
+    {
+      "start": 515.96,
+      "duration": 0.0,
+      "text": "a full recovery of the slow variables. Even<00:08:36.200><c> if</c><00:08:36.520><c> we</c><00:08:36.640><c> don't</c><00:08:36.880><c> have</c><00:08:37.080><c> a</c><00:08:37.159><c> complete</c>"
+    },
+    {
+      "start": 517.51,
+      "duration": 0.0,
+      "text": "Even if we don't have a complete"
+    },
+    {
+      "start": 517.52,
+      "duration": 0.0,
+      "text": "Even if we don't have a complete observables<00:08:38.400><c> for</c><00:08:38.560><c> the</c><00:08:38.680><c> slow</c><00:08:38.919><c> variables,</c><00:08:39.599><c> we</c>"
+    },
+    {
+      "start": 519.79,
+      "duration": 0.0,
+      "text": "observables for the slow variables, we"
+    },
+    {
+      "start": 519.8,
+      "duration": 0.0,
+      "text": "observables for the slow variables, we can<00:08:40.240><c> still</c><00:08:40.719><c> augment</c><00:08:41.400><c> the</c><00:08:41.560><c> state</c><00:08:41.880><c> space</c><00:08:42.960><c> using</c>"
+    },
+    {
+      "start": 523.469,
+      "duration": 0.0,
+      "text": "can still augment the state space using"
+    },
+    {
+      "start": 523.479,
+      "duration": 0.0,
+      "text": "can still augment the state space using some<00:08:43.760><c> delay</c><00:08:44.720><c> embedded</c><00:08:45.240><c> version</c><00:08:45.800><c> of</c><00:08:45.920><c> the</c><00:08:46.000><c> data</c>"
+    },
+    {
+      "start": 526.27,
+      "duration": 0.0,
+      "text": "some delay embedded version of the data"
+    },
+    {
+      "start": 526.28,
+      "duration": 0.0,
+      "text": "some delay embedded version of the data set."
+    },
+    {
+      "start": 527.83,
+      "duration": 0.0,
+      "text": "set."
+    },
+    {
+      "start": 527.84,
+      "duration": 0.0,
+      "text": "set. And<00:08:48.040><c> then</c><00:08:48.400><c> finally,</c><00:08:49.400><c> even</c><00:08:49.760><c> if</c>"
+    },
+    {
+      "start": 531.15,
+      "duration": 0.0,
+      "text": "And then finally, even if"
+    },
+    {
+      "start": 531.16,
+      "duration": 0.0,
+      "text": "And then finally, even if uh<00:08:51.760><c> we</c><00:08:52.400><c> will</c><00:08:52.560><c> not</c><00:08:52.840><c> have</c><00:08:53.240><c> probably</c><00:08:53.960><c> some</c><00:08:54.840><c> very</c>"
+    },
+    {
+      "start": 535.15,
+      "duration": 0.0,
+      "text": "uh we will not have probably some very"
+    },
+    {
+      "start": 535.16,
+      "duration": 0.0,
+      "text": "uh we will not have probably some very fine<00:08:55.600><c> resolution</c>"
+    },
+    {
+      "start": 537.23,
+      "duration": 0.0,
+      "text": "fine resolution"
+    },
+    {
+      "start": 537.24,
+      "duration": 0.0,
+      "text": "fine resolution of<00:08:57.440><c> the</c><00:08:57.560><c> data</c><00:08:57.840><c> sample,</c>"
+    },
+    {
+      "start": 539.31,
+      "duration": 0.0,
+      "text": "of the data sample,"
+    },
+    {
+      "start": 539.32,
+      "duration": 0.0,
+      "text": "of the data sample, we<00:08:59.480><c> still</c><00:08:59.800><c> assume</c><00:09:00.440><c> to</c><00:09:00.600><c> have</c><00:09:01.480><c> enough</c>"
+    },
+    {
+      "start": 541.95,
+      "duration": 0.0,
+      "text": "we still assume to have enough"
+    },
+    {
+      "start": 541.96,
+      "duration": 0.0,
+      "text": "we still assume to have enough resolution<00:09:02.760><c> to</c><00:09:02.880><c> be</c><00:09:03.000><c> able</c><00:09:03.280><c> to</c><00:09:03.400><c> define</c>"
+    },
+    {
+      "start": 544.03,
+      "duration": 0.0,
+      "text": "resolution to be able to define"
+    },
+    {
+      "start": 544.04,
+      "duration": 0.0,
+      "text": "resolution to be able to define correlation<00:09:04.640><c> functions.</c><00:09:05.440><c> So,</c><00:09:05.600><c> we</c><00:09:05.680><c> don't</c><00:09:05.880><c> have</c>"
+    },
+    {
+      "start": 546.03,
+      "duration": 0.0,
+      "text": "correlation functions. So, we don't have"
+    },
+    {
+      "start": 546.04,
+      "duration": 0.0,
+      "text": "correlation functions. So, we don't have enough<00:09:06.320><c> resolution</c><00:09:07.200><c> to</c>"
+    },
+    {
+      "start": 548.67,
+      "duration": 0.0,
+      "text": "enough resolution to"
+    },
+    {
+      "start": 548.68,
+      "duration": 0.0,
+      "text": "enough resolution to evaluate<00:09:09.720><c> the</c><00:09:09.840><c> velocity</c><00:09:10.440><c> at</c><00:09:10.640><c> every</c>"
+    },
+    {
+      "start": 552.51,
+      "duration": 0.0,
+      "text": "evaluate the velocity at every"
+    },
+    {
+      "start": 552.52,
+      "duration": 0.0,
+      "text": "evaluate the velocity at every time<00:09:12.880><c> point</c><00:09:13.840><c> by</c><00:09:14.000><c> using</c><00:09:14.320><c> finite</c><00:09:14.600><c> difference,</c>"
+    },
+    {
+      "start": 555.27,
+      "duration": 0.0,
+      "text": "time point by using finite difference,"
+    },
+    {
+      "start": 555.28,
+      "duration": 0.0,
+      "text": "time point by using finite difference, but<00:09:15.480><c> still</c><00:09:15.880><c> we</c><00:09:16.000><c> have</c><00:09:16.200><c> enough</c><00:09:16.400><c> data</c>"
+    },
+    {
+      "start": 556.91,
+      "duration": 0.0,
+      "text": "but still we have enough data"
+    },
+    {
+      "start": 556.92,
+      "duration": 0.0,
+      "text": "but still we have enough data essentially<00:09:17.480><c> to</c><00:09:17.840><c> define</c><00:09:18.560><c> a</c><00:09:18.640><c> correlation</c>"
+    },
+    {
+      "start": 559.15,
+      "duration": 0.0,
+      "text": "essentially to define a correlation"
+    },
+    {
+      "start": 559.16,
+      "duration": 0.0,
+      "text": "essentially to define a correlation function."
+    },
+    {
+      "start": 560.23,
+      "duration": 0.0,
+      "text": "function."
+    },
+    {
+      "start": 560.24,
+      "duration": 0.0,
+      "text": "function. So,<00:09:20.320><c> these</c><00:09:20.600><c> are</c><00:09:20.960><c> the</c><00:09:21.080><c> goals</c><00:09:21.640><c> that</c><00:09:21.840><c> I</c><00:09:21.920><c> would</c>"
+    },
+    {
+      "start": 562.11,
+      "duration": 0.0,
+      "text": "So, these are the goals that I would"
+    },
+    {
+      "start": 562.12,
+      "duration": 0.0,
+      "text": "So, these are the goals that I would like"
+    },
+    {
+      "start": 563.35,
+      "duration": 0.0,
+      "text": "like"
+    },
+    {
+      "start": 563.36,
+      "duration": 0.0,
+      "text": "like my<00:09:23.520><c> modeling</c><00:09:23.960><c> strategy</c><00:09:24.840><c> to</c><00:09:25.000><c> achieve,</c>"
+    },
+    {
+      "start": 566.43,
+      "duration": 0.0,
+      "text": "my modeling strategy to achieve,"
+    },
+    {
+      "start": 566.44,
+      "duration": 0.0,
+      "text": "my modeling strategy to achieve, and<00:09:26.760><c> also</c><00:09:27.760><c> those</c><00:09:28.080><c> are</c><00:09:28.400><c> the</c><00:09:28.520><c> main</c><00:09:28.680><c> assumptions</c>"
+    },
+    {
+      "start": 569.39,
+      "duration": 0.0,
+      "text": "and also those are the main assumptions"
+    },
+    {
+      "start": 569.4,
+      "duration": 0.0,
+      "text": "and also those are the main assumptions that<00:09:29.640><c> I'm</c><00:09:29.760><c> doing</c><00:09:30.400><c> on</c><00:09:31.080><c> the</c>"
+    },
+    {
+      "start": 571.83,
+      "duration": 0.0,
+      "text": "that I'm doing on the"
+    },
+    {
+      "start": 571.84,
+      "duration": 0.0,
+      "text": "that I'm doing on the system<00:09:32.320><c> that</c><00:09:32.520><c> I'm</c><00:09:32.640><c> studying.</c>"
+    },
+    {
+      "start": 574.35,
+      "duration": 0.0,
+      "text": "system that I'm studying."
+    },
+    {
+      "start": 574.36,
+      "duration": 0.0,
+      "text": "system that I'm studying. So,<00:09:34.600><c> in</c><00:09:34.760><c> this</c><00:09:34.920><c> talk,</c><00:09:35.480><c> I</c><00:09:35.600><c> will</c><00:09:35.800><c> present</c><00:09:36.480><c> two</c>"
+    },
+    {
+      "start": 576.63,
+      "duration": 0.0,
+      "text": "So, in this talk, I will present two"
+    },
+    {
+      "start": 576.64,
+      "duration": 0.0,
+      "text": "So, in this talk, I will present two different<00:09:37.080><c> directions.</c><00:09:37.800><c> So,</c><00:09:38.040><c> two</c><00:09:38.200><c> different</c>"
+    },
+    {
+      "start": 578.87,
+      "duration": 0.0,
+      "text": "different directions. So, two different"
+    },
+    {
+      "start": 578.88,
+      "duration": 0.0,
+      "text": "different directions. So, two different modeling<00:09:39.280><c> strategies</c><00:09:40.280><c> that</c><00:09:40.560><c> I'm</c><00:09:40.760><c> currently</c>"
+    },
+    {
+      "start": 581.87,
+      "duration": 0.0,
+      "text": "modeling strategies that I'm currently"
+    },
+    {
+      "start": 581.88,
+      "duration": 0.0,
+      "text": "modeling strategies that I'm currently pursuing."
+    },
+    {
+      "start": 583.19,
+      "duration": 0.0,
+      "text": "pursuing."
+    },
+    {
+      "start": 583.2,
+      "duration": 0.0,
+      "text": "pursuing. The<00:09:43.320><c> first</c><00:09:43.640><c> one</c><00:09:44.040><c> So,</c><00:09:44.160><c> in</c><00:09:44.240><c> the</c><00:09:44.320><c> first</c><00:09:44.600><c> one,</c><00:09:45.120><c> I'm</c>"
+    },
+    {
+      "start": 585.27,
+      "duration": 0.0,
+      "text": "The first one So, in the first one, I'm"
+    },
+    {
+      "start": 585.28,
+      "duration": 0.0,
+      "text": "The first one So, in the first one, I'm assuming"
+    },
+    {
+      "start": 586.51,
+      "duration": 0.0,
+      "text": "assuming"
+    },
+    {
+      "start": 586.52,
+      "duration": 0.0,
+      "text": "assuming to<00:09:46.640><c> have</c><00:09:47.520><c> a</c><00:09:47.600><c> model</c><00:09:48.000><c> ansatz.</c><00:09:48.800><c> So,</c><00:09:48.920><c> essentially</c>"
+    },
+    {
+      "start": 590.07,
+      "duration": 0.0,
+      "text": "to have a model ansatz. So, essentially"
+    },
+    {
+      "start": 590.08,
+      "duration": 0.0,
+      "text": "to have a model ansatz. So, essentially I<00:09:50.400><c> have</c>"
+    },
+    {
+      "start": 591.79,
+      "duration": 0.0,
+      "text": "I have"
+    },
+    {
+      "start": 591.8,
+      "duration": 0.0,
+      "text": "I have a<00:09:51.920><c> knowledge</c><00:09:52.600><c> of</c><00:09:52.760><c> the</c><00:09:52.880><c> functional</c><00:09:53.560><c> form</c><00:09:54.200><c> of</c><00:09:54.360><c> my</c>"
+    },
+    {
+      "start": 594.51,
+      "duration": 0.0,
+      "text": "a knowledge of the functional form of my"
+    },
+    {
+      "start": 594.52,
+      "duration": 0.0,
+      "text": "a knowledge of the functional form of my model,<00:09:55.080><c> so</c><00:09:55.200><c> a</c><00:09:55.240><c> knowledge</c><00:09:55.680><c> that</c><00:09:55.920><c> be</c><00:09:56.120><c> derived</c>"
+    },
+    {
+      "start": 597.07,
+      "duration": 0.0,
+      "text": "model, so a knowledge that be derived"
+    },
+    {
+      "start": 597.08,
+      "duration": 0.0,
+      "text": "model, so a knowledge that be derived directly<00:09:57.600><c> from</c><00:09:57.800><c> physics.</c>"
+    },
+    {
+      "start": 599.67,
+      "duration": 0.0,
+      "text": "directly from physics."
+    },
+    {
+      "start": 599.68,
+      "duration": 0.0,
+      "text": "directly from physics. And<00:10:00.000><c> this</c><00:10:00.600><c> model</c><00:10:00.960><c> answers</c><00:10:01.520><c> depends</c><00:10:02.120><c> on</c><00:10:02.240><c> a</c><00:10:02.280><c> set</c>"
+    },
+    {
+      "start": 602.59,
+      "duration": 0.0,
+      "text": "And this model answers depends on a set"
+    },
+    {
+      "start": 602.6,
+      "duration": 0.0,
+      "text": "And this model answers depends on a set of<00:10:02.800><c> parameters</c><00:10:03.760><c> that</c><00:10:03.960><c> I</c><00:10:04.000><c> would</c><00:10:04.240><c> like</c><00:10:04.600><c> to</c>"
+    },
+    {
+      "start": 604.71,
+      "duration": 0.0,
+      "text": "of parameters that I would like to"
+    },
+    {
+      "start": 604.72,
+      "duration": 0.0,
+      "text": "of parameters that I would like to determine."
+    },
+    {
+      "start": 605.83,
+      "duration": 0.0,
+      "text": "determine."
+    },
+    {
+      "start": 605.84,
+      "duration": 0.0,
+      "text": "determine. I<00:10:05.960><c> have</c><00:10:06.720><c> an</c><00:10:06.880><c> initial</c><00:10:07.200><c> guess</c><00:10:07.920><c> for</c><00:10:08.080><c> those</c>"
+    },
+    {
+      "start": 608.27,
+      "duration": 0.0,
+      "text": "I have an initial guess for those"
+    },
+    {
+      "start": 608.28,
+      "duration": 0.0,
+      "text": "I have an initial guess for those parameters,<00:10:09.520><c> and</c><00:10:09.760><c> I</c><00:10:09.840><c> would</c><00:10:10.040><c> like</c><00:10:10.400><c> to</c>"
+    },
+    {
+      "start": 610.55,
+      "duration": 0.0,
+      "text": "parameters, and I would like to"
+    },
+    {
+      "start": 610.56,
+      "duration": 0.0,
+      "text": "parameters, and I would like to calibrate<00:10:11.280><c> those</c><00:10:11.520><c> model</c><00:10:11.840><c> parameters</c><00:10:13.000><c> in</c>"
+    },
+    {
+      "start": 613.15,
+      "duration": 0.0,
+      "text": "calibrate those model parameters in"
+    },
+    {
+      "start": 613.16,
+      "duration": 0.0,
+      "text": "calibrate those model parameters in order<00:10:13.520><c> to</c><00:10:13.680><c> reproduce</c><00:10:14.520><c> a</c><00:10:14.640><c> set</c><00:10:15.000><c> of</c><00:10:15.120><c> statistical</c>"
+    },
+    {
+      "start": 615.71,
+      "duration": 0.0,
+      "text": "order to reproduce a set of statistical"
+    },
+    {
+      "start": 615.72,
+      "duration": 0.0,
+      "text": "order to reproduce a set of statistical observables"
+    },
+    {
+      "start": 617.27,
+      "duration": 0.0,
+      "text": "observables"
+    },
+    {
+      "start": 617.28,
+      "duration": 0.0,
+      "text": "observables that<00:10:17.600><c> I</c><00:10:17.760><c> derive</c><00:10:18.480><c> from</c><00:10:18.800><c> my</c><00:10:18.960><c> observations.</c>"
+    },
+    {
+      "start": 620.83,
+      "duration": 0.0,
+      "text": "that I derive from my observations."
+    },
+    {
+      "start": 620.84,
+      "duration": 0.0,
+      "text": "that I derive from my observations. The<00:10:21.000><c> second</c><00:10:21.320><c> direction</c><00:10:22.080><c> instead</c><00:10:22.760><c> is</c><00:10:22.920><c> a</c><00:10:22.960><c> bit</c>"
+    },
+    {
+      "start": 623.19,
+      "duration": 0.0,
+      "text": "The second direction instead is a bit"
+    },
+    {
+      "start": 623.2,
+      "duration": 0.0,
+      "text": "The second direction instead is a bit more<00:10:23.360><c> ambitious.</c>"
+    },
+    {
+      "start": 624.71,
+      "duration": 0.0,
+      "text": "more ambitious."
+    },
+    {
+      "start": 624.72,
+      "duration": 0.0,
+      "text": "more ambitious. I<00:10:25.360><c> assume</c><00:10:26.440><c> not</c><00:10:26.720><c> to</c><00:10:26.880><c> know</c>"
+    },
+    {
+      "start": 628.11,
+      "duration": 0.0,
+      "text": "I assume not to know"
+    },
+    {
+      "start": 628.12,
+      "duration": 0.0,
+      "text": "I assume not to know any<00:10:28.560><c> model</c><00:10:28.880><c> answers.</c><00:10:29.400><c> So,</c><00:10:29.560><c> I</c><00:10:29.720><c> don't</c><00:10:30.000><c> start</c>"
+    },
+    {
+      "start": 630.43,
+      "duration": 0.0,
+      "text": "any model answers. So, I don't start"
+    },
+    {
+      "start": 630.44,
+      "duration": 0.0,
+      "text": "any model answers. So, I don't start from"
+    },
+    {
+      "start": 631.31,
+      "duration": 0.0,
+      "text": "from"
+    },
+    {
+      "start": 631.32,
+      "duration": 0.0,
+      "text": "from model<00:10:31.680><c> answers</c><00:10:32.280><c> assumption,</c><00:10:33.480><c> and</c><00:10:33.720><c> I</c><00:10:33.800><c> want</c><00:10:34.160><c> to</c>"
+    },
+    {
+      "start": 634.27,
+      "duration": 0.0,
+      "text": "model answers assumption, and I want to"
+    },
+    {
+      "start": 634.28,
+      "duration": 0.0,
+      "text": "model answers assumption, and I want to derive<00:10:35.160><c> from</c><00:10:35.440><c> my</c><00:10:35.600><c> data</c>"
+    },
+    {
+      "start": 636.87,
+      "duration": 0.0,
+      "text": "derive from my data"
+    },
+    {
+      "start": 636.88,
+      "duration": 0.0,
+      "text": "derive from my data the<00:10:37.040><c> whole</c><00:10:37.600><c> functional</c><00:10:38.320><c> form</c><00:10:38.680><c> of</c><00:10:38.800><c> the</c><00:10:38.920><c> model.</c>"
+    },
+    {
+      "start": 640.31,
+      "duration": 0.0,
+      "text": "the whole functional form of the model."
+    },
+    {
+      "start": 640.32,
+      "duration": 0.0,
+      "text": "the whole functional form of the model. In<00:10:40.520><c> in</c><00:10:40.680><c> this</c><00:10:40.840><c> case,</c><00:10:41.240><c> what</c><00:10:41.560><c> I'm</c><00:10:42.160><c> interested</c><00:10:42.960><c> to</c>"
+    },
+    {
+      "start": 643.07,
+      "duration": 0.0,
+      "text": "In in this case, what I'm interested to"
+    },
+    {
+      "start": 643.08,
+      "duration": 0.0,
+      "text": "In in this case, what I'm interested to do<00:10:43.200><c> precisely</c><00:10:44.160><c> is</c><00:10:44.400><c> to</c><00:10:44.560><c> start</c><00:10:45.520><c> from</c><00:10:46.480><c> the</c><00:10:46.800><c> set</c><00:10:47.480><c> of</c>"
+    },
+    {
+      "start": 647.87,
+      "duration": 0.0,
+      "text": "do precisely is to start from the set of"
+    },
+    {
+      "start": 647.88,
+      "duration": 0.0,
+      "text": "do precisely is to start from the set of statistical<00:10:49.160><c> and</c><00:10:49.800><c> dynamical</c><00:10:50.560><c> target</c>"
+    },
+    {
+      "start": 650.99,
+      "duration": 0.0,
+      "text": "statistical and dynamical target"
+    },
+    {
+      "start": 651.0,
+      "duration": 0.0,
+      "text": "statistical and dynamical target observables,"
+    },
+    {
+      "start": 652.39,
+      "duration": 0.0,
+      "text": "observables,"
+    },
+    {
+      "start": 652.4,
+      "duration": 0.0,
+      "text": "observables, and<00:10:52.640><c> I</c><00:10:52.680><c> would</c><00:10:52.920><c> like</c><00:10:53.320><c> to</c><00:10:53.440><c> infer</c><00:10:54.000><c> from</c><00:10:54.280><c> them</c>"
+    },
+    {
+      "start": 655.55,
+      "duration": 0.0,
+      "text": "and I would like to infer from them"
+    },
+    {
+      "start": 655.56,
+      "duration": 0.0,
+      "text": "and I would like to infer from them the<00:10:55.680><c> most</c><00:10:55.960><c> general</c><00:10:56.480><c> class</c><00:10:57.560><c> of</c><00:10:58.240><c> dynamical</c>"
+    },
+    {
+      "start": 658.79,
+      "duration": 0.0,
+      "text": "the most general class of dynamical"
+    },
+    {
+      "start": 658.8,
+      "duration": 0.0,
+      "text": "the most general class of dynamical systems<00:10:59.640><c> that</c><00:10:59.960><c> by</c><00:11:00.120><c> construction</c><00:11:01.480><c> reproduces</c>"
+    },
+    {
+      "start": 662.43,
+      "duration": 0.0,
+      "text": "systems that by construction reproduces"
+    },
+    {
+      "start": 662.44,
+      "duration": 0.0,
+      "text": "systems that by construction reproduces those<00:11:03.160><c> dynamical</c><00:11:03.640><c> observables.</c>"
+    },
+    {
+      "start": 665.63,
+      "duration": 0.0,
+      "text": "those dynamical observables."
+    },
+    {
+      "start": 665.64,
+      "duration": 0.0,
+      "text": "those dynamical observables. And<00:11:06.080><c> I</c><00:11:06.160><c> would</c><00:11:06.360><c> like</c><00:11:06.640><c> to</c><00:11:06.760><c> do</c><00:11:06.920><c> it.</c><00:11:07.320><c> So,</c><00:11:07.480><c> this</c><00:11:07.680><c> is</c>"
+    },
+    {
+      "start": 667.87,
+      "duration": 0.0,
+      "text": "And I would like to do it. So, this is"
+    },
+    {
+      "start": 667.88,
+      "duration": 0.0,
+      "text": "And I would like to do it. So, this is the<00:11:08.240><c> most</c><00:11:09.040><c> ambitious</c><00:11:09.520><c> part</c><00:11:09.920><c> of</c><00:11:10.120><c> the</c>"
+    },
+    {
+      "start": 671.63,
+      "duration": 0.0,
+      "text": "the most ambitious part of the"
+    },
+    {
+      "start": 671.64,
+      "duration": 0.0,
+      "text": "the most ambitious part of the of<00:11:11.800><c> the</c><00:11:11.920><c> method</c><00:11:12.440><c> because</c><00:11:12.760><c> I</c><00:11:12.839><c> would</c><00:11:13.000><c> like</c><00:11:13.240><c> to</c><00:11:13.360><c> do</c>"
+    },
+    {
+      "start": 673.51,
+      "duration": 0.0,
+      "text": "of the method because I would like to do"
+    },
+    {
+      "start": 673.52,
+      "duration": 0.0,
+      "text": "of the method because I would like to do it<00:11:14.080><c> without</c><00:11:14.760><c> ever</c><00:11:15.280><c> integrating</c><00:11:16.080><c> my</c><00:11:16.280><c> model</c>"
+    },
+    {
+      "start": 676.59,
+      "duration": 0.0,
+      "text": "it without ever integrating my model"
+    },
+    {
+      "start": 676.6,
+      "duration": 0.0,
+      "text": "it without ever integrating my model forward.<00:11:17.400><c> So,</c><00:11:17.600><c> just</c><00:11:18.120><c> from</c><00:11:18.360><c> the</c><00:11:18.480><c> knowledge</c><00:11:19.280><c> of</c>"
+    },
+    {
+      "start": 679.47,
+      "duration": 0.0,
+      "text": "forward. So, just from the knowledge of"
+    },
+    {
+      "start": 679.48,
+      "duration": 0.0,
+      "text": "forward. So, just from the knowledge of the<00:11:19.600><c> target</c><00:11:20.160><c> statistical</c><00:11:20.760><c> and</c><00:11:20.839><c> dynamical</c>"
+    },
+    {
+      "start": 681.95,
+      "duration": 0.0,
+      "text": "the target statistical and dynamical"
+    },
+    {
+      "start": 681.96,
+      "duration": 0.0,
+      "text": "the target statistical and dynamical observable,<00:11:23.200><c> I</c><00:11:23.320><c> would</c><00:11:23.520><c> like</c><00:11:23.880><c> to</c><00:11:23.960><c> be</c><00:11:24.080><c> able</c><00:11:24.360><c> to</c>"
+    },
+    {
+      "start": 684.47,
+      "duration": 0.0,
+      "text": "observable, I would like to be able to"
+    },
+    {
+      "start": 684.48,
+      "duration": 0.0,
+      "text": "observable, I would like to be able to infer<00:11:24.760><c> the</c><00:11:24.880><c> model</c><00:11:25.480><c> without</c><00:11:25.960><c> ever</c><00:11:26.520><c> integrating</c>"
+    },
+    {
+      "start": 687.31,
+      "duration": 0.0,
+      "text": "infer the model without ever integrating"
+    },
+    {
+      "start": 687.32,
+      "duration": 0.0,
+      "text": "infer the model without ever integrating my<00:11:27.440><c> model</c><00:11:27.760><c> forward</c><00:11:28.160><c> in</c><00:11:28.280><c> time.</c>"
+    },
+    {
+      "start": 689.35,
+      "duration": 0.0,
+      "text": "my model forward in time."
+    },
+    {
+      "start": 689.36,
+      "duration": 0.0,
+      "text": "my model forward in time. So,<00:11:29.440><c> these</c><00:11:29.640><c> are</c><00:11:29.760><c> the</c><00:11:29.960><c> two</c><00:11:30.160><c> directions</c><00:11:30.720><c> that</c><00:11:30.920><c> I</c>"
+    },
+    {
+      "start": 690.99,
+      "duration": 0.0,
+      "text": "So, these are the two directions that I"
+    },
+    {
+      "start": 691.0,
+      "duration": 0.0,
+      "text": "So, these are the two directions that I will<00:11:31.160><c> present</c><00:11:31.760><c> today.</c>"
+    },
+    {
+      "start": 693.03,
+      "duration": 0.0,
+      "text": "will present today."
+    },
+    {
+      "start": 693.04,
+      "duration": 0.0,
+      "text": "will present today. Let's<00:11:33.960><c> um</c><00:11:34.320><c> for</c><00:11:34.520><c> both</c><00:11:34.720><c> directions,</c><00:11:36.080><c> the</c><00:11:36.320><c> key</c>"
+    },
+    {
+      "start": 696.63,
+      "duration": 0.0,
+      "text": "Let's um for both directions, the key"
+    },
+    {
+      "start": 696.64,
+      "duration": 0.0,
+      "text": "Let's um for both directions, the key element<00:11:37.400><c> that</c><00:11:37.720><c> allow</c>"
+    },
+    {
+      "start": 699.23,
+      "duration": 0.0,
+      "text": "element that allow"
+    },
+    {
+      "start": 699.24,
+      "duration": 0.0,
+      "text": "element that allow essentially"
+    },
+    {
+      "start": 701.07,
+      "duration": 0.0,
+      "text": "essentially"
+    },
+    {
+      "start": 701.08,
+      "duration": 0.0,
+      "text": "essentially those<00:11:41.360><c> two</c><00:11:41.560><c> directions</c>"
+    },
+    {
+      "start": 703.11,
+      "duration": 0.0,
+      "text": "those two directions"
+    },
+    {
+      "start": 703.12,
+      "duration": 0.0,
+      "text": "those two directions is<00:11:43.760><c> the</c><00:11:43.960><c> score</c><00:11:44.240><c> function.</c><00:11:44.960><c> So,</c><00:11:45.120><c> essentially,</c>"
+    },
+    {
+      "start": 705.75,
+      "duration": 0.0,
+      "text": "is the score function. So, essentially,"
+    },
+    {
+      "start": 705.76,
+      "duration": 0.0,
+      "text": "is the score function. So, essentially, I<00:11:45.839><c> will</c><00:11:46.080><c> show</c><00:11:46.680><c> how</c><00:11:47.440><c> from</c><00:11:47.760><c> the</c><00:11:47.880><c> knowledge</c><00:11:48.240><c> of</c>"
+    },
+    {
+      "start": 708.31,
+      "duration": 0.0,
+      "text": "I will show how from the knowledge of"
+    },
+    {
+      "start": 708.32,
+      "duration": 0.0,
+      "text": "I will show how from the knowledge of the<00:11:48.440><c> score</c><00:11:48.640><c> function,</c>"
+    },
+    {
+      "start": 709.91,
+      "duration": 0.0,
+      "text": "the score function,"
+    },
+    {
+      "start": 709.92,
+      "duration": 0.0,
+      "text": "the score function, we<00:11:50.040><c> are</c><00:11:50.200><c> able</c><00:11:51.160><c> to</c>"
+    },
+    {
+      "start": 712.31,
+      "duration": 0.0,
+      "text": "we are able to"
+    },
+    {
+      "start": 712.32,
+      "duration": 0.0,
+      "text": "we are able to build<00:11:53.080><c> those</c><00:11:53.360><c> two</c><00:11:53.520><c> modeling</c><00:11:54.000><c> strategies.</c><00:11:54.839><c> So,</c>"
+    },
+    {
+      "start": 714.95,
+      "duration": 0.0,
+      "text": "build those two modeling strategies. So,"
+    },
+    {
+      "start": 714.96,
+      "duration": 0.0,
+      "text": "build those two modeling strategies. So, the<00:11:55.040><c> score</c><00:11:55.320><c> function</c><00:11:56.040><c> is</c><00:11:56.160><c> defined</c><00:11:56.760><c> as</c><00:11:56.960><c> the</c>"
+    },
+    {
+      "start": 717.03,
+      "duration": 0.0,
+      "text": "the score function is defined as the"
+    },
+    {
+      "start": 717.04,
+      "duration": 0.0,
+      "text": "the score function is defined as the gradient<00:11:57.480><c> of</c><00:11:57.600><c> the</c><00:11:57.680><c> logarithm</c><00:11:58.760><c> of</c><00:11:58.880><c> the</c><00:11:59.000><c> steady</c>"
+    },
+    {
+      "start": 719.27,
+      "duration": 0.0,
+      "text": "gradient of the logarithm of the steady"
+    },
+    {
+      "start": 719.28,
+      "duration": 0.0,
+      "text": "gradient of the logarithm of the steady state<00:11:59.800><c> distribution.</c>"
+    },
+    {
+      "start": 721.67,
+      "duration": 0.0,
+      "text": "state distribution."
+    },
+    {
+      "start": 721.68,
+      "duration": 0.0,
+      "text": "state distribution. Um"
+    },
+    {
+      "start": 723.91,
+      "duration": 0.0,
+      "text": "Um"
+    },
+    {
+      "start": 723.92,
+      "duration": 0.0,
+      "text": "Um Um<00:12:04.560><c> so,</c><00:12:04.720><c> the</c><00:12:04.839><c> score</c><00:12:05.120><c> function</c><00:12:06.120><c> is</c><00:12:06.520><c> one</c><00:12:06.760><c> of</c><00:12:06.880><c> the</c>"
+    },
+    {
+      "start": 726.99,
+      "duration": 0.0,
+      "text": "Um so, the score function is one of the"
+    },
+    {
+      "start": 727.0,
+      "duration": 0.0,
+      "text": "Um so, the score function is one of the central<00:12:07.760><c> So,</c><00:12:07.880><c> it's</c><00:12:08.080><c> a</c><00:12:08.200><c> central</c><00:12:08.600><c> quantity</c><00:12:09.200><c> in</c>"
+    },
+    {
+      "start": 729.35,
+      "duration": 0.0,
+      "text": "central So, it's a central quantity in"
+    },
+    {
+      "start": 729.36,
+      "duration": 0.0,
+      "text": "central So, it's a central quantity in machine<00:12:09.640><c> learning</c>"
+    },
+    {
+      "start": 731.07,
+      "duration": 0.0,
+      "text": "machine learning"
+    },
+    {
+      "start": 731.08,
+      "duration": 0.0,
+      "text": "machine learning specifically<00:12:12.000><c> in</c><00:12:12.200><c> score-based</c><00:12:13.160><c> generative</c>"
+    },
+    {
+      "start": 733.63,
+      "duration": 0.0,
+      "text": "specifically in score-based generative"
+    },
+    {
+      "start": 733.64,
+      "duration": 0.0,
+      "text": "specifically in score-based generative modeling"
+    },
+    {
+      "start": 734.91,
+      "duration": 0.0,
+      "text": "modeling"
+    },
+    {
+      "start": 734.92,
+      "duration": 0.0,
+      "text": "modeling because<00:12:15.200><c> essentially</c><00:12:16.120><c> it's</c><00:12:16.360><c> a</c><00:12:16.440><c> quantity</c><00:12:16.920><c> that</c>"
+    },
+    {
+      "start": 737.11,
+      "duration": 0.0,
+      "text": "because essentially it's a quantity that"
+    },
+    {
+      "start": 737.12,
+      "duration": 0.0,
+      "text": "because essentially it's a quantity that allow"
+    },
+    {
+      "start": 738.51,
+      "duration": 0.0,
+      "text": "allow"
+    },
+    {
+      "start": 738.52,
+      "duration": 0.0,
+      "text": "allow uh"
+    },
+    {
+      "start": 739.15,
+      "duration": 0.0,
+      "text": "uh"
+    },
+    {
+      "start": 739.16,
+      "duration": 0.0,
+      "text": "uh the<00:12:19.320><c> generation</c><00:12:20.160><c> of</c><00:12:20.360><c> new</c><00:12:20.560><c> data</c><00:12:20.839><c> sample</c>"
+    },
+    {
+      "start": 741.63,
+      "duration": 0.0,
+      "text": "the generation of new data sample"
+    },
+    {
+      "start": 741.64,
+      "duration": 0.0,
+      "text": "the generation of new data sample according<00:12:22.200><c> to</c><00:12:22.320><c> a</c><00:12:22.400><c> specific</c><00:12:23.080><c> probability</c>"
+    },
+    {
+      "start": 743.75,
+      "duration": 0.0,
+      "text": "according to a specific probability"
+    },
+    {
+      "start": 743.76,
+      "duration": 0.0,
+      "text": "according to a specific probability density<00:12:24.760><c> without</c><00:12:25.560><c> the</c><00:12:25.680><c> knowledge</c><00:12:26.280><c> of</c><00:12:26.560><c> the</c>"
+    },
+    {
+      "start": 746.67,
+      "duration": 0.0,
+      "text": "density without the knowledge of the"
+    },
+    {
+      "start": 746.68,
+      "duration": 0.0,
+      "text": "density without the knowledge of the normalization<00:12:27.400><c> constant</c><00:12:28.120><c> that</c>"
+    },
+    {
+      "start": 748.87,
+      "duration": 0.0,
+      "text": "normalization constant that"
+    },
+    {
+      "start": 748.88,
+      "duration": 0.0,
+      "text": "normalization constant that that<00:12:29.080><c> otherwise</c><00:12:30.080><c> someone</c><00:12:30.520><c> needed</c><00:12:30.960><c> to</c><00:12:31.360><c> to</c>"
+    },
+    {
+      "start": 751.71,
+      "duration": 0.0,
+      "text": "that otherwise someone needed to to"
+    },
+    {
+      "start": 751.72,
+      "duration": 0.0,
+      "text": "that otherwise someone needed to to evaluate<00:12:32.800><c> in</c><00:12:33.000><c> order</c><00:12:33.360><c> to</c><00:12:33.520><c> to</c><00:12:33.680><c> define</c><00:12:34.440><c> a</c>"
+    },
+    {
+      "start": 754.51,
+      "duration": 0.0,
+      "text": "evaluate in order to to define a"
+    },
+    {
+      "start": 754.52,
+      "duration": 0.0,
+      "text": "evaluate in order to to define a normalized<00:12:35.440><c> probability</c><00:12:36.040><c> density</c><00:12:36.560><c> function.</c>"
+    },
+    {
+      "start": 757.19,
+      "duration": 0.0,
+      "text": "normalized probability density function."
+    },
+    {
+      "start": 757.2,
+      "duration": 0.0,
+      "text": "normalized probability density function. So,<00:12:37.280><c> essentially,</c><00:12:37.760><c> it's</c><00:12:37.880><c> a</c><00:12:37.960><c> way</c><00:12:38.480><c> to</c><00:12:38.600><c> sample</c>"
+    },
+    {
+      "start": 759.23,
+      "duration": 0.0,
+      "text": "So, essentially, it's a way to sample"
+    },
+    {
+      "start": 759.24,
+      "duration": 0.0,
+      "text": "So, essentially, it's a way to sample new<00:12:39.400><c> data</c><00:12:39.640><c> sample</c><00:12:40.520><c> according</c><00:12:41.080><c> to</c><00:12:41.160><c> a</c><00:12:41.240><c> specific</c>"
+    },
+    {
+      "start": 761.71,
+      "duration": 0.0,
+      "text": "new data sample according to a specific"
+    },
+    {
+      "start": 761.72,
+      "duration": 0.0,
+      "text": "new data sample according to a specific probability<00:12:42.280><c> density</c><00:12:43.080><c> without</c><00:12:43.800><c> the</c>"
+    },
+    {
+      "start": 763.91,
+      "duration": 0.0,
+      "text": "probability density without the"
+    },
+    {
+      "start": 763.92,
+      "duration": 0.0,
+      "text": "probability density without the knowledge<00:12:44.920><c> of</c><00:12:45.200><c> the</c><00:12:45.320><c> steady</c><00:12:45.560><c> state</c>"
+    },
+    {
+      "start": 765.87,
+      "duration": 0.0,
+      "text": "knowledge of the steady state"
+    },
+    {
+      "start": 765.88,
+      "duration": 0.0,
+      "text": "knowledge of the steady state distribution,<00:12:46.720><c> which</c><00:12:46.960><c> can</c><00:12:47.160><c> be</c><00:12:47.280><c> extremely</c>"
+    },
+    {
+      "start": 767.67,
+      "duration": 0.0,
+      "text": "distribution, which can be extremely"
+    },
+    {
+      "start": 767.68,
+      "duration": 0.0,
+      "text": "distribution, which can be extremely challenging<00:12:48.720><c> to</c><00:12:48.920><c> be</c><00:12:49.240><c> constructed</c><00:12:50.120><c> from</c><00:12:50.360><c> high</c>"
+    },
+    {
+      "start": 770.55,
+      "duration": 0.0,
+      "text": "challenging to be constructed from high"
+    },
+    {
+      "start": 770.56,
+      "duration": 0.0,
+      "text": "challenging to be constructed from high dimensional<00:12:51.120><c> systems.</c>"
+    },
+    {
+      "start": 773.96,
+      "duration": 0.0,
+      "text": "In<00:12:54.520><c> the</c><00:12:54.880><c> In</c><00:12:55.000><c> this</c><00:12:55.160><c> talk,</c><00:12:56.120><c> I</c><00:12:56.320><c> will</c><00:12:57.160><c> So,</c><00:12:57.320><c> there</c>"
+    },
+    {
+      "start": 777.43,
+      "duration": 0.0,
+      "text": "In the In this talk, I will So, there"
+    },
+    {
+      "start": 777.44,
+      "duration": 0.0,
+      "text": "In the In this talk, I will So, there are<00:12:57.560><c> of</c><00:12:57.760><c> course</c><00:12:58.120><c> many</c><00:12:58.360><c> different</c><00:12:58.880><c> methodology</c>"
+    },
+    {
+      "start": 779.75,
+      "duration": 0.0,
+      "text": "are of course many different methodology"
+    },
+    {
+      "start": 779.76,
+      "duration": 0.0,
+      "text": "are of course many different methodology to<00:12:59.920><c> estimate</c><00:13:00.400><c> the</c><00:13:00.480><c> score</c><00:13:00.760><c> function</c><00:13:01.200><c> from</c>"
+    },
+    {
+      "start": 781.39,
+      "duration": 0.0,
+      "text": "to estimate the score function from"
+    },
+    {
+      "start": 781.4,
+      "duration": 0.0,
+      "text": "to estimate the score function from data.<00:13:02.200><c> In</c><00:13:02.320><c> this</c><00:13:02.480><c> talk,</c><00:13:03.080><c> I</c><00:13:03.240><c> will</c><00:13:03.480><c> use</c><00:13:04.120><c> the</c>"
+    },
+    {
+      "start": 784.23,
+      "duration": 0.0,
+      "text": "data. In this talk, I will use the"
+    },
+    {
+      "start": 784.24,
+      "duration": 0.0,
+      "text": "data. In this talk, I will use the denoising<00:13:04.760><c> score</c><00:13:05.000><c> matching</c><00:13:05.400><c> method,</c>"
+    },
+    {
+      "start": 786.79,
+      "duration": 0.0,
+      "text": "denoising score matching method,"
+    },
+    {
+      "start": 786.8,
+      "duration": 0.0,
+      "text": "denoising score matching method, which<00:13:06.920><c> essentially</c><00:13:08.040><c> consists</c><00:13:08.800><c> in</c><00:13:09.800><c> taking</c><00:13:10.800><c> the</c>"
+    },
+    {
+      "start": 790.87,
+      "duration": 0.0,
+      "text": "which essentially consists in taking the"
+    },
+    {
+      "start": 790.88,
+      "duration": 0.0,
+      "text": "which essentially consists in taking the data<00:13:11.160><c> set</c><00:13:11.720><c> that</c><00:13:11.960><c> I</c><00:13:12.000><c> would</c><00:13:12.240><c> like</c><00:13:12.560><c> to</c><00:13:12.680><c> use</c><00:13:13.240><c> to</c>"
+    },
+    {
+      "start": 793.31,
+      "duration": 0.0,
+      "text": "data set that I would like to use to"
+    },
+    {
+      "start": 793.32,
+      "duration": 0.0,
+      "text": "data set that I would like to use to estimate<00:13:13.920><c> the</c><00:13:14.040><c> score</c><00:13:14.280><c> function,</c><00:13:15.200><c> I</c><00:13:15.400><c> perturb</c>"
+    },
+    {
+      "start": 795.87,
+      "duration": 0.0,
+      "text": "estimate the score function, I perturb"
+    },
+    {
+      "start": 795.88,
+      "duration": 0.0,
+      "text": "estimate the score function, I perturb it<00:13:16.560><c> by</c><00:13:16.880><c> adding</c><00:13:17.560><c> a</c><00:13:17.640><c> tiny</c><00:13:18.720><c> Gaussian</c><00:13:19.400><c> white</c><00:13:19.640><c> noise</c>"
+    },
+    {
+      "start": 800.11,
+      "duration": 0.0,
+      "text": "it by adding a tiny Gaussian white noise"
+    },
+    {
+      "start": 800.12,
+      "duration": 0.0,
+      "text": "it by adding a tiny Gaussian white noise with<00:13:20.320><c> amplitude</c><00:13:20.800><c> sigma,</c>"
+    },
+    {
+      "start": 802.39,
+      "duration": 0.0,
+      "text": "with amplitude sigma,"
+    },
+    {
+      "start": 802.4,
+      "duration": 0.0,
+      "text": "with amplitude sigma, and<00:13:22.560><c> then</c><00:13:23.360><c> I</c><00:13:23.680><c> will</c><00:13:23.880><c> use</c><00:13:24.320><c> the</c><00:13:25.000><c> the</c><00:13:25.320><c> denoising</c>"
+    },
+    {
+      "start": 805.79,
+      "duration": 0.0,
+      "text": "and then I will use the the denoising"
+    },
+    {
+      "start": 805.8,
+      "duration": 0.0,
+      "text": "and then I will use the the denoising score<00:13:26.040><c> matching</c><00:13:26.480><c> identity,</c><00:13:27.400><c> according</c><00:13:27.839><c> to</c>"
+    },
+    {
+      "start": 807.95,
+      "duration": 0.0,
+      "text": "score matching identity, according to"
+    },
+    {
+      "start": 807.96,
+      "duration": 0.0,
+      "text": "score matching identity, according to which<00:13:28.560><c> I</c><00:13:28.800><c> can</c><00:13:29.160><c> write</c><00:13:29.680><c> the</c><00:13:29.800><c> score</c><00:13:30.160><c> function</c><00:13:30.800><c> of</c>"
+    },
+    {
+      "start": 810.99,
+      "duration": 0.0,
+      "text": "which I can write the score function of"
+    },
+    {
+      "start": 811.0,
+      "duration": 0.0,
+      "text": "which I can write the score function of this<00:13:31.240><c> perturbed</c><00:13:32.040><c> probability</c><00:13:32.640><c> density</c>"
+    },
+    {
+      "start": 814.15,
+      "duration": 0.0,
+      "text": "this perturbed probability density"
+    },
+    {
+      "start": 814.16,
+      "duration": 0.0,
+      "text": "this perturbed probability density in<00:13:34.480><c> fun</c><00:13:34.880><c> So,</c><00:13:35.200><c> in</c>"
+    },
+    {
+      "start": 817.07,
+      "duration": 0.0,
+      "text": "in fun So, in"
+    },
+    {
+      "start": 817.08,
+      "duration": 0.0,
+      "text": "in fun So, in in<00:13:37.240><c> terms</c><00:13:38.000><c> of</c><00:13:38.560><c> the</c><00:13:38.960><c> expected</c><00:13:39.520><c> value</c><00:13:40.520><c> of</c><00:13:41.440><c> zeta</c>"
+    },
+    {
+      "start": 822.829,
+      "duration": 0.0,
+      "text": "in terms of the expected value of zeta"
+    },
+    {
+      "start": 822.839,
+      "duration": 0.0,
+      "text": "in terms of the expected value of zeta conditioned<00:13:43.880><c> on</c><00:13:44.280><c> the</c><00:13:44.440><c> perturbed</c><00:13:45.000><c> value</c><00:13:45.600><c> x</c>"
+    },
+    {
+      "start": 826.23,
+      "duration": 0.0,
+      "text": "conditioned on the perturbed value x"
+    },
+    {
+      "start": 826.24,
+      "duration": 0.0,
+      "text": "conditioned on the perturbed value x sigma.<00:13:46.960><c> So,</c><00:13:47.120><c> given</c><00:13:47.839><c> So,</c><00:13:48.200><c> I</c><00:13:48.320><c> am</c><00:13:48.839><c> at</c><00:13:48.960><c> the</c><00:13:49.040><c> end</c>"
+    },
+    {
+      "start": 830.15,
+      "duration": 0.0,
+      "text": "sigma. So, given So, I am at the end"
+    },
+    {
+      "start": 830.16,
+      "duration": 0.0,
+      "text": "sigma. So, given So, I am at the end what<00:13:50.440><c> I'm</c><00:13:50.920><c> doing</c><00:13:51.520><c> is</c><00:13:51.640><c> to</c><00:13:51.760><c> train</c><00:13:52.080><c> a</c><00:13:52.120><c> neural</c>"
+    },
+    {
+      "start": 832.39,
+      "duration": 0.0,
+      "text": "what I'm doing is to train a neural"
+    },
+    {
+      "start": 832.4,
+      "duration": 0.0,
+      "text": "what I'm doing is to train a neural network<00:13:53.600><c> to</c><00:13:53.920><c> predict</c>"
+    },
+    {
+      "start": 835.55,
+      "duration": 0.0,
+      "text": "network to predict"
+    },
+    {
+      "start": 835.56,
+      "duration": 0.0,
+      "text": "network to predict the<00:13:56.280><c> value</c><00:13:57.120><c> of</c><00:13:57.280><c> the</c><00:13:57.400><c> noise</c><00:13:58.120><c> zeta</c>"
+    },
+    {
+      "start": 839.23,
+      "duration": 0.0,
+      "text": "the value of the noise zeta"
+    },
+    {
+      "start": 839.24,
+      "duration": 0.0,
+      "text": "the value of the noise zeta given<00:13:59.640><c> an</c><00:13:59.760><c> input</c><00:14:00.880><c> the</c><00:14:01.240><c> perturbed</c><00:14:02.280><c> value</c><00:14:02.880><c> of</c>"
+    },
+    {
+      "start": 843.03,
+      "duration": 0.0,
+      "text": "given an input the perturbed value of"
+    },
+    {
+      "start": 843.04,
+      "duration": 0.0,
+      "text": "given an input the perturbed value of the<00:14:03.160><c> data</c><00:14:03.400><c> set</c><00:14:03.600><c> point</c><00:14:04.280><c> x</c><00:14:04.760><c> sigma.</c>"
+    },
+    {
+      "start": 846.87,
+      "duration": 0.0,
+      "text": "the data set point x sigma."
+    },
+    {
+      "start": 846.88,
+      "duration": 0.0,
+      "text": "the data set point x sigma. Um<00:14:07.240><c> so,</c><00:14:07.480><c> if</c><00:14:07.960><c> sigma</c><00:14:08.800><c> So,</c><00:14:09.000><c> by</c><00:14:09.160><c> considering</c><00:14:09.839><c> sigma</c>"
+    },
+    {
+      "start": 850.59,
+      "duration": 0.0,
+      "text": "Um so, if sigma So, by considering sigma"
+    },
+    {
+      "start": 850.6,
+      "duration": 0.0,
+      "text": "Um so, if sigma So, by considering sigma So,<00:14:10.720><c> the</c><00:14:10.880><c> perturbation</c><00:14:12.360><c> amplitude</c><00:14:13.640><c> very</c>"
+    },
+    {
+      "start": 853.87,
+      "duration": 0.0,
+      "text": "So, the perturbation amplitude very"
+    },
+    {
+      "start": 853.88,
+      "duration": 0.0,
+      "text": "So, the perturbation amplitude very small,<00:14:14.800><c> I'm</c><00:14:15.360><c> essentially</c><00:14:16.280><c> estimating</c><00:14:17.040><c> the</c>"
+    },
+    {
+      "start": 857.15,
+      "duration": 0.0,
+      "text": "small, I'm essentially estimating the"
+    },
+    {
+      "start": 857.16,
+      "duration": 0.0,
+      "text": "small, I'm essentially estimating the score<00:14:17.440><c> function</c><00:14:18.079><c> of</c><00:14:18.320><c> a</c><00:14:18.400><c> perturbed</c><00:14:18.959><c> density,</c>"
+    },
+    {
+      "start": 859.51,
+      "duration": 0.0,
+      "text": "score function of a perturbed density,"
+    },
+    {
+      "start": 859.52,
+      "duration": 0.0,
+      "text": "score function of a perturbed density, which<00:14:19.720><c> is</c><00:14:19.880><c> extremely</c><00:14:20.360><c> close</c>"
+    },
+    {
+      "start": 861.59,
+      "duration": 0.0,
+      "text": "which is extremely close"
+    },
+    {
+      "start": 861.6,
+      "duration": 0.0,
+      "text": "which is extremely close to<00:14:21.800><c> the</c><00:14:22.360><c> observed</c><00:14:22.920><c> one.</c><00:14:23.720><c> And</c><00:14:23.880><c> so,</c><00:14:24.360><c> the</c><00:14:24.480><c> score</c>"
+    },
+    {
+      "start": 864.75,
+      "duration": 0.0,
+      "text": "to the observed one. And so, the score"
+    },
+    {
+      "start": 864.76,
+      "duration": 0.0,
+      "text": "to the observed one. And so, the score function<00:14:25.360><c> that</c><00:14:25.560><c> I</c><00:14:25.640><c> derive</c><00:14:26.360><c> using</c><00:14:26.760><c> this</c><00:14:27.000><c> method</c>"
+    },
+    {
+      "start": 868.63,
+      "duration": 0.0,
+      "text": "function that I derive using this method"
+    },
+    {
+      "start": 868.64,
+      "duration": 0.0,
+      "text": "function that I derive using this method it<00:14:28.760><c> will</c><00:14:28.959><c> be</c>"
+    },
+    {
+      "start": 870.19,
+      "duration": 0.0,
+      "text": "it will be"
+    },
+    {
+      "start": 870.2,
+      "duration": 0.0,
+      "text": "it will be very<00:14:30.480><c> close</c>"
+    },
+    {
+      "start": 871.59,
+      "duration": 0.0,
+      "text": "very close"
+    },
+    {
+      "start": 871.6,
+      "duration": 0.0,
+      "text": "very close to<00:14:31.800><c> the</c><00:14:32.920><c> correct</c><00:14:33.480><c> score</c><00:14:33.760><c> function.</c>"
+    },
+    {
+      "start": 874.99,
+      "duration": 0.0,
+      "text": "to the correct score function."
+    },
+    {
+      "start": 875.0,
+      "duration": 0.0,
+      "text": "to the correct score function. The<00:14:35.120><c> advantage</c><00:14:35.880><c> of</c><00:14:36.079><c> this</c><00:14:36.280><c> algo</c><00:14:37.000><c> of</c><00:14:37.160><c> this</c>"
+    },
+    {
+      "start": 877.39,
+      "duration": 0.0,
+      "text": "The advantage of this algo of this"
+    },
+    {
+      "start": 877.4,
+      "duration": 0.0,
+      "text": "The advantage of this algo of this algorithm<00:14:38.360><c> is</c><00:14:38.520><c> that</c><00:14:38.839><c> it</c><00:14:39.000><c> scales</c><00:14:39.640><c> extremely</c>"
+    },
+    {
+      "start": 880.35,
+      "duration": 0.0,
+      "text": "algorithm is that it scales extremely"
+    },
+    {
+      "start": 880.36,
+      "duration": 0.0,
+      "text": "algorithm is that it scales extremely well<00:14:40.720><c> with</c><00:14:40.880><c> the</c><00:14:40.959><c> dimension</c><00:14:41.680><c> because</c>"
+    },
+    {
+      "start": 881.949,
+      "duration": 0.0,
+      "text": "well with the dimension because"
+    },
+    {
+      "start": 881.959,
+      "duration": 0.0,
+      "text": "well with the dimension because essentially<00:14:42.560><c> we</c><00:14:42.680><c> are</c><00:14:42.800><c> recasting</c>"
+    },
+    {
+      "start": 884.19,
+      "duration": 0.0,
+      "text": "essentially we are recasting"
+    },
+    {
+      "start": 884.2,
+      "duration": 0.0,
+      "text": "essentially we are recasting the<00:14:44.320><c> score</c><00:14:44.680><c> estimation</c><00:14:45.360><c> problem,</c><00:14:46.200><c> which</c><00:14:46.520><c> will</c>"
+    },
+    {
+      "start": 887.35,
+      "duration": 0.0,
+      "text": "the score estimation problem, which will"
+    },
+    {
+      "start": 887.36,
+      "duration": 0.0,
+      "text": "the score estimation problem, which will otherwise<00:14:47.959><c> imply</c><00:14:48.680><c> the</c><00:14:48.839><c> differentiation</c><00:14:49.839><c> of</c>"
+    },
+    {
+      "start": 889.99,
+      "duration": 0.0,
+      "text": "otherwise imply the differentiation of"
+    },
+    {
+      "start": 890.0,
+      "duration": 0.0,
+      "text": "otherwise imply the differentiation of the<00:14:50.120><c> logarithm</c><00:14:50.720><c> of</c><00:14:50.800><c> the</c><00:14:50.920><c> steady</c><00:14:51.160><c> state</c>"
+    },
+    {
+      "start": 891.51,
+      "duration": 0.0,
+      "text": "the logarithm of the steady state"
+    },
+    {
+      "start": 891.52,
+      "duration": 0.0,
+      "text": "the logarithm of the steady state density,"
+    },
+    {
+      "start": 893.03,
+      "duration": 0.0,
+      "text": "density,"
+    },
+    {
+      "start": 893.04,
+      "duration": 0.0,
+      "text": "density, with<00:14:53.880><c> a</c><00:14:54.000><c> regression</c><00:14:54.640><c> problem.</c>"
+    },
+    {
+      "start": 895.71,
+      "duration": 0.0,
+      "text": "with a regression problem."
+    },
+    {
+      "start": 895.72,
+      "duration": 0.0,
+      "text": "with a regression problem. And<00:14:56.280><c> this</c><00:14:56.520><c> regression</c><00:14:56.959><c> problem</c><00:14:57.360><c> scales</c><00:14:57.839><c> very</c>"
+    },
+    {
+      "start": 898.03,
+      "duration": 0.0,
+      "text": "And this regression problem scales very"
+    },
+    {
+      "start": 898.04,
+      "duration": 0.0,
+      "text": "And this regression problem scales very well<00:14:58.200><c> with</c>"
+    },
+    {
+      "start": 898.87,
+      "duration": 0.0,
+      "text": "well with"
+    },
+    {
+      "start": 898.88,
+      "duration": 0.0,
+      "text": "well with with<00:14:59.000><c> the</c><00:14:59.079><c> dimension.</c><00:15:00.320><c> This</c><00:15:00.520><c> essentially</c>"
+    },
+    {
+      "start": 900.91,
+      "duration": 0.0,
+      "text": "with the dimension. This essentially"
+    },
+    {
+      "start": 900.92,
+      "duration": 0.0,
+      "text": "with the dimension. This essentially means<00:15:01.280><c> that</c><00:15:01.480><c> we</c><00:15:01.600><c> can</c><00:15:01.839><c> estimate</c><00:15:02.920><c> the</c><00:15:03.079><c> score</c>"
+    },
+    {
+      "start": 903.35,
+      "duration": 0.0,
+      "text": "means that we can estimate the score"
+    },
+    {
+      "start": 903.36,
+      "duration": 0.0,
+      "text": "means that we can estimate the score function"
+    },
+    {
+      "start": 905.11,
+      "duration": 0.0,
+      "text": "function"
+    },
+    {
+      "start": 905.12,
+      "duration": 0.0,
+      "text": "function quite<00:15:05.360><c> efficiently</c><00:15:06.480><c> also</c><00:15:06.959><c> for</c><00:15:07.200><c> very</c><00:15:07.600><c> high</c>"
+    },
+    {
+      "start": 907.79,
+      "duration": 0.0,
+      "text": "quite efficiently also for very high"
+    },
+    {
+      "start": 907.8,
+      "duration": 0.0,
+      "text": "quite efficiently also for very high dimensional<00:15:08.360><c> systems.</c><00:15:09.400><c> And</c><00:15:09.520><c> then</c><00:15:09.640><c> we'll</c><00:15:09.800><c> use</c>"
+    },
+    {
+      "start": 910.51,
+      "duration": 0.0,
+      "text": "dimensional systems. And then we'll use"
+    },
+    {
+      "start": 910.52,
+      "duration": 0.0,
+      "text": "dimensional systems. And then we'll use the<00:15:10.640><c> knowledge</c><00:15:11.480><c> of</c><00:15:11.839><c> the</c><00:15:11.959><c> score</c><00:15:12.200><c> function</c>"
+    },
+    {
+      "start": 913.55,
+      "duration": 0.0,
+      "text": "the knowledge of the score function"
+    },
+    {
+      "start": 913.56,
+      "duration": 0.0,
+      "text": "the knowledge of the score function to<00:15:13.640><c> infer</c><00:15:14.680><c> the</c><00:15:14.839><c> full</c><00:15:15.760><c> mathematical</c><00:15:16.360><c> model</c>"
+    },
+    {
+      "start": 917.51,
+      "duration": 0.0,
+      "text": "to infer the full mathematical model"
+    },
+    {
+      "start": 917.52,
+      "duration": 0.0,
+      "text": "to infer the full mathematical model that<00:15:17.760><c> explain</c><00:15:18.800><c> that</c><00:15:18.920><c> is</c><00:15:19.120><c> able</c><00:15:19.480><c> to</c><00:15:19.600><c> reproduce</c>"
+    },
+    {
+      "start": 921.069,
+      "duration": 0.0,
+      "text": "that explain that is able to reproduce"
+    },
+    {
+      "start": 921.079,
+      "duration": 0.0,
+      "text": "that explain that is able to reproduce this<00:15:21.400><c> set</c><00:15:21.720><c> of</c><00:15:21.839><c> target</c><00:15:22.320><c> observables</c><00:15:23.120><c> that</c><00:15:23.360><c> I</c>"
+    },
+    {
+      "start": 923.39,
+      "duration": 0.0,
+      "text": "this set of target observables that I"
+    },
+    {
+      "start": 923.4,
+      "duration": 0.0,
+      "text": "this set of target observables that I mentioned"
+    },
+    {
+      "start": 924.55,
+      "duration": 0.0,
+      "text": "mentioned"
+    },
+    {
+      "start": 924.56,
+      "duration": 0.0,
+      "text": "mentioned at<00:15:24.640><c> the</c><00:15:24.760><c> beginning.</c>"
+    },
+    {
+      "start": 926.27,
+      "duration": 0.0,
+      "text": "at the beginning."
+    },
+    {
+      "start": 926.28,
+      "duration": 0.0,
+      "text": "at the beginning. So,<00:15:26.360><c> let's</c><00:15:26.560><c> start</c><00:15:27.120><c> from</c><00:15:28.000><c> the</c><00:15:28.160><c> first</c><00:15:28.920><c> modeling</c>"
+    },
+    {
+      "start": 929.35,
+      "duration": 0.0,
+      "text": "So, let's start from the first modeling"
+    },
+    {
+      "start": 929.36,
+      "duration": 0.0,
+      "text": "So, let's start from the first modeling strategy."
+    },
+    {
+      "start": 933.0,
+      "duration": 0.0,
+      "text": "So,<00:15:33.240><c> as</c><00:15:33.440><c> I</c><00:15:33.480><c> said,</c>"
+    },
+    {
+      "start": 934.91,
+      "duration": 0.0,
+      "text": "So, as I said,"
+    },
+    {
+      "start": 934.92,
+      "duration": 0.0,
+      "text": "So, as I said, I'm<00:15:35.400><c> assuming</c><00:15:36.360><c> to</c><00:15:36.520><c> have</c><00:15:37.400><c> a</c><00:15:37.480><c> model</c><00:15:37.839><c> answers,</c><00:15:38.480><c> so</c>"
+    },
+    {
+      "start": 938.829,
+      "duration": 0.0,
+      "text": "I'm assuming to have a model answers, so"
+    },
+    {
+      "start": 938.839,
+      "duration": 0.0,
+      "text": "I'm assuming to have a model answers, so to<00:15:39.040><c> have</c><00:15:39.240><c> an</c><00:15:39.360><c> answers</c><00:15:39.839><c> for</c><00:15:40.040><c> the</c><00:15:40.160><c> functional</c>"
+    },
+    {
+      "start": 940.75,
+      "duration": 0.0,
+      "text": "to have an answers for the functional"
+    },
+    {
+      "start": 940.76,
+      "duration": 0.0,
+      "text": "to have an answers for the functional form<00:15:41.760><c> of</c><00:15:42.040><c> my</c><00:15:42.920><c> Langevin</c><00:15:43.440><c> equation</c><00:15:44.320><c> with</c>"
+    },
+    {
+      "start": 944.55,
+      "duration": 0.0,
+      "text": "form of my Langevin equation with"
+    },
+    {
+      "start": 944.56,
+      "duration": 0.0,
+      "text": "form of my Langevin equation with multiplicative<00:15:45.200><c> noise.</c>"
+    },
+    {
+      "start": 946.949,
+      "duration": 0.0,
+      "text": "multiplicative noise."
+    },
+    {
+      "start": 946.959,
+      "duration": 0.0,
+      "text": "multiplicative noise. And<00:15:47.200><c> this</c><00:15:47.360><c> model</c><00:15:47.680><c> answers</c><00:15:48.120><c> depends</c><00:15:48.640><c> on</c><00:15:48.760><c> a</c><00:15:48.800><c> set</c>"
+    },
+    {
+      "start": 949.069,
+      "duration": 0.0,
+      "text": "And this model answers depends on a set"
+    },
+    {
+      "start": 949.079,
+      "duration": 0.0,
+      "text": "And this model answers depends on a set of<00:15:49.200><c> parameters</c><00:15:50.160><c> alpha</c><00:15:51.040><c> and</c><00:15:51.240><c> beta.</c>"
+    },
+    {
+      "start": 952.51,
+      "duration": 0.0,
+      "text": "of parameters alpha and beta."
+    },
+    {
+      "start": 952.52,
+      "duration": 0.0,
+      "text": "of parameters alpha and beta. We<00:15:52.839><c> call</c><00:15:53.200><c> alpha</c><00:15:54.079><c> all</c><00:15:54.240><c> the</c><00:15:54.360><c> parameters</c><00:15:55.320><c> inside</c>"
+    },
+    {
+      "start": 956.03,
+      "duration": 0.0,
+      "text": "We call alpha all the parameters inside"
+    },
+    {
+      "start": 956.04,
+      "duration": 0.0,
+      "text": "We call alpha all the parameters inside the<00:15:56.120><c> drift</c><00:15:56.440><c> term,</c><00:15:57.040><c> and</c><00:15:57.200><c> beta</c><00:15:57.480><c> all</c><00:15:57.600><c> the</c>"
+    },
+    {
+      "start": 957.71,
+      "duration": 0.0,
+      "text": "the drift term, and beta all the"
+    },
+    {
+      "start": 957.72,
+      "duration": 0.0,
+      "text": "the drift term, and beta all the parameters<00:15:58.720><c> inside</c><00:15:59.280><c> the</c><00:15:59.440><c> diffusion</c><00:15:59.880><c> term.</c>"
+    },
+    {
+      "start": 960.91,
+      "duration": 0.0,
+      "text": "parameters inside the diffusion term."
+    },
+    {
+      "start": 960.92,
+      "duration": 0.0,
+      "text": "parameters inside the diffusion term. So,<00:16:00.959><c> the</c><00:16:01.079><c> problem</c><00:16:01.560><c> consists</c><00:16:02.040><c> now</c><00:16:02.560><c> in</c><00:16:02.839><c> finding</c>"
+    },
+    {
+      "start": 963.829,
+      "duration": 0.0,
+      "text": "So, the problem consists now in finding"
+    },
+    {
+      "start": 963.839,
+      "duration": 0.0,
+      "text": "So, the problem consists now in finding the<00:16:03.959><c> values</c><00:16:04.560><c> for</c><00:16:04.760><c> alpha</c><00:16:04.959><c> and</c><00:16:05.079><c> beta</c>"
+    },
+    {
+      "start": 966.19,
+      "duration": 0.0,
+      "text": "the values for alpha and beta"
+    },
+    {
+      "start": 966.2,
+      "duration": 0.0,
+      "text": "the values for alpha and beta that<00:16:06.800><c> reproduces</c><00:16:07.680><c> the</c><00:16:07.839><c> target</c><00:16:08.360><c> statistical</c>"
+    },
+    {
+      "start": 968.91,
+      "duration": 0.0,
+      "text": "that reproduces the target statistical"
+    },
+    {
+      "start": 968.92,
+      "duration": 0.0,
+      "text": "that reproduces the target statistical observables."
+    },
+    {
+      "start": 971.55,
+      "duration": 0.0,
+      "text": "observables."
+    },
+    {
+      "start": 971.56,
+      "duration": 0.0,
+      "text": "observables. So,<00:16:11.880><c> learning</c>"
+    },
+    {
+      "start": 973.23,
+      "duration": 0.0,
+      "text": "So, learning"
+    },
+    {
+      "start": 973.24,
+      "duration": 0.0,
+      "text": "So, learning So,"
+    },
+    {
+      "start": 974.43,
+      "duration": 0.0,
+      "text": "So,"
+    },
+    {
+      "start": 974.44,
+      "duration": 0.0,
+      "text": "So, in"
+    },
+    {
+      "start": 975.91,
+      "duration": 0.0,
+      "text": "in"
+    },
+    {
+      "start": 975.92,
+      "duration": 0.0,
+      "text": "in in<00:16:16.079><c> order</c><00:16:16.360><c> to</c><00:16:16.600><c> solve</c><00:16:17.079><c> this</c><00:16:17.240><c> calibration</c>"
+    },
+    {
+      "start": 978.51,
+      "duration": 0.0,
+      "text": "in order to solve this calibration"
+    },
+    {
+      "start": 978.52,
+      "duration": 0.0,
+      "text": "in order to solve this calibration problem,<00:16:19.600><c> what</c><00:16:19.880><c> we</c><00:16:20.000><c> need</c><00:16:20.640><c> is</c><00:16:20.800><c> the</c><00:16:20.920><c> parameter</c>"
+    },
+    {
+      "start": 981.43,
+      "duration": 0.0,
+      "text": "problem, what we need is the parameter"
+    },
+    {
+      "start": 981.44,
+      "duration": 0.0,
+      "text": "problem, what we need is the parameter sensitivity.<00:16:22.520><c> So,</c><00:16:23.040><c> if</c><00:16:23.240><c> we</c><00:16:23.360><c> call</c><00:16:24.120><c> phi</c><00:16:24.400><c> m</c><00:16:25.040><c> the</c>"
+    },
+    {
+      "start": 985.15,
+      "duration": 0.0,
+      "text": "sensitivity. So, if we call phi m the"
+    },
+    {
+      "start": 985.16,
+      "duration": 0.0,
+      "text": "sensitivity. So, if we call phi m the set<00:16:26.280><c> of</c><00:16:26.640><c> observables</c><00:16:27.600><c> that</c><00:16:27.760><c> we</c><00:16:27.880><c> want</c><00:16:28.240><c> our</c>"
+    },
+    {
+      "start": 988.43,
+      "duration": 0.0,
+      "text": "set of observables that we want our"
+    },
+    {
+      "start": 988.44,
+      "duration": 0.0,
+      "text": "set of observables that we want our model<00:16:28.920><c> to</c><00:16:29.079><c> reproduce,</c>"
+    },
+    {
+      "start": 990.71,
+      "duration": 0.0,
+      "text": "model to reproduce,"
+    },
+    {
+      "start": 990.72,
+      "duration": 0.0,
+      "text": "model to reproduce, what<00:16:31.040><c> we</c><00:16:31.160><c> need</c><00:16:31.480><c> to</c><00:16:31.600><c> know</c><00:16:32.160><c> is</c><00:16:32.600><c> how</c>"
+    },
+    {
+      "start": 994.15,
+      "duration": 0.0,
+      "text": "what we need to know is how"
+    },
+    {
+      "start": 994.16,
+      "duration": 0.0,
+      "text": "what we need to know is how those<00:16:34.560><c> observable</c><00:16:35.760><c> phi</c><00:16:35.959><c> of</c><00:16:36.120><c> m</c><00:16:36.839><c> will</c><00:16:37.160><c> change</c><00:16:38.320><c> by</c>"
+    },
+    {
+      "start": 999.31,
+      "duration": 0.0,
+      "text": "those observable phi of m will change by"
+    },
+    {
+      "start": 999.32,
+      "duration": 0.0,
+      "text": "those observable phi of m will change by changing<00:16:40.280><c> by</c><00:16:40.480><c> a</c><00:16:40.560><c> tiny</c><00:16:40.839><c> amount</c><00:16:41.640><c> each</c><00:16:42.160><c> of</c><00:16:42.520><c> the</c>"
+    },
+    {
+      "start": 1002.63,
+      "duration": 0.0,
+      "text": "changing by a tiny amount each of the"
+    },
+    {
+      "start": 1002.64,
+      "duration": 0.0,
+      "text": "changing by a tiny amount each of the model<00:16:42.959><c> parameters.</c>"
+    },
+    {
+      "start": 1004.43,
+      "duration": 0.0,
+      "text": "model parameters."
+    },
+    {
+      "start": 1004.44,
+      "duration": 0.0,
+      "text": "model parameters. So,<00:16:44.600><c> if</c><00:16:45.240><c> we</c><00:16:45.360><c> know</c><00:16:45.560><c> these</c><00:16:45.880><c> these</c><00:16:46.160><c> quantities,</c>"
+    },
+    {
+      "start": 1006.829,
+      "duration": 0.0,
+      "text": "So, if we know these these quantities,"
+    },
+    {
+      "start": 1006.839,
+      "duration": 0.0,
+      "text": "So, if we know these these quantities, we<00:16:47.000><c> can</c><00:16:47.280><c> essentially</c><00:16:48.000><c> update</c>"
+    },
+    {
+      "start": 1009.35,
+      "duration": 0.0,
+      "text": "we can essentially update"
+    },
+    {
+      "start": 1009.36,
+      "duration": 0.0,
+      "text": "we can essentially update the<00:16:49.520><c> parameters</c><00:16:50.560><c> accordingly</c><00:16:51.600><c> in</c><00:16:51.800><c> order</c><00:16:52.440><c> to</c>"
+    },
+    {
+      "start": 1012.59,
+      "duration": 0.0,
+      "text": "the parameters accordingly in order to"
+    },
+    {
+      "start": 1012.6,
+      "duration": 0.0,
+      "text": "the parameters accordingly in order to minimize<00:16:53.440><c> the</c><00:16:53.600><c> distance</c><00:16:54.440><c> between</c><00:16:55.360><c> the</c><00:16:55.680><c> target</c>"
+    },
+    {
+      "start": 1016.55,
+      "duration": 0.0,
+      "text": "minimize the distance between the target"
+    },
+    {
+      "start": 1016.56,
+      "duration": 0.0,
+      "text": "minimize the distance between the target observables<00:16:57.720><c> and</c><00:16:57.920><c> the</c><00:16:58.000><c> observables</c>"
+    },
+    {
+      "start": 1018.87,
+      "duration": 0.0,
+      "text": "observables and the observables"
+    },
+    {
+      "start": 1018.88,
+      "duration": 0.0,
+      "text": "observables and the observables predicted<00:16:59.480><c> by</c><00:16:59.600><c> the</c><00:16:59.720><c> model.</c>"
+    },
+    {
+      "start": 1021.75,
+      "duration": 0.0,
+      "text": "predicted by the model."
+    },
+    {
+      "start": 1021.76,
+      "duration": 0.0,
+      "text": "predicted by the model. We<00:17:01.880><c> can</c><00:17:02.079><c> estimate</c><00:17:02.680><c> the</c><00:17:02.800><c> statistical</c>"
+    },
+    {
+      "start": 1023.55,
+      "duration": 0.0,
+      "text": "We can estimate the statistical"
+    },
+    {
+      "start": 1023.56,
+      "duration": 0.0,
+      "text": "We can estimate the statistical Jacobians"
+    },
+    {
+      "start": 1025.11,
+      "duration": 0.0,
+      "text": "Jacobians"
+    },
+    {
+      "start": 1025.12,
+      "duration": 0.0,
+      "text": "Jacobians using<00:17:05.720><c> a</c><00:17:05.800><c> naive</c><00:17:06.160><c> approach</c><00:17:06.959><c> by</c><00:17:07.360><c> just</c>"
+    },
+    {
+      "start": 1028.189,
+      "duration": 0.0,
+      "text": "using a naive approach by just"
+    },
+    {
+      "start": 1028.199,
+      "duration": 0.0,
+      "text": "using a naive approach by just integrating<00:17:09.160><c> the</c><00:17:09.280><c> model</c><00:17:09.600><c> forward</c><00:17:10.160><c> many</c>"
+    },
+    {
+      "start": 1030.39,
+      "duration": 0.0,
+      "text": "integrating the model forward many"
+    },
+    {
+      "start": 1030.4,
+      "duration": 0.0,
+      "text": "integrating the model forward many times.<00:17:11.360><c> Every</c><00:17:11.640><c> time</c><00:17:12.160><c> we</c><00:17:12.320><c> perturb</c><00:17:13.439><c> just</c><00:17:13.760><c> one</c>"
+    },
+    {
+      "start": 1034.47,
+      "duration": 0.0,
+      "text": "times. Every time we perturb just one"
+    },
+    {
+      "start": 1034.48,
+      "duration": 0.0,
+      "text": "times. Every time we perturb just one parameter<00:17:15.280><c> by</c><00:17:15.480><c> a</c><00:17:15.560><c> tiny</c><00:17:15.800><c> amount,</c>"
+    },
+    {
+      "start": 1037.11,
+      "duration": 0.0,
+      "text": "parameter by a tiny amount,"
+    },
+    {
+      "start": 1037.12,
+      "duration": 0.0,
+      "text": "parameter by a tiny amount, and<00:17:17.280><c> then</c><00:17:17.560><c> using</c><00:17:17.880><c> finite</c><00:17:18.199><c> difference,</c><00:17:18.880><c> we</c>"
+    },
+    {
+      "start": 1038.99,
+      "duration": 0.0,
+      "text": "and then using finite difference, we"
+    },
+    {
+      "start": 1039.0,
+      "duration": 0.0,
+      "text": "and then using finite difference, we estimate<00:17:19.720><c> this</c><00:17:20.480><c> statistical</c><00:17:21.000><c> Jacobians.</c>"
+    },
+    {
+      "start": 1043.189,
+      "duration": 0.0,
+      "text": "estimate this statistical Jacobians."
+    },
+    {
+      "start": 1043.199,
+      "duration": 0.0,
+      "text": "estimate this statistical Jacobians. This<00:17:23.480><c> method</c><00:17:24.000><c> of</c><00:17:24.400><c> course</c><00:17:24.839><c> works,</c><00:17:25.640><c> but</c><00:17:26.439><c> it</c><00:17:26.679><c> has</c>"
+    },
+    {
+      "start": 1047.27,
+      "duration": 0.0,
+      "text": "This method of course works, but it has"
+    },
+    {
+      "start": 1047.28,
+      "duration": 0.0,
+      "text": "This method of course works, but it has the<00:17:27.920><c> problem</c><00:17:29.040><c> that</c><00:17:29.560><c> it</c><00:17:29.800><c> requires</c><00:17:31.120><c> an</c>"
+    },
+    {
+      "start": 1051.27,
+      "duration": 0.0,
+      "text": "the problem that it requires an"
+    },
+    {
+      "start": 1051.28,
+      "duration": 0.0,
+      "text": "the problem that it requires an extremely<00:17:31.760><c> large</c><00:17:32.200><c> number</c><00:17:32.679><c> of</c><00:17:32.840><c> model</c>"
+    },
+    {
+      "start": 1053.15,
+      "duration": 0.0,
+      "text": "extremely large number of model"
+    },
+    {
+      "start": 1053.16,
+      "duration": 0.0,
+      "text": "extremely large number of model integrations.<00:17:34.160><c> So,</c><00:17:34.320><c> we</c><00:17:34.440><c> need</c><00:17:35.080><c> at</c><00:17:35.240><c> least</c>"
+    },
+    {
+      "start": 1056.63,
+      "duration": 0.0,
+      "text": "integrations. So, we need at least"
+    },
+    {
+      "start": 1056.64,
+      "duration": 0.0,
+      "text": "integrations. So, we need at least one<00:17:36.880><c> model</c><00:17:37.200><c> integration</c><00:17:38.000><c> for</c><00:17:38.200><c> every</c>"
+    },
+    {
+      "start": 1058.43,
+      "duration": 0.0,
+      "text": "one model integration for every"
+    },
+    {
+      "start": 1058.44,
+      "duration": 0.0,
+      "text": "one model integration for every parameter.<00:17:39.360><c> And</c><00:17:40.000><c> if</c><00:17:40.200><c> the</c><00:17:40.280><c> model</c><00:17:40.720><c> is</c><00:17:40.920><c> very</c>"
+    },
+    {
+      "start": 1061.11,
+      "duration": 0.0,
+      "text": "parameter. And if the model is very"
+    },
+    {
+      "start": 1061.12,
+      "duration": 0.0,
+      "text": "parameter. And if the model is very large,"
+    },
+    {
+      "start": 1062.31,
+      "duration": 0.0,
+      "text": "large,"
+    },
+    {
+      "start": 1062.32,
+      "duration": 0.0,
+      "text": "large, it<00:17:42.480><c> becomes</c><00:17:43.080><c> extremely</c><00:17:43.560><c> computationally</c>"
+    },
+    {
+      "start": 1064.19,
+      "duration": 0.0,
+      "text": "it becomes extremely computationally"
+    },
+    {
+      "start": 1064.2,
+      "duration": 0.0,
+      "text": "it becomes extremely computationally expensive<00:17:44.880><c> to</c><00:17:45.000><c> integrate,</c><00:17:45.960><c> in</c><00:17:46.080><c> particular,</c>"
+    },
+    {
+      "start": 1066.95,
+      "duration": 0.0,
+      "text": "expensive to integrate, in particular,"
+    },
+    {
+      "start": 1066.96,
+      "duration": 0.0,
+      "text": "expensive to integrate, in particular, if<00:17:47.120><c> we</c><00:17:47.240><c> have</c><00:17:47.600><c> many</c><00:17:48.200><c> parameters</c><00:17:48.840><c> that</c><00:17:49.040><c> we</c><00:17:49.160><c> want</c>"
+    },
+    {
+      "start": 1069.39,
+      "duration": 0.0,
+      "text": "if we have many parameters that we want"
+    },
+    {
+      "start": 1069.4,
+      "duration": 0.0,
+      "text": "if we have many parameters that we want to<00:17:49.480><c> calibrate.</c>"
+    },
+    {
+      "start": 1071.59,
+      "duration": 0.0,
+      "text": "to calibrate."
+    },
+    {
+      "start": 1071.6,
+      "duration": 0.0,
+      "text": "to calibrate. So,<00:17:51.679><c> the</c>"
+    },
+    {
+      "start": 1072.35,
+      "duration": 0.0,
+      "text": "So, the"
+    },
+    {
+      "start": 1072.36,
+      "duration": 0.0,
+      "text": "So, the the<00:17:52.560><c> key</c><00:17:52.720><c> idea</c><00:17:53.280><c> here</c><00:17:54.120><c> is</c><00:17:54.440><c> to</c><00:17:54.640><c> recast</c><00:17:55.760><c> this</c>"
+    },
+    {
+      "start": 1076.31,
+      "duration": 0.0,
+      "text": "the key idea here is to recast this"
+    },
+    {
+      "start": 1076.32,
+      "duration": 0.0,
+      "text": "the key idea here is to recast this calibration<00:17:56.880><c> problem</c>"
+    },
+    {
+      "start": 1078.19,
+      "duration": 0.0,
+      "text": "calibration problem"
+    },
+    {
+      "start": 1078.2,
+      "duration": 0.0,
+      "text": "calibration problem to<00:17:58.320><c> a</c><00:17:58.440><c> perturbation</c><00:17:59.200><c> problem.</c><00:18:00.160><c> So,</c><00:18:00.440><c> if</c><00:18:00.679><c> we</c>"
+    },
+    {
+      "start": 1081.39,
+      "duration": 0.0,
+      "text": "to a perturbation problem. So, if we"
+    },
+    {
+      "start": 1081.4,
+      "duration": 0.0,
+      "text": "to a perturbation problem. So, if we imagine<00:18:02.320><c> to</c><00:18:02.520><c> Taylor</c><00:18:02.880><c> expand</c><00:18:03.840><c> at</c><00:18:04.120><c> first</c><00:18:04.480><c> order</c>"
+    },
+    {
+      "start": 1085.63,
+      "duration": 0.0,
+      "text": "imagine to Taylor expand at first order"
+    },
+    {
+      "start": 1085.64,
+      "duration": 0.0,
+      "text": "imagine to Taylor expand at first order the<00:18:06.160><c> drift</c><00:18:06.720><c> and</c><00:18:07.120><c> and</c><00:18:07.240><c> the</c><00:18:07.320><c> diffusion</c>"
+    },
+    {
+      "start": 1087.83,
+      "duration": 0.0,
+      "text": "the drift and and the diffusion"
+    },
+    {
+      "start": 1087.84,
+      "duration": 0.0,
+      "text": "the drift and and the diffusion coefficients<00:18:08.800><c> after</c><00:18:09.679><c> we</c><00:18:10.280><c> perturb</c><00:18:11.120><c> by</c><00:18:11.280><c> a</c><00:18:11.360><c> tiny</c>"
+    },
+    {
+      "start": 1091.59,
+      "duration": 0.0,
+      "text": "coefficients after we perturb by a tiny"
+    },
+    {
+      "start": 1091.6,
+      "duration": 0.0,
+      "text": "coefficients after we perturb by a tiny amount<00:18:12.520><c> each</c><00:18:12.760><c> of</c><00:18:12.880><c> those</c><00:18:13.120><c> parameters,</c>"
+    },
+    {
+      "start": 1094.71,
+      "duration": 0.0,
+      "text": "amount each of those parameters,"
+    },
+    {
+      "start": 1094.72,
+      "duration": 0.0,
+      "text": "amount each of those parameters, so<00:18:14.840><c> essentially,</c><00:18:15.400><c> what</c><00:18:15.600><c> we</c><00:18:15.720><c> obtain</c><00:18:16.600><c> is</c><00:18:17.000><c> a</c>"
+    },
+    {
+      "start": 1097.07,
+      "duration": 0.0,
+      "text": "so essentially, what we obtain is a"
+    },
+    {
+      "start": 1097.08,
+      "duration": 0.0,
+      "text": "so essentially, what we obtain is a perturbed<00:18:18.000><c> version</c><00:18:18.640><c> of</c><00:18:18.880><c> this</c><00:18:19.720><c> Langevin</c>"
+    },
+    {
+      "start": 1100.15,
+      "duration": 0.0,
+      "text": "perturbed version of this Langevin"
+    },
+    {
+      "start": 1100.16,
+      "duration": 0.0,
+      "text": "perturbed version of this Langevin equation,<00:18:21.120><c> where</c><00:18:21.560><c> we</c><00:18:21.720><c> have</c><00:18:22.440><c> an</c><00:18:22.600><c> additional</c>"
+    },
+    {
+      "start": 1103.07,
+      "duration": 0.0,
+      "text": "equation, where we have an additional"
+    },
+    {
+      "start": 1103.08,
+      "duration": 0.0,
+      "text": "equation, where we have an additional term<00:18:24.080><c> in</c><00:18:24.240><c> the</c><00:18:24.360><c> drift</c><00:18:25.200><c> and</c><00:18:25.360><c> in</c><00:18:25.480><c> the</c><00:18:25.560><c> diffusion</c>"
+    },
+    {
+      "start": 1105.99,
+      "duration": 0.0,
+      "text": "term in the drift and in the diffusion"
+    },
+    {
+      "start": 1106.0,
+      "duration": 0.0,
+      "text": "term in the drift and in the diffusion coefficients,<00:18:27.440><c> which</c><00:18:27.600><c> is</c><00:18:27.720><c> given</c><00:18:28.159><c> by</c><00:18:28.480><c> the</c>"
+    },
+    {
+      "start": 1108.59,
+      "duration": 0.0,
+      "text": "coefficients, which is given by the"
+    },
+    {
+      "start": 1108.6,
+      "duration": 0.0,
+      "text": "coefficients, which is given by the perturbation<00:18:29.240><c> amplitude</c><00:18:29.720><c> of</c><00:18:29.840><c> the</c><00:18:29.919><c> parameter</c>"
+    },
+    {
+      "start": 1110.669,
+      "duration": 0.0,
+      "text": "perturbation amplitude of the parameter"
+    },
+    {
+      "start": 1110.679,
+      "duration": 0.0,
+      "text": "perturbation amplitude of the parameter multiplied<00:18:31.480><c> by</c><00:18:31.760><c> the</c><00:18:31.919><c> partial</c><00:18:32.280><c> derivatives</c><00:18:33.400><c> of</c>"
+    },
+    {
+      "start": 1114.07,
+      "duration": 0.0,
+      "text": "multiplied by the partial derivatives of"
+    },
+    {
+      "start": 1114.08,
+      "duration": 0.0,
+      "text": "multiplied by the partial derivatives of the<00:18:34.320><c> drift</c><00:18:35.120><c> and</c><00:18:35.360><c> the</c><00:18:35.440><c> diffusion</c><00:18:36.280><c> term</c><00:18:37.080><c> with</c>"
+    },
+    {
+      "start": 1117.31,
+      "duration": 0.0,
+      "text": "the drift and the diffusion term with"
+    },
+    {
+      "start": 1117.32,
+      "duration": 0.0,
+      "text": "the drift and the diffusion term with respect<00:18:38.000><c> to</c><00:18:38.120><c> that</c><00:18:38.360><c> specific</c><00:18:38.919><c> parameter.</c><00:18:39.919><c> So,</c>"
+    },
+    {
+      "start": 1120.03,
+      "duration": 0.0,
+      "text": "respect to that specific parameter. So,"
+    },
+    {
+      "start": 1120.04,
+      "duration": 0.0,
+      "text": "respect to that specific parameter. So, essentially,<00:18:40.679><c> it's</c><00:18:40.840><c> like</c><00:18:41.080><c> if</c><00:18:41.360><c> we</c><00:18:41.560><c> are</c>"
+    },
+    {
+      "start": 1121.71,
+      "duration": 0.0,
+      "text": "essentially, it's like if we are"
+    },
+    {
+      "start": 1121.72,
+      "duration": 0.0,
+      "text": "essentially, it's like if we are studying<00:18:42.440><c> a</c><00:18:42.560><c> perturbed</c><00:18:43.280><c> version</c>"
+    },
+    {
+      "start": 1124.51,
+      "duration": 0.0,
+      "text": "studying a perturbed version"
+    },
+    {
+      "start": 1124.52,
+      "duration": 0.0,
+      "text": "studying a perturbed version of<00:18:44.760><c> our</c><00:18:45.120><c> model,</c><00:18:45.800><c> and</c><00:18:45.960><c> we</c><00:18:46.080><c> would</c><00:18:46.280><c> like</c><00:18:46.760><c> to</c>"
+    },
+    {
+      "start": 1126.87,
+      "duration": 0.0,
+      "text": "of our model, and we would like to"
+    },
+    {
+      "start": 1126.88,
+      "duration": 0.0,
+      "text": "of our model, and we would like to predict<00:18:47.679><c> how</c><00:18:48.800><c> all</c><00:18:49.240><c> those</c><00:18:49.880><c> statistical</c>"
+    },
+    {
+      "start": 1130.35,
+      "duration": 0.0,
+      "text": "predict how all those statistical"
+    },
+    {
+      "start": 1130.36,
+      "duration": 0.0,
+      "text": "predict how all those statistical observables<00:18:51.280><c> will</c><00:18:51.520><c> change</c><00:18:52.520><c> after</c><00:18:53.520><c> we</c><00:18:53.800><c> add</c>"
+    },
+    {
+      "start": 1134.19,
+      "duration": 0.0,
+      "text": "observables will change after we add"
+    },
+    {
+      "start": 1134.2,
+      "duration": 0.0,
+      "text": "observables will change after we add this<00:18:54.520><c> perturbation.</c>"
+    },
+    {
+      "start": 1136.83,
+      "duration": 0.0,
+      "text": "this perturbation."
+    },
+    {
+      "start": 1136.84,
+      "duration": 0.0,
+      "text": "this perturbation. Okay.<00:18:57.240><c> So,</c><00:18:57.400><c> now</c><00:18:57.640><c> we</c><00:18:57.800><c> have</c><00:18:58.600><c> this</c>"
+    },
+    {
+      "start": 1140.07,
+      "duration": 0.0,
+      "text": "Okay. So, now we have this"
+    },
+    {
+      "start": 1140.08,
+      "duration": 0.0,
+      "text": "Okay. So, now we have this perturbation<00:19:00.720><c> problem,</c>"
+    },
+    {
+      "start": 1142.43,
+      "duration": 0.0,
+      "text": "perturbation problem,"
+    },
+    {
+      "start": 1142.44,
+      "duration": 0.0,
+      "text": "perturbation problem, which<00:19:02.679><c> can</c><00:19:03.000><c> be</c><00:19:03.280><c> addressed</c><00:19:04.240><c> using</c><00:19:05.000><c> a</c><00:19:05.080><c> very</c>"
+    },
+    {
+      "start": 1145.35,
+      "duration": 0.0,
+      "text": "which can be addressed using a very"
+    },
+    {
+      "start": 1145.36,
+      "duration": 0.0,
+      "text": "which can be addressed using a very powerful"
+    },
+    {
+      "start": 1146.71,
+      "duration": 0.0,
+      "text": "powerful"
+    },
+    {
+      "start": 1146.72,
+      "duration": 0.0,
+      "text": "powerful tool<00:19:07.240><c> from</c>"
+    },
+    {
+      "start": 1148.63,
+      "duration": 0.0,
+      "text": "tool from"
+    },
+    {
+      "start": 1148.64,
+      "duration": 0.0,
+      "text": "tool from statistical<00:19:09.200><c> physics,</c>"
+    },
+    {
+      "start": 1150.87,
+      "duration": 0.0,
+      "text": "statistical physics,"
+    },
+    {
+      "start": 1150.88,
+      "duration": 0.0,
+      "text": "statistical physics, which<00:19:11.200><c> is</c><00:19:11.800><c> the</c><00:19:12.640><c> generalized</c>"
+    },
+    {
+      "start": 1153.71,
+      "duration": 0.0,
+      "text": "which is the generalized"
+    },
+    {
+      "start": 1153.72,
+      "duration": 0.0,
+      "text": "which is the generalized fluctuation-dissipation<00:19:15.159><c> theorem</c><00:19:15.760><c> or</c><00:19:15.960><c> GFT.</c>"
+    },
+    {
+      "start": 1157.47,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem or GFT."
+    },
+    {
+      "start": 1157.48,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem or GFT. So,<00:19:17.640><c> GFT</c>"
+    },
+    {
+      "start": 1159.63,
+      "duration": 0.0,
+      "text": "So, GFT"
+    },
+    {
+      "start": 1159.64,
+      "duration": 0.0,
+      "text": "So, GFT is"
+    },
+    {
+      "start": 1162.159,
+      "duration": 0.0,
+      "text": "a<00:19:22.200><c> mathematical</c><00:19:22.760><c> machinery</c><00:19:23.400><c> that</c><00:19:23.840><c> allows</c><00:19:24.600><c> to</c>"
+    },
+    {
+      "start": 1164.75,
+      "duration": 0.0,
+      "text": "a mathematical machinery that allows to"
+    },
+    {
+      "start": 1164.76,
+      "duration": 0.0,
+      "text": "a mathematical machinery that allows to predict<00:19:25.720><c> how</c><00:19:26.320><c> a</c><00:19:26.400><c> dynamical</c><00:19:26.919><c> system</c><00:19:27.600><c> respond</c>"
+    },
+    {
+      "start": 1168.43,
+      "duration": 0.0,
+      "text": "predict how a dynamical system respond"
+    },
+    {
+      "start": 1168.44,
+      "duration": 0.0,
+      "text": "predict how a dynamical system respond to<00:19:28.919><c> a</c><00:19:29.000><c> perturbation</c><00:19:30.240><c> without</c><00:19:30.840><c> actually</c>"
+    },
+    {
+      "start": 1171.27,
+      "duration": 0.0,
+      "text": "to a perturbation without actually"
+    },
+    {
+      "start": 1171.28,
+      "duration": 0.0,
+      "text": "to a perturbation without actually perturbing<00:19:31.760><c> it,</c><00:19:32.120><c> just</c><00:19:32.800><c> from</c><00:19:33.000><c> the</c><00:19:33.120><c> knowledge</c>"
+    },
+    {
+      "start": 1173.99,
+      "duration": 0.0,
+      "text": "perturbing it, just from the knowledge"
+    },
+    {
+      "start": 1174.0,
+      "duration": 0.0,
+      "text": "perturbing it, just from the knowledge of<00:19:34.360><c> its</c><00:19:34.880><c> statistics.</c>"
+    },
+    {
+      "start": 1176.19,
+      "duration": 0.0,
+      "text": "of its statistics."
+    },
+    {
+      "start": 1176.2,
+      "duration": 0.0,
+      "text": "of its statistics. So,<00:19:36.280><c> without</c><00:19:36.679><c> entering</c><00:19:37.560><c> into</c><00:19:37.840><c> the</c>"
+    },
+    {
+      "start": 1177.95,
+      "duration": 0.0,
+      "text": "So, without entering into the"
+    },
+    {
+      "start": 1177.96,
+      "duration": 0.0,
+      "text": "So, without entering into the mathematical<00:19:38.560><c> details</c><00:19:39.560><c> of</c><00:19:39.720><c> this</c><00:19:39.880><c> problem,</c><00:19:41.000><c> we</c>"
+    },
+    {
+      "start": 1181.11,
+      "duration": 0.0,
+      "text": "mathematical details of this problem, we"
+    },
+    {
+      "start": 1181.12,
+      "duration": 0.0,
+      "text": "mathematical details of this problem, we can<00:19:41.840><c> essentially</c><00:19:42.400><c> write</c><00:19:43.240><c> the</c><00:19:43.480><c> response</c><00:19:44.400><c> of</c><00:19:45.000><c> an</c>"
+    },
+    {
+      "start": 1185.149,
+      "duration": 0.0,
+      "text": "can essentially write the response of an"
+    },
+    {
+      "start": 1185.159,
+      "duration": 0.0,
+      "text": "can essentially write the response of an observable<00:19:46.080><c> A</c>"
+    },
+    {
+      "start": 1187.11,
+      "duration": 0.0,
+      "text": "observable A"
+    },
+    {
+      "start": 1187.12,
+      "duration": 0.0,
+      "text": "observable A to<00:19:47.400><c> a</c><00:19:47.480><c> perturbation</c><00:19:48.840><c> given</c><00:19:49.840><c> by</c>"
+    },
+    {
+      "start": 1191.149,
+      "duration": 0.0,
+      "text": "to a perturbation given by"
+    },
+    {
+      "start": 1191.159,
+      "duration": 0.0,
+      "text": "to a perturbation given by uh<00:19:51.760><c> U</c><00:19:52.080><c> of</c><00:19:52.600><c> so</c><00:19:52.840><c> a</c><00:19:52.920><c> drift</c><00:19:53.400><c> perturbation</c><00:19:54.280><c> given</c><00:19:54.720><c> by</c>"
+    },
+    {
+      "start": 1194.99,
+      "duration": 0.0,
+      "text": "uh U of so a drift perturbation given by"
+    },
+    {
+      "start": 1195.0,
+      "duration": 0.0,
+      "text": "uh U of so a drift perturbation given by U<00:19:55.200><c> of</c><00:19:55.320><c> X</c><00:19:56.080><c> and</c><00:19:56.400><c> a</c><00:19:56.480><c> diffusion</c><00:19:57.000><c> perturbation</c>"
+    },
+    {
+      "start": 1197.79,
+      "duration": 0.0,
+      "text": "U of X and a diffusion perturbation"
+    },
+    {
+      "start": 1197.8,
+      "duration": 0.0,
+      "text": "U of X and a diffusion perturbation given<00:19:58.200><c> by</c><00:19:58.520><c> V</c><00:19:59.120><c> of</c><00:19:59.280><c> X.</c>"
+    },
+    {
+      "start": 1200.11,
+      "duration": 0.0,
+      "text": "given by V of X."
+    },
+    {
+      "start": 1200.12,
+      "duration": 0.0,
+      "text": "given by V of X. So,<00:20:00.240><c> those</c><00:20:00.760><c> are</c><00:20:00.920><c> nothing</c><00:20:01.520><c> but</c>"
+    },
+    {
+      "start": 1202.87,
+      "duration": 0.0,
+      "text": "So, those are nothing but"
+    },
+    {
+      "start": 1202.88,
+      "duration": 0.0,
+      "text": "So, those are nothing but these<00:20:03.680><c> perturbations</c>"
+    },
+    {
+      "start": 1205.27,
+      "duration": 0.0,
+      "text": "these perturbations"
+    },
+    {
+      "start": 1205.28,
+      "duration": 0.0,
+      "text": "these perturbations that<00:20:05.520><c> we</c><00:20:06.040><c> defined</c><00:20:06.680><c> before.</c>"
+    },
+    {
+      "start": 1211.16,
+      "duration": 0.0,
+      "text": "And<00:20:11.320><c> we</c><00:20:11.400><c> can</c><00:20:11.600><c> write</c><00:20:12.080><c> the</c><00:20:12.920><c> response</c><00:20:14.320><c> of</c><00:20:14.720><c> this</c>"
+    },
+    {
+      "start": 1214.95,
+      "duration": 0.0,
+      "text": "And we can write the response of this"
+    },
+    {
+      "start": 1214.96,
+      "duration": 0.0,
+      "text": "And we can write the response of this observable<00:20:15.680><c> A</c><00:20:16.160><c> in</c><00:20:16.360><c> terms</c><00:20:16.920><c> of</c><00:20:17.160><c> an</c><00:20:17.240><c> integral</c><00:20:18.160><c> of</c>"
+    },
+    {
+      "start": 1218.39,
+      "duration": 0.0,
+      "text": "observable A in terms of an integral of"
+    },
+    {
+      "start": 1218.4,
+      "duration": 0.0,
+      "text": "observable A in terms of an integral of the<00:20:18.520><c> correlation</c><00:20:19.320><c> between</c><00:20:20.040><c> the</c><00:20:20.160><c> observable</c>"
+    },
+    {
+      "start": 1220.75,
+      "duration": 0.0,
+      "text": "the correlation between the observable"
+    },
+    {
+      "start": 1220.76,
+      "duration": 0.0,
+      "text": "the correlation between the observable itself<00:20:22.000><c> and</c><00:20:22.400><c> this</c><00:20:22.640><c> conjugate</c><00:20:23.240><c> variable</c><00:20:23.800><c> B,</c>"
+    },
+    {
+      "start": 1224.51,
+      "duration": 0.0,
+      "text": "itself and this conjugate variable B,"
+    },
+    {
+      "start": 1224.52,
+      "duration": 0.0,
+      "text": "itself and this conjugate variable B, which<00:20:24.800><c> depends</c><00:20:25.480><c> on</c><00:20:26.280><c> the</c>"
+    },
+    {
+      "start": 1227.39,
+      "duration": 0.0,
+      "text": "which depends on the"
+    },
+    {
+      "start": 1227.4,
+      "duration": 0.0,
+      "text": "which depends on the perturbation<00:20:28.520><c> that</c><00:20:28.720><c> we</c><00:20:28.880><c> are</c><00:20:29.440><c> applying.</c><00:20:30.200><c> So,</c><00:20:30.480><c> U</c>"
+    },
+    {
+      "start": 1230.67,
+      "duration": 0.0,
+      "text": "perturbation that we are applying. So, U"
+    },
+    {
+      "start": 1230.68,
+      "duration": 0.0,
+      "text": "perturbation that we are applying. So, U of<00:20:30.840><c> X</c>"
+    },
+    {
+      "start": 1231.75,
+      "duration": 0.0,
+      "text": "of X"
+    },
+    {
+      "start": 1231.76,
+      "duration": 0.0,
+      "text": "of X and<00:20:32.200><c> V</c><00:20:32.520><c> of</c><00:20:32.760><c> X.</c>"
+    },
+    {
+      "start": 1233.83,
+      "duration": 0.0,
+      "text": "and V of X."
+    },
+    {
+      "start": 1233.84,
+      "duration": 0.0,
+      "text": "and V of X. And<00:20:34.440><c> importantly</c><00:20:35.560><c> on</c><00:20:36.000><c> the</c><00:20:36.120><c> score</c><00:20:36.480><c> function</c><00:20:37.320><c> S.</c>"
+    },
+    {
+      "start": 1239.19,
+      "duration": 0.0,
+      "text": "And importantly on the score function S."
+    },
+    {
+      "start": 1239.2,
+      "duration": 0.0,
+      "text": "And importantly on the score function S. So,<00:20:39.320><c> essentially</c><00:20:40.080><c> if</c><00:20:40.960><c> we</c><00:20:41.160><c> have</c><00:20:41.640><c> the</c><00:20:41.760><c> knowledge</c>"
+    },
+    {
+      "start": 1243.11,
+      "duration": 0.0,
+      "text": "So, essentially if we have the knowledge"
+    },
+    {
+      "start": 1243.12,
+      "duration": 0.0,
+      "text": "So, essentially if we have the knowledge of<00:20:44.320><c> the</c><00:20:44.480><c> score</c><00:20:44.760><c> function</c><00:20:45.560><c> since</c>"
+    },
+    {
+      "start": 1246.87,
+      "duration": 0.0,
+      "text": "of the score function since"
+    },
+    {
+      "start": 1246.88,
+      "duration": 0.0,
+      "text": "of the score function since we<00:20:47.080><c> know</c><00:20:47.360><c> the</c><00:20:47.720><c> analytic</c><00:20:48.520><c> expression</c><00:20:49.440><c> for</c><00:20:49.680><c> our</c>"
+    },
+    {
+      "start": 1249.87,
+      "duration": 0.0,
+      "text": "we know the analytic expression for our"
+    },
+    {
+      "start": 1249.88,
+      "duration": 0.0,
+      "text": "we know the analytic expression for our model.<00:20:50.920><c> And</c><00:20:51.120><c> also</c><00:20:51.400><c> we</c><00:20:51.520><c> know</c><00:20:51.800><c> the</c><00:20:51.920><c> analytic</c>"
+    },
+    {
+      "start": 1252.35,
+      "duration": 0.0,
+      "text": "model. And also we know the analytic"
+    },
+    {
+      "start": 1252.36,
+      "duration": 0.0,
+      "text": "model. And also we know the analytic expression<00:20:53.360><c> for</c><00:20:53.520><c> the</c><00:20:53.640><c> perturbation</c><00:20:54.360><c> that</c><00:20:54.520><c> we</c>"
+    },
+    {
+      "start": 1254.59,
+      "duration": 0.0,
+      "text": "expression for the perturbation that we"
+    },
+    {
+      "start": 1254.6,
+      "duration": 0.0,
+      "text": "expression for the perturbation that we are<00:20:54.720><c> applying</c><00:20:55.360><c> U</c><00:20:55.560><c> and</c><00:20:55.760><c> V.</c>"
+    },
+    {
+      "start": 1256.79,
+      "duration": 0.0,
+      "text": "are applying U and V."
+    },
+    {
+      "start": 1256.8,
+      "duration": 0.0,
+      "text": "are applying U and V. We<00:20:57.000><c> can</c><00:20:57.480><c> predict</c><00:20:58.400><c> how</c><00:20:59.360><c> our</c><00:20:59.640><c> system,</c><00:21:00.520><c> so</c><00:21:00.760><c> in</c>"
+    },
+    {
+      "start": 1261.07,
+      "duration": 0.0,
+      "text": "We can predict how our system, so in"
+    },
+    {
+      "start": 1261.08,
+      "duration": 0.0,
+      "text": "We can predict how our system, so in this<00:21:01.280><c> case</c><00:21:01.880><c> how</c><00:21:02.680><c> the</c><00:21:02.800><c> observable</c><00:21:03.520><c> A</c><00:21:04.360><c> will</c>"
+    },
+    {
+      "start": 1264.59,
+      "duration": 0.0,
+      "text": "this case how the observable A will"
+    },
+    {
+      "start": 1264.6,
+      "duration": 0.0,
+      "text": "this case how the observable A will respond<00:21:05.560><c> to</c><00:21:05.720><c> an</c><00:21:05.840><c> external</c><00:21:06.360><c> perturbation.</c>"
+    },
+    {
+      "start": 1267.99,
+      "duration": 0.0,
+      "text": "respond to an external perturbation."
+    },
+    {
+      "start": 1268.0,
+      "duration": 0.0,
+      "text": "respond to an external perturbation. So,<00:21:08.200><c> if</c><00:21:08.400><c> we</c><00:21:08.560><c> identify</c><00:21:10.000><c> the</c><00:21:10.120><c> observable</c><00:21:10.880><c> A</c><00:21:11.560><c> with</c>"
+    },
+    {
+      "start": 1271.75,
+      "duration": 0.0,
+      "text": "So, if we identify the observable A with"
+    },
+    {
+      "start": 1271.76,
+      "duration": 0.0,
+      "text": "So, if we identify the observable A with the<00:21:11.880><c> set</c><00:21:12.440><c> of</c><00:21:12.640><c> observables</c><00:21:13.600><c> that</c><00:21:13.760><c> we</c><00:21:13.880><c> want</c><00:21:14.240><c> our</c>"
+    },
+    {
+      "start": 1274.43,
+      "duration": 0.0,
+      "text": "the set of observables that we want our"
+    },
+    {
+      "start": 1274.44,
+      "duration": 0.0,
+      "text": "the set of observables that we want our model<00:21:14.800><c> to</c><00:21:14.920><c> reproduce.</c>"
+    },
+    {
+      "start": 1276.39,
+      "duration": 0.0,
+      "text": "model to reproduce."
+    },
+    {
+      "start": 1276.4,
+      "duration": 0.0,
+      "text": "model to reproduce. We<00:21:16.520><c> can</c><00:21:16.680><c> essentially</c><00:21:17.160><c> construct</c><00:21:18.280><c> using</c><00:21:19.160><c> the</c>"
+    },
+    {
+      "start": 1279.27,
+      "duration": 0.0,
+      "text": "We can essentially construct using the"
+    },
+    {
+      "start": 1279.28,
+      "duration": 0.0,
+      "text": "We can essentially construct using the generalized<00:21:19.960><c> fluctuation-dissipation</c>"
+    },
+    {
+      "start": 1281.15,
+      "duration": 0.0,
+      "text": "generalized fluctuation-dissipation"
+    },
+    {
+      "start": 1281.16,
+      "duration": 0.0,
+      "text": "generalized fluctuation-dissipation theorem<00:21:22.200><c> the</c><00:21:22.360><c> parameter</c><00:21:22.960><c> sensitivities</c><00:21:24.000><c> that</c>"
+    },
+    {
+      "start": 1284.11,
+      "duration": 0.0,
+      "text": "theorem the parameter sensitivities that"
+    },
+    {
+      "start": 1284.12,
+      "duration": 0.0,
+      "text": "theorem the parameter sensitivities that we<00:21:24.200><c> are</c><00:21:24.320><c> interested</c><00:21:24.840><c> in.</c>"
+    },
+    {
+      "start": 1286.03,
+      "duration": 0.0,
+      "text": "we are interested in."
+    },
+    {
+      "start": 1286.04,
+      "duration": 0.0,
+      "text": "we are interested in. Just<00:21:26.680><c> from</c><00:21:26.960><c> the</c><00:21:27.080><c> knowledge</c><00:21:27.600><c> of</c><00:21:27.720><c> the</c><00:21:27.800><c> score</c>"
+    },
+    {
+      "start": 1288.03,
+      "duration": 0.0,
+      "text": "Just from the knowledge of the score"
+    },
+    {
+      "start": 1288.04,
+      "duration": 0.0,
+      "text": "Just from the knowledge of the score function<00:21:28.480><c> itself.</c>"
+    },
+    {
+      "start": 1291.4,
+      "duration": 0.0,
+      "text": "So,<00:21:31.560><c> this</c><00:21:31.760><c> is</c><00:21:31.880><c> the</c><00:21:32.000><c> main</c><00:21:32.920><c> connection.</c><00:21:33.760><c> So,</c>"
+    },
+    {
+      "start": 1293.99,
+      "duration": 0.0,
+      "text": "So, this is the main connection. So,"
+    },
+    {
+      "start": 1294.0,
+      "duration": 0.0,
+      "text": "So, this is the main connection. So, using<00:21:34.560><c> GFDT</c><00:21:35.480><c> we</c><00:21:35.640><c> are</c><00:21:35.800><c> able</c><00:21:36.240><c> to</c><00:21:36.360><c> estimate</c><00:21:37.000><c> the</c>"
+    },
+    {
+      "start": 1297.15,
+      "duration": 0.0,
+      "text": "using GFDT we are able to estimate the"
+    },
+    {
+      "start": 1297.16,
+      "duration": 0.0,
+      "text": "using GFDT we are able to estimate the parameter<00:21:37.680><c> sensitivities</c>"
+    },
+    {
+      "start": 1299.95,
+      "duration": 0.0,
+      "text": "parameter sensitivities"
+    },
+    {
+      "start": 1299.96,
+      "duration": 0.0,
+      "text": "parameter sensitivities without<00:21:40.640><c> the</c><00:21:40.720><c> running</c><00:21:41.680><c> the</c><00:21:41.800><c> model</c><00:21:42.160><c> forward</c>"
+    },
+    {
+      "start": 1302.95,
+      "duration": 0.0,
+      "text": "without the running the model forward"
+    },
+    {
+      "start": 1302.96,
+      "duration": 0.0,
+      "text": "without the running the model forward for<00:21:43.200><c> every</c><00:21:43.760><c> parameter.</c>"
+    },
+    {
+      "start": 1305.07,
+      "duration": 0.0,
+      "text": "for every parameter."
+    },
+    {
+      "start": 1305.08,
+      "duration": 0.0,
+      "text": "for every parameter. We<00:21:45.240><c> just</c><00:21:45.560><c> needed</c><00:21:46.000><c> to</c><00:21:46.120><c> run</c><00:21:46.520><c> the</c><00:21:46.680><c> model</c><00:21:47.040><c> forward</c>"
+    },
+    {
+      "start": 1307.59,
+      "duration": 0.0,
+      "text": "We just needed to run the model forward"
+    },
+    {
+      "start": 1307.6,
+      "duration": 0.0,
+      "text": "We just needed to run the model forward one<00:21:47.880><c> time</c><00:21:48.640><c> to</c><00:21:49.520><c> estimate</c><00:21:50.280><c> the</c><00:21:50.400><c> score</c><00:21:50.680><c> function.</c>"
+    },
+    {
+      "start": 1311.75,
+      "duration": 0.0,
+      "text": "one time to estimate the score function."
+    },
+    {
+      "start": 1311.76,
+      "duration": 0.0,
+      "text": "one time to estimate the score function. And<00:21:52.000><c> from</c><00:21:52.240><c> this</c><00:21:52.520><c> single</c><00:21:52.960><c> model</c><00:21:53.240><c> integration</c>"
+    },
+    {
+      "start": 1314.03,
+      "duration": 0.0,
+      "text": "And from this single model integration"
+    },
+    {
+      "start": 1314.04,
+      "duration": 0.0,
+      "text": "And from this single model integration we<00:21:54.160><c> are</c><00:21:54.360><c> able</c><00:21:54.880><c> to</c><00:21:55.000><c> estimate</c><00:21:55.720><c> all</c><00:21:55.920><c> the</c>"
+    },
+    {
+      "start": 1316.03,
+      "duration": 0.0,
+      "text": "we are able to estimate all the"
+    },
+    {
+      "start": 1316.04,
+      "duration": 0.0,
+      "text": "we are able to estimate all the parameter<00:21:56.560><c> sensitivities</c><00:21:57.680><c> that</c><00:21:57.840><c> we</c><00:21:57.960><c> can</c><00:21:58.120><c> use</c>"
+    },
+    {
+      "start": 1318.39,
+      "duration": 0.0,
+      "text": "parameter sensitivities that we can use"
+    },
+    {
+      "start": 1318.4,
+      "duration": 0.0,
+      "text": "parameter sensitivities that we can use for<00:21:58.560><c> calibration.</c>"
+    },
+    {
+      "start": 1320.91,
+      "duration": 0.0,
+      "text": "for calibration."
+    },
+    {
+      "start": 1320.92,
+      "duration": 0.0,
+      "text": "for calibration. Okay,<00:22:01.160><c> so</c><00:22:01.280><c> as</c><00:22:01.400><c> we</c><00:22:01.480><c> have</c><00:22:01.600><c> seen</c><00:22:02.120><c> this</c><00:22:02.400><c> entirely</c>"
+    },
+    {
+      "start": 1323.47,
+      "duration": 0.0,
+      "text": "Okay, so as we have seen this entirely"
+    },
+    {
+      "start": 1323.48,
+      "duration": 0.0,
+      "text": "Okay, so as we have seen this entirely so<00:22:03.640><c> this</c><00:22:04.480><c> so</c><00:22:04.640><c> the</c><00:22:04.880><c> applicability</c><00:22:05.640><c> of</c><00:22:05.760><c> the</c>"
+    },
+    {
+      "start": 1325.83,
+      "duration": 0.0,
+      "text": "so this so the applicability of the"
+    },
+    {
+      "start": 1325.84,
+      "duration": 0.0,
+      "text": "so this so the applicability of the generalized<00:22:06.560><c> fluctuation-dissipation</c>"
+    },
+    {
+      "start": 1327.55,
+      "duration": 0.0,
+      "text": "generalized fluctuation-dissipation"
+    },
+    {
+      "start": 1327.56,
+      "duration": 0.0,
+      "text": "generalized fluctuation-dissipation theorem<00:22:08.000><c> to</c><00:22:08.120><c> this</c><00:22:08.240><c> specific</c><00:22:08.720><c> problem</c><00:22:09.679><c> relies</c>"
+    },
+    {
+      "start": 1330.71,
+      "duration": 0.0,
+      "text": "theorem to this specific problem relies"
+    },
+    {
+      "start": 1330.72,
+      "duration": 0.0,
+      "text": "theorem to this specific problem relies only<00:22:11.280><c> on</c><00:22:11.800><c> the</c><00:22:12.080><c> estimation</c><00:22:12.720><c> of</c><00:22:12.800><c> the</c><00:22:12.920><c> score</c>"
+    },
+    {
+      "start": 1333.15,
+      "duration": 0.0,
+      "text": "only on the estimation of the score"
+    },
+    {
+      "start": 1333.16,
+      "duration": 0.0,
+      "text": "only on the estimation of the score function."
+    },
+    {
+      "start": 1334.83,
+      "duration": 0.0,
+      "text": "function."
+    },
+    {
+      "start": 1334.84,
+      "duration": 0.0,
+      "text": "function. So,<00:22:15.400><c> at</c><00:22:15.480><c> this</c><00:22:15.600><c> point</c><00:22:16.120><c> you</c><00:22:16.240><c> may</c><00:22:16.400><c> ask</c><00:22:16.640><c> okay,</c><00:22:16.840><c> but</c>"
+    },
+    {
+      "start": 1337.19,
+      "duration": 0.0,
+      "text": "So, at this point you may ask okay, but"
+    },
+    {
+      "start": 1337.2,
+      "duration": 0.0,
+      "text": "So, at this point you may ask okay, but why<00:22:17.520><c> nobody</c><00:22:18.200><c> like</c><00:22:19.040><c> followed</c><00:22:19.640><c> this</c><00:22:19.960><c> direction?</c>"
+    },
+    {
+      "start": 1341.31,
+      "duration": 0.0,
+      "text": "why nobody like followed this direction?"
+    },
+    {
+      "start": 1341.32,
+      "duration": 0.0,
+      "text": "why nobody like followed this direction? And<00:22:21.480><c> the</c><00:22:21.560><c> reason</c><00:22:22.280><c> is</c><00:22:22.520><c> that</c><00:22:23.320><c> estimating</c><00:22:23.920><c> the</c>"
+    },
+    {
+      "start": 1344.03,
+      "duration": 0.0,
+      "text": "And the reason is that estimating the"
+    },
+    {
+      "start": 1344.04,
+      "duration": 0.0,
+      "text": "And the reason is that estimating the score<00:22:24.320><c> function</c><00:22:24.880><c> for</c><00:22:25.040><c> high-dimensional</c>"
+    },
+    {
+      "start": 1345.83,
+      "duration": 0.0,
+      "text": "score function for high-dimensional"
+    },
+    {
+      "start": 1345.84,
+      "duration": 0.0,
+      "text": "score function for high-dimensional system<00:22:26.679><c> has</c><00:22:27.040><c> always</c><00:22:27.520><c> been</c><00:22:28.360><c> the</c><00:22:28.480><c> main</c>"
+    },
+    {
+      "start": 1348.83,
+      "duration": 0.0,
+      "text": "system has always been the main"
+    },
+    {
+      "start": 1348.84,
+      "duration": 0.0,
+      "text": "system has always been the main bottleneck<00:22:30.200><c> to</c><00:22:30.679><c> use</c><00:22:31.200><c> this</c><00:22:31.560><c> generalized</c>"
+    },
+    {
+      "start": 1352.23,
+      "duration": 0.0,
+      "text": "bottleneck to use this generalized"
+    },
+    {
+      "start": 1352.24,
+      "duration": 0.0,
+      "text": "bottleneck to use this generalized fluctuation-dissipation<00:22:33.720><c> theorem</c><00:22:34.480><c> inside</c>"
+    },
+    {
+      "start": 1355.59,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem inside"
+    },
+    {
+      "start": 1355.6,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem inside realistic<00:22:36.400><c> and</c><00:22:36.600><c> practical</c><00:22:37.120><c> problems.</c>"
+    },
+    {
+      "start": 1358.63,
+      "duration": 0.0,
+      "text": "realistic and practical problems."
+    },
+    {
+      "start": 1358.64,
+      "duration": 0.0,
+      "text": "realistic and practical problems. So,<00:22:38.760><c> usually</c><00:22:39.240><c> people</c><00:22:39.920><c> in</c><00:22:40.080><c> the</c><00:22:40.200><c> literature</c>"
+    },
+    {
+      "start": 1361.39,
+      "duration": 0.0,
+      "text": "So, usually people in the literature"
+    },
+    {
+      "start": 1361.4,
+      "duration": 0.0,
+      "text": "So, usually people in the literature have<00:22:41.600><c> always</c><00:22:41.880><c> struggled</c><00:22:42.520><c> in</c><00:22:42.920><c> estimating</c><00:22:43.520><c> the</c>"
+    },
+    {
+      "start": 1363.63,
+      "duration": 0.0,
+      "text": "have always struggled in estimating the"
+    },
+    {
+      "start": 1363.64,
+      "duration": 0.0,
+      "text": "have always struggled in estimating the score<00:22:43.880><c> function.</c><00:22:45.080><c> And</c><00:22:45.600><c> so,</c><00:22:45.720><c> it</c><00:22:45.840><c> was</c><00:22:46.000><c> possible</c>"
+    },
+    {
+      "start": 1366.669,
+      "duration": 0.0,
+      "text": "score function. And so, it was possible"
+    },
+    {
+      "start": 1366.679,
+      "duration": 0.0,
+      "text": "score function. And so, it was possible for<00:22:46.840><c> low-dimensional</c><00:22:47.480><c> systems.</c><00:22:48.560><c> Instead</c><00:22:49.080><c> for</c>"
+    },
+    {
+      "start": 1369.27,
+      "duration": 0.0,
+      "text": "for low-dimensional systems. Instead for"
+    },
+    {
+      "start": 1369.28,
+      "duration": 0.0,
+      "text": "for low-dimensional systems. Instead for high-dimensional<00:22:50.000><c> systems,</c>"
+    },
+    {
+      "start": 1371.39,
+      "duration": 0.0,
+      "text": "high-dimensional systems,"
+    },
+    {
+      "start": 1371.4,
+      "duration": 0.0,
+      "text": "high-dimensional systems, what<00:22:51.600><c> has</c><00:22:51.720><c> been</c><00:22:52.000><c> usually</c><00:22:52.320><c> done</c><00:22:52.640><c> in</c><00:22:52.800><c> the</c>"
+    },
+    {
+      "start": 1372.91,
+      "duration": 0.0,
+      "text": "what has been usually done in the"
+    },
+    {
+      "start": 1372.92,
+      "duration": 0.0,
+      "text": "what has been usually done in the literature<00:22:53.720><c> was</c><00:22:53.960><c> to</c><00:22:54.080><c> use</c><00:22:54.560><c> the</c><00:22:54.960><c> so-called</c>"
+    },
+    {
+      "start": 1375.59,
+      "duration": 0.0,
+      "text": "literature was to use the so-called"
+    },
+    {
+      "start": 1375.6,
+      "duration": 0.0,
+      "text": "literature was to use the so-called quasi-Gaussian<00:22:56.240><c> approximations,</c>"
+    },
+    {
+      "start": 1377.71,
+      "duration": 0.0,
+      "text": "quasi-Gaussian approximations,"
+    },
+    {
+      "start": 1377.72,
+      "duration": 0.0,
+      "text": "quasi-Gaussian approximations, which<00:22:58.040><c> consists</c><00:22:58.800><c> in</c><00:22:59.080><c> approximating</c><00:23:00.000><c> the</c>"
+    },
+    {
+      "start": 1380.11,
+      "duration": 0.0,
+      "text": "which consists in approximating the"
+    },
+    {
+      "start": 1380.12,
+      "duration": 0.0,
+      "text": "which consists in approximating the steady<00:23:00.400><c> state</c><00:23:00.800><c> distribution</c><00:23:01.679><c> with</c><00:23:01.920><c> a</c>"
+    },
+    {
+      "start": 1381.95,
+      "duration": 0.0,
+      "text": "steady state distribution with a"
+    },
+    {
+      "start": 1381.96,
+      "duration": 0.0,
+      "text": "steady state distribution with a multivariate<00:23:02.480><c> Gaussian,</c><00:23:03.440><c> which</c><00:23:03.720><c> allows</c><00:23:04.640><c> to</c>"
+    },
+    {
+      "start": 1384.83,
+      "duration": 0.0,
+      "text": "multivariate Gaussian, which allows to"
+    },
+    {
+      "start": 1384.84,
+      "duration": 0.0,
+      "text": "multivariate Gaussian, which allows to write<00:23:05.360><c> the</c><00:23:05.440><c> score</c><00:23:05.720><c> function</c><00:23:06.280><c> in</c><00:23:06.400><c> terms</c><00:23:06.800><c> of</c><00:23:06.960><c> a</c>"
+    },
+    {
+      "start": 1386.99,
+      "duration": 0.0,
+      "text": "write the score function in terms of a"
+    },
+    {
+      "start": 1387.0,
+      "duration": 0.0,
+      "text": "write the score function in terms of a linear<00:23:07.280><c> function</c><00:23:08.080><c> that</c><00:23:08.240><c> depends</c><00:23:08.640><c> on</c><00:23:08.720><c> the</c>"
+    },
+    {
+      "start": 1388.83,
+      "duration": 0.0,
+      "text": "linear function that depends on the"
+    },
+    {
+      "start": 1388.84,
+      "duration": 0.0,
+      "text": "linear function that depends on the covariance<00:23:09.360><c> matrix</c><00:23:09.800><c> of</c><00:23:09.960><c> the</c><00:23:10.040><c> data</c><00:23:10.280><c> set,</c><00:23:10.960><c> which</c>"
+    },
+    {
+      "start": 1391.15,
+      "duration": 0.0,
+      "text": "covariance matrix of the data set, which"
+    },
+    {
+      "start": 1391.16,
+      "duration": 0.0,
+      "text": "covariance matrix of the data set, which of<00:23:11.280><c> course</c><00:23:11.720><c> is</c><00:23:11.880><c> extremely</c><00:23:12.400><c> easy</c><00:23:12.920><c> to</c><00:23:13.040><c> be</c>"
+    },
+    {
+      "start": 1393.11,
+      "duration": 0.0,
+      "text": "of course is extremely easy to be"
+    },
+    {
+      "start": 1393.12,
+      "duration": 0.0,
+      "text": "of course is extremely easy to be estimated."
+    },
+    {
+      "start": 1394.59,
+      "duration": 0.0,
+      "text": "estimated."
+    },
+    {
+      "start": 1394.6,
+      "duration": 0.0,
+      "text": "estimated. But<00:23:14.720><c> the</c><00:23:14.800><c> main</c><00:23:15.000><c> problem</c><00:23:15.480><c> is</c><00:23:15.560><c> that</c>"
+    },
+    {
+      "start": 1396.63,
+      "duration": 0.0,
+      "text": "But the main problem is that"
+    },
+    {
+      "start": 1396.64,
+      "duration": 0.0,
+      "text": "But the main problem is that when<00:23:17.080><c> the</c><00:23:17.200><c> dynamics</c><00:23:17.880><c> is</c><00:23:18.120><c> highly</c><00:23:18.440><c> nonlinear,</c>"
+    },
+    {
+      "start": 1399.31,
+      "duration": 0.0,
+      "text": "when the dynamics is highly nonlinear,"
+    },
+    {
+      "start": 1399.32,
+      "duration": 0.0,
+      "text": "when the dynamics is highly nonlinear, so<00:23:19.440><c> when</c><00:23:19.640><c> the</c><00:23:20.000><c> steady</c><00:23:20.120><c> state</c><00:23:20.440><c> distribution</c>"
+    },
+    {
+      "start": 1401.11,
+      "duration": 0.0,
+      "text": "so when the steady state distribution"
+    },
+    {
+      "start": 1401.12,
+      "duration": 0.0,
+      "text": "so when the steady state distribution also<00:23:21.360><c> is</c><00:23:21.560><c> highly</c>"
+    },
+    {
+      "start": 1402.95,
+      "duration": 0.0,
+      "text": "also is highly"
+    },
+    {
+      "start": 1402.96,
+      "duration": 0.0,
+      "text": "also is highly non-Gaussian,"
+    },
+    {
+      "start": 1404.39,
+      "duration": 0.0,
+      "text": "non-Gaussian,"
+    },
+    {
+      "start": 1404.4,
+      "duration": 0.0,
+      "text": "non-Gaussian, then<00:23:24.920><c> this</c><00:23:25.200><c> approach</c><00:23:25.640><c> introduces</c><00:23:26.400><c> some</c>"
+    },
+    {
+      "start": 1406.63,
+      "duration": 0.0,
+      "text": "then this approach introduces some"
+    },
+    {
+      "start": 1406.64,
+      "duration": 0.0,
+      "text": "then this approach introduces some strong<00:23:26.960><c> biases,</c>"
+    },
+    {
+      "start": 1408.35,
+      "duration": 0.0,
+      "text": "strong biases,"
+    },
+    {
+      "start": 1408.36,
+      "duration": 0.0,
+      "text": "strong biases, which"
+    },
+    {
+      "start": 1410.31,
+      "duration": 0.0,
+      "text": "which"
+    },
+    {
+      "start": 1410.32,
+      "duration": 0.0,
+      "text": "which doesn't<00:23:30.679><c> allow</c><00:23:31.200><c> to</c><00:23:31.360><c> get</c><00:23:32.200><c> quite</c><00:23:32.520><c> precise</c>"
+    },
+    {
+      "start": 1413.99,
+      "duration": 0.0,
+      "text": "doesn't allow to get quite precise"
+    },
+    {
+      "start": 1414.0,
+      "duration": 0.0,
+      "text": "doesn't allow to get quite precise prediction<00:23:34.880><c> of</c><00:23:35.040><c> the</c><00:23:35.120><c> system</c><00:23:35.880><c> responses</c><00:23:36.880><c> using</c>"
+    },
+    {
+      "start": 1417.43,
+      "duration": 0.0,
+      "text": "prediction of the system responses using"
+    },
+    {
+      "start": 1417.44,
+      "duration": 0.0,
+      "text": "prediction of the system responses using GFDT."
+    },
+    {
+      "start": 1418.79,
+      "duration": 0.0,
+      "text": "GFDT."
+    },
+    {
+      "start": 1418.8,
+      "duration": 0.0,
+      "text": "GFDT. But<00:23:38.960><c> we</c><00:23:39.320><c> we</c><00:23:39.440><c> have</c><00:23:39.560><c> seen</c><00:23:39.840><c> before</c><00:23:40.600><c> how</c><00:23:41.280><c> yeah,</c>"
+    },
+    {
+      "start": 1421.51,
+      "duration": 0.0,
+      "text": "But we we have seen before how yeah,"
+    },
+    {
+      "start": 1421.52,
+      "duration": 0.0,
+      "text": "But we we have seen before how yeah, recent<00:23:42.160><c> advances</c><00:23:42.880><c> in</c><00:23:43.440><c> score</c><00:23:43.760><c> estimation</c>"
+    },
+    {
+      "start": 1424.43,
+      "duration": 0.0,
+      "text": "recent advances in score estimation"
+    },
+    {
+      "start": 1424.44,
+      "duration": 0.0,
+      "text": "recent advances in score estimation methods<00:23:45.080><c> using</c><00:23:45.520><c> a</c><00:23:45.600><c> neural</c><00:23:45.880><c> network</c>"
+    },
+    {
+      "start": 1427.39,
+      "duration": 0.0,
+      "text": "methods using a neural network"
+    },
+    {
+      "start": 1427.4,
+      "duration": 0.0,
+      "text": "methods using a neural network allow<00:23:47.760><c> us</c><00:23:48.040><c> to</c><00:23:48.200><c> get</c><00:23:48.480><c> a</c><00:23:48.560><c> very</c><00:23:48.800><c> precise</c><00:23:49.960><c> and</c>"
+    },
+    {
+      "start": 1431.71,
+      "duration": 0.0,
+      "text": "allow us to get a very precise and"
+    },
+    {
+      "start": 1431.72,
+      "duration": 0.0,
+      "text": "allow us to get a very precise and efficient<00:23:52.640><c> estimation</c><00:23:53.440><c> for</c><00:23:53.600><c> the</c><00:23:53.720><c> score</c>"
+    },
+    {
+      "start": 1433.99,
+      "duration": 0.0,
+      "text": "efficient estimation for the score"
+    },
+    {
+      "start": 1434.0,
+      "duration": 0.0,
+      "text": "efficient estimation for the score function<00:23:54.560><c> also</c><00:23:55.080><c> for</c><00:23:55.280><c> very</c><00:23:55.520><c> high-dimensional</c>"
+    },
+    {
+      "start": 1436.23,
+      "duration": 0.0,
+      "text": "function also for very high-dimensional"
+    },
+    {
+      "start": 1436.24,
+      "duration": 0.0,
+      "text": "function also for very high-dimensional systems."
+    },
+    {
+      "start": 1437.87,
+      "duration": 0.0,
+      "text": "systems."
+    },
+    {
+      "start": 1437.88,
+      "duration": 0.0,
+      "text": "systems. And<00:23:58.080><c> so,</c><00:23:58.240><c> this</c><00:23:58.480><c> knowledge</c><00:23:59.440><c> allow</c><00:23:59.840><c> us</c><00:24:00.600><c> to</c>"
+    },
+    {
+      "start": 1441.63,
+      "duration": 0.0,
+      "text": "And so, this knowledge allow us to"
+    },
+    {
+      "start": 1441.64,
+      "duration": 0.0,
+      "text": "And so, this knowledge allow us to construct<00:24:02.600><c> and</c><00:24:02.840><c> to</c><00:24:03.000><c> estimate</c><00:24:03.679><c> the</c><00:24:03.760><c> system</c>"
+    },
+    {
+      "start": 1444.11,
+      "duration": 0.0,
+      "text": "construct and to estimate the system"
+    },
+    {
+      "start": 1444.12,
+      "duration": 0.0,
+      "text": "construct and to estimate the system responses<00:24:04.880><c> using</c><00:24:05.880><c> the</c><00:24:06.000><c> generalized</c>"
+    },
+    {
+      "start": 1446.669,
+      "duration": 0.0,
+      "text": "responses using the generalized"
+    },
+    {
+      "start": 1446.679,
+      "duration": 0.0,
+      "text": "responses using the generalized fluctuation-dissipation<00:24:07.720><c> theorem.</c>"
+    },
+    {
+      "start": 1448.99,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem."
+    },
+    {
+      "start": 1449.0,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem. So,<00:24:09.120><c> we</c><00:24:09.280><c> applied</c><00:24:09.840><c> those</c><00:24:10.080><c> ideas</c><00:24:10.960><c> already</c>"
+    },
+    {
+      "start": 1452.35,
+      "duration": 0.0,
+      "text": "So, we applied those ideas already"
+    },
+    {
+      "start": 1452.36,
+      "duration": 0.0,
+      "text": "So, we applied those ideas already to<00:24:12.920><c> evaluate</c>"
+    },
+    {
+      "start": 1454.51,
+      "duration": 0.0,
+      "text": "to evaluate"
+    },
+    {
+      "start": 1454.52,
+      "duration": 0.0,
+      "text": "to evaluate and<00:24:14.679><c> to</c><00:24:14.800><c> predict</c><00:24:15.480><c> system</c><00:24:16.080><c> responses</c><00:24:17.200><c> for</c>"
+    },
+    {
+      "start": 1457.59,
+      "duration": 0.0,
+      "text": "and to predict system responses for"
+    },
+    {
+      "start": 1457.6,
+      "duration": 0.0,
+      "text": "and to predict system responses for quite<00:24:18.000><c> high-dimensional</c><00:24:18.720><c> systems.</c><00:24:19.480><c> We</c>"
+    },
+    {
+      "start": 1459.59,
+      "duration": 0.0,
+      "text": "quite high-dimensional systems. We"
+    },
+    {
+      "start": 1459.6,
+      "duration": 0.0,
+      "text": "quite high-dimensional systems. We started<00:24:20.360><c> PDEs</c><00:24:21.080><c> discretized</c><00:24:22.080><c> on</c><00:24:22.840><c> around</c><00:24:23.400><c> 10</c><00:24:23.600><c> at</c>"
+    },
+    {
+      "start": 1463.71,
+      "duration": 0.0,
+      "text": "started PDEs discretized on around 10 at"
+    },
+    {
+      "start": 1463.72,
+      "duration": 0.0,
+      "text": "started PDEs discretized on around 10 at the<00:24:23.840><c> third</c><00:24:24.640><c> grid</c><00:24:24.920><c> points.</c>"
+    },
+    {
+      "start": 1466.51,
+      "duration": 0.0,
+      "text": "the third grid points."
+    },
+    {
+      "start": 1466.52,
+      "duration": 0.0,
+      "text": "the third grid points. And<00:24:26.960><c> we</c><00:24:27.160><c> considered</c><00:24:28.120><c> more</c><00:24:28.280><c> specifically</c>"
+    },
+    {
+      "start": 1469.59,
+      "duration": 0.0,
+      "text": "And we considered more specifically"
+    },
+    {
+      "start": 1469.6,
+      "duration": 0.0,
+      "text": "And we considered more specifically two-dimensional<00:24:30.280><c> turbulent</c><00:24:30.880><c> data</c><00:24:31.720><c> and</c>"
+    },
+    {
+      "start": 1471.99,
+      "duration": 0.0,
+      "text": "two-dimensional turbulent data and"
+    },
+    {
+      "start": 1472.0,
+      "duration": 0.0,
+      "text": "two-dimensional turbulent data and Alan-Khan<00:24:32.640><c> reaction-diffusion</c><00:24:33.920><c> data.</c><00:24:34.640><c> So,</c>"
+    },
+    {
+      "start": 1474.79,
+      "duration": 0.0,
+      "text": "Alan-Khan reaction-diffusion data. So,"
+    },
+    {
+      "start": 1474.8,
+      "duration": 0.0,
+      "text": "Alan-Khan reaction-diffusion data. So, these<00:24:35.000><c> are</c><00:24:35.360><c> the</c><00:24:35.480><c> two</c><00:24:35.640><c> papers</c><00:24:36.160><c> where</c><00:24:36.320><c> we</c>"
+    },
+    {
+      "start": 1476.47,
+      "duration": 0.0,
+      "text": "these are the two papers where we"
+    },
+    {
+      "start": 1476.48,
+      "duration": 0.0,
+      "text": "these are the two papers where we published<00:24:37.040><c> this</c>"
+    },
+    {
+      "start": 1477.95,
+      "duration": 0.0,
+      "text": "published this"
+    },
+    {
+      "start": 1477.96,
+      "duration": 0.0,
+      "text": "published this connection<00:24:38.679><c> between</c><00:24:39.120><c> the</c><00:24:39.160><c> score-based</c>"
+    },
+    {
+      "start": 1479.79,
+      "duration": 0.0,
+      "text": "connection between the score-based"
+    },
+    {
+      "start": 1479.8,
+      "duration": 0.0,
+      "text": "connection between the score-based generative<00:24:40.320><c> modeling</c><00:24:41.480><c> and</c><00:24:41.640><c> the</c><00:24:41.720><c> generalized</c>"
+    },
+    {
+      "start": 1482.51,
+      "duration": 0.0,
+      "text": "generative modeling and the generalized"
+    },
+    {
+      "start": 1482.52,
+      "duration": 0.0,
+      "text": "generative modeling and the generalized fluctuation-dissipation<00:24:43.560><c> theorem.</c><00:24:44.600><c> And</c><00:24:44.760><c> so,</c>"
+    },
+    {
+      "start": 1484.87,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem. And so,"
+    },
+    {
+      "start": 1484.88,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem. And so, now<00:24:45.240><c> the</c><00:24:45.360><c> idea</c><00:24:46.080><c> is</c><00:24:46.280><c> to</c><00:24:46.400><c> use</c><00:24:47.040><c> this</c><00:24:47.600><c> mathematical</c>"
+    },
+    {
+      "start": 1488.15,
+      "duration": 0.0,
+      "text": "now the idea is to use this mathematical"
+    },
+    {
+      "start": 1488.16,
+      "duration": 0.0,
+      "text": "now the idea is to use this mathematical machinery<00:24:49.200><c> to</c><00:24:49.600><c> evaluate</c><00:24:50.880><c> the</c><00:24:51.000><c> parameter</c>"
+    },
+    {
+      "start": 1491.51,
+      "duration": 0.0,
+      "text": "machinery to evaluate the parameter"
+    },
+    {
+      "start": 1491.52,
+      "duration": 0.0,
+      "text": "machinery to evaluate the parameter sensitivities."
+    },
+    {
+      "start": 1494.07,
+      "duration": 0.0,
+      "text": "sensitivities."
+    },
+    {
+      "start": 1494.08,
+      "duration": 0.0,
+      "text": "sensitivities. And"
+    },
+    {
+      "start": 1495.23,
+      "duration": 0.0,
+      "text": "And"
+    },
+    {
+      "start": 1495.24,
+      "duration": 0.0,
+      "text": "And so,<00:24:55.560><c> to</c><00:24:55.679><c> do</c><00:24:55.800><c> that</c><00:24:56.560><c> I</c>"
+    },
+    {
+      "start": 1497.71,
+      "duration": 0.0,
+      "text": "so, to do that I"
+    },
+    {
+      "start": 1497.72,
+      "duration": 0.0,
+      "text": "so, to do that I so,<00:24:58.080><c> I</c><00:24:58.200><c> described</c><00:24:59.200><c> how</c><00:24:59.880><c> we</c><00:25:00.000><c> can</c><00:25:00.200><c> do</c><00:25:00.360><c> it.</c><00:25:00.880><c> So,</c>"
+    },
+    {
+      "start": 1501.19,
+      "duration": 0.0,
+      "text": "so, I described how we can do it. So,"
+    },
+    {
+      "start": 1501.2,
+      "duration": 0.0,
+      "text": "so, I described how we can do it. So, from<00:25:01.840><c> the</c><00:25:02.320><c> knowledge</c><00:25:02.720><c> of</c><00:25:02.800><c> the</c><00:25:02.920><c> response</c>"
+    },
+    {
+      "start": 1503.27,
+      "duration": 0.0,
+      "text": "from the knowledge of the response"
+    },
+    {
+      "start": 1503.28,
+      "duration": 0.0,
+      "text": "from the knowledge of the response function<00:25:03.760><c> we</c><00:25:03.880><c> can</c><00:25:04.480><c> estimate</c><00:25:05.160><c> the</c><00:25:05.280><c> parameter</c>"
+    },
+    {
+      "start": 1505.71,
+      "duration": 0.0,
+      "text": "function we can estimate the parameter"
+    },
+    {
+      "start": 1505.72,
+      "duration": 0.0,
+      "text": "function we can estimate the parameter sensitivities.<00:25:06.960><c> Now,</c><00:25:07.160><c> let's</c><00:25:07.400><c> see</c><00:25:07.600><c> how</c><00:25:07.840><c> to</c><00:25:07.960><c> do</c>"
+    },
+    {
+      "start": 1508.07,
+      "duration": 0.0,
+      "text": "sensitivities. Now, let's see how to do"
+    },
+    {
+      "start": 1508.08,
+      "duration": 0.0,
+      "text": "sensitivities. Now, let's see how to do that<00:25:08.320><c> in</c><00:25:08.400><c> practice.</c>"
+    },
+    {
+      "start": 1509.83,
+      "duration": 0.0,
+      "text": "that in practice."
+    },
+    {
+      "start": 1509.84,
+      "duration": 0.0,
+      "text": "that in practice. More<00:25:09.960><c> specifically,</c><00:25:10.720><c> let's</c><00:25:11.000><c> consider</c><00:25:11.520><c> two</c>"
+    },
+    {
+      "start": 1511.63,
+      "duration": 0.0,
+      "text": "More specifically, let's consider two"
+    },
+    {
+      "start": 1511.64,
+      "duration": 0.0,
+      "text": "More specifically, let's consider two examples<00:25:12.320><c> now.</c><00:25:12.960><c> So,</c><00:25:13.120><c> the</c><00:25:13.240><c> first</c><00:25:13.560><c> one</c><00:25:14.120><c> is</c><00:25:14.320><c> a</c>"
+    },
+    {
+      "start": 1514.39,
+      "duration": 0.0,
+      "text": "examples now. So, the first one is a"
+    },
+    {
+      "start": 1514.4,
+      "duration": 0.0,
+      "text": "examples now. So, the first one is a very<00:25:14.760><c> low-dimensional</c><00:25:15.600><c> model.</c><00:25:16.640><c> We</c><00:25:16.760><c> have</c><00:25:17.160><c> a</c>"
+    },
+    {
+      "start": 1517.23,
+      "duration": 0.0,
+      "text": "very low-dimensional model. We have a"
+    },
+    {
+      "start": 1517.24,
+      "duration": 0.0,
+      "text": "very low-dimensional model. We have a three-dimensional<00:25:18.480><c> SDE</c><00:25:19.600><c> with</c><00:25:20.120><c> a</c>"
+    },
+    {
+      "start": 1520.19,
+      "duration": 0.0,
+      "text": "three-dimensional SDE with a"
+    },
+    {
+      "start": 1520.2,
+      "duration": 0.0,
+      "text": "three-dimensional SDE with a multiplicative<00:25:20.840><c> noise.</c><00:25:21.560><c> So,</c><00:25:21.720><c> this</c><00:25:21.920><c> model</c><00:25:22.280><c> is</c>"
+    },
+    {
+      "start": 1522.43,
+      "duration": 0.0,
+      "text": "multiplicative noise. So, this model is"
+    },
+    {
+      "start": 1522.44,
+      "duration": 0.0,
+      "text": "multiplicative noise. So, this model is used<00:25:22.840><c> in</c><00:25:23.040><c> geophysical</c><00:25:23.640><c> fluid</c><00:25:23.840><c> dynamics</c><00:25:24.520><c> to</c>"
+    },
+    {
+      "start": 1524.669,
+      "duration": 0.0,
+      "text": "used in geophysical fluid dynamics to"
+    },
+    {
+      "start": 1524.679,
+      "duration": 0.0,
+      "text": "used in geophysical fluid dynamics to describe"
+    },
+    {
+      "start": 1525.95,
+      "duration": 0.0,
+      "text": "describe"
+    },
+    {
+      "start": 1525.96,
+      "duration": 0.0,
+      "text": "describe El<00:25:26.080><c> Niño-Southern</c><00:25:26.840><c> Oscillation,</c><00:25:27.600><c> which</c><00:25:27.840><c> is</c><00:25:28.040><c> a</c>"
+    },
+    {
+      "start": 1529.23,
+      "duration": 0.0,
+      "text": "El Niño-Southern Oscillation, which is a"
+    },
+    {
+      "start": 1529.24,
+      "duration": 0.0,
+      "text": "El Niño-Southern Oscillation, which is a interannual<00:25:30.080><c> so,</c><00:25:30.200><c> it's</c><00:25:30.400><c> a</c><00:25:31.240><c> annual</c>"
+    },
+    {
+      "start": 1531.669,
+      "duration": 0.0,
+      "text": "interannual so, it's a annual"
+    },
+    {
+      "start": 1531.679,
+      "duration": 0.0,
+      "text": "interannual so, it's a annual variability<00:25:32.280><c> phenomenon</c><00:25:33.360><c> of</c><00:25:33.480><c> the</c><00:25:33.760><c> sea</c>"
+    },
+    {
+      "start": 1533.95,
+      "duration": 0.0,
+      "text": "variability phenomenon of the sea"
+    },
+    {
+      "start": 1533.96,
+      "duration": 0.0,
+      "text": "variability phenomenon of the sea surface<00:25:34.320><c> temperature</c><00:25:35.040><c> in</c><00:25:35.120><c> the</c><00:25:35.240><c> tropical</c>"
+    },
+    {
+      "start": 1535.63,
+      "duration": 0.0,
+      "text": "surface temperature in the tropical"
+    },
+    {
+      "start": 1535.64,
+      "duration": 0.0,
+      "text": "surface temperature in the tropical Pacific.<00:25:36.760><c> We</c><00:25:36.920><c> have</c><00:25:37.440><c> two</c><00:25:37.600><c> slow</c><00:25:37.920><c> variables,</c><00:25:38.679><c> one</c>"
+    },
+    {
+      "start": 1539.15,
+      "duration": 0.0,
+      "text": "Pacific. We have two slow variables, one"
+    },
+    {
+      "start": 1539.16,
+      "duration": 0.0,
+      "text": "Pacific. We have two slow variables, one fast<00:25:39.520><c> variable,</c><00:25:40.240><c> which</c><00:25:40.440><c> are</c><00:25:40.520><c> coupled.</c>"
+    },
+    {
+      "start": 1542.03,
+      "duration": 0.0,
+      "text": "fast variable, which are coupled."
+    },
+    {
+      "start": 1542.04,
+      "duration": 0.0,
+      "text": "fast variable, which are coupled. This<00:25:42.360><c> model</c><00:25:42.760><c> depends</c><00:25:43.400><c> on</c><00:25:43.679><c> six</c><00:25:44.600><c> coefficients.</c>"
+    },
+    {
+      "start": 1546.83,
+      "duration": 0.0,
+      "text": "This model depends on six coefficients."
+    },
+    {
+      "start": 1546.84,
+      "duration": 0.0,
+      "text": "This model depends on six coefficients. And<00:25:47.840><c> so,</c><00:25:48.000><c> what</c><00:25:48.200><c> we</c><00:25:48.520><c> are</c><00:25:48.720><c> going</c><00:25:48.920><c> to</c><00:25:49.000><c> do</c><00:25:49.440><c> is</c><00:25:49.720><c> to</c>"
+    },
+    {
+      "start": 1550.43,
+      "duration": 0.0,
+      "text": "And so, what we are going to do is to"
+    },
+    {
+      "start": 1550.44,
+      "duration": 0.0,
+      "text": "And so, what we are going to do is to start<00:25:51.480><c> so,</c><00:25:51.800><c> is</c><00:25:51.920><c> to</c><00:25:52.520><c> first</c><00:25:52.960><c> run</c><00:25:53.600><c> this</c><00:25:53.800><c> model</c>"
+    },
+    {
+      "start": 1554.31,
+      "duration": 0.0,
+      "text": "start so, is to first run this model"
+    },
+    {
+      "start": 1554.32,
+      "duration": 0.0,
+      "text": "start so, is to first run this model with<00:25:54.480><c> the</c><00:25:54.560><c> correct</c><00:25:54.960><c> values</c><00:25:55.440><c> of</c><00:25:55.520><c> these</c>"
+    },
+    {
+      "start": 1555.669,
+      "duration": 0.0,
+      "text": "with the correct values of these"
+    },
+    {
+      "start": 1555.679,
+      "duration": 0.0,
+      "text": "with the correct values of these coefficients<00:25:56.360><c> to</c><00:25:56.480><c> have</c>"
+    },
+    {
+      "start": 1558.47,
+      "duration": 0.0,
+      "text": "coefficients to have"
+    },
+    {
+      "start": 1558.48,
+      "duration": 0.0,
+      "text": "coefficients to have an<00:25:58.640><c> obser</c><00:25:59.120><c> so,</c><00:25:59.320><c> to</c><00:25:59.880><c> build</c><00:26:00.560><c> our</c><00:26:00.880><c> observations.</c>"
+    },
+    {
+      "start": 1562.43,
+      "duration": 0.0,
+      "text": "an obser so, to build our observations."
+    },
+    {
+      "start": 1562.44,
+      "duration": 0.0,
+      "text": "an obser so, to build our observations. Then"
+    },
+    {
+      "start": 1563.71,
+      "duration": 0.0,
+      "text": "Then"
+    },
+    {
+      "start": 1563.72,
+      "duration": 0.0,
+      "text": "Then we<00:26:03.920><c> will</c>"
+    },
+    {
+      "start": 1564.91,
+      "duration": 0.0,
+      "text": "we will"
+    },
+    {
+      "start": 1564.92,
+      "duration": 0.0,
+      "text": "we will try<00:26:05.720><c> to</c><00:26:06.360><c> recover</c><00:26:07.200><c> the</c><00:26:07.360><c> correct</c><00:26:08.200><c> values</c><00:26:08.679><c> of</c>"
+    },
+    {
+      "start": 1568.79,
+      "duration": 0.0,
+      "text": "try to recover the correct values of"
+    },
+    {
+      "start": 1568.8,
+      "duration": 0.0,
+      "text": "try to recover the correct values of those<00:26:09.040><c> coefficients</c><00:26:09.840><c> and</c><00:26:10.000><c> starting</c><00:26:10.640><c> from</c><00:26:10.880><c> an</c>"
+    },
+    {
+      "start": 1570.95,
+      "duration": 0.0,
+      "text": "those coefficients and starting from an"
+    },
+    {
+      "start": 1570.96,
+      "duration": 0.0,
+      "text": "those coefficients and starting from an initial<00:26:11.320><c> guess</c>"
+    },
+    {
+      "start": 1572.79,
+      "duration": 0.0,
+      "text": "initial guess"
+    },
+    {
+      "start": 1572.8,
+      "duration": 0.0,
+      "text": "initial guess obtained<00:26:13.520><c> by</c><00:26:13.880><c> perturbing</c><00:26:15.000><c> by</c><00:26:16.120><c> around</c><00:26:17.000><c> 20%</c>"
+    },
+    {
+      "start": 1578.31,
+      "duration": 0.0,
+      "text": "obtained by perturbing by around 20%"
+    },
+    {
+      "start": 1578.32,
+      "duration": 0.0,
+      "text": "obtained by perturbing by around 20% each<00:26:18.520><c> of</c><00:26:18.640><c> those</c><00:26:18.880><c> coefficients</c><00:26:20.200><c> and</c><00:26:20.400><c> running</c>"
+    },
+    {
+      "start": 1580.99,
+      "duration": 0.0,
+      "text": "each of those coefficients and running"
+    },
+    {
+      "start": 1581.0,
+      "duration": 0.0,
+      "text": "each of those coefficients and running our<00:26:22.040><c> calibration</c><00:26:22.600><c> algorithm.</c><00:26:23.400><c> So,</c><00:26:23.600><c> using</c>"
+    },
+    {
+      "start": 1584.07,
+      "duration": 0.0,
+      "text": "our calibration algorithm. So, using"
+    },
+    {
+      "start": 1584.08,
+      "duration": 0.0,
+      "text": "our calibration algorithm. So, using GFDT<00:26:24.760><c> to</c><00:26:24.880><c> estimate</c><00:26:25.480><c> the</c>"
+    },
+    {
+      "start": 1586.669,
+      "duration": 0.0,
+      "text": "GFDT to estimate the"
+    },
+    {
+      "start": 1586.679,
+      "duration": 0.0,
+      "text": "GFDT to estimate the parameter<00:26:27.120><c> sensitivities</c>"
+    },
+    {
+      "start": 1588.55,
+      "duration": 0.0,
+      "text": "parameter sensitivities"
+    },
+    {
+      "start": 1588.56,
+      "duration": 0.0,
+      "text": "parameter sensitivities and<00:26:28.760><c> use</c>"
+    },
+    {
+      "start": 1589.71,
+      "duration": 0.0,
+      "text": "and use"
+    },
+    {
+      "start": 1589.72,
+      "duration": 0.0,
+      "text": "and use this<00:26:30.000><c> knowledge</c><00:26:30.640><c> inside</c><00:26:31.679><c> a</c><00:26:31.760><c> Newton</c><00:26:32.160><c> algorithm</c>"
+    },
+    {
+      "start": 1593.27,
+      "duration": 0.0,
+      "text": "this knowledge inside a Newton algorithm"
+    },
+    {
+      "start": 1593.28,
+      "duration": 0.0,
+      "text": "this knowledge inside a Newton algorithm to<00:26:33.880><c> estimate</c><00:26:34.600><c> the</c><00:26:34.720><c> correct</c><00:26:35.040><c> values</c><00:26:35.440><c> of</c><00:26:35.520><c> the</c>"
+    },
+    {
+      "start": 1595.63,
+      "duration": 0.0,
+      "text": "to estimate the correct values of the"
+    },
+    {
+      "start": 1595.64,
+      "duration": 0.0,
+      "text": "to estimate the correct values of the coefficient."
+    },
+    {
+      "start": 1597.55,
+      "duration": 0.0,
+      "text": "coefficient."
+    },
+    {
+      "start": 1597.56,
+      "duration": 0.0,
+      "text": "coefficient. Those<00:26:38.000><c> are</c><00:26:38.480><c> the</c><00:26:38.640><c> six</c><00:26:38.920><c> observables</c><00:26:39.960><c> that</c><00:26:40.160><c> we</c>"
+    },
+    {
+      "start": 1600.27,
+      "duration": 0.0,
+      "text": "Those are the six observables that we"
+    },
+    {
+      "start": 1600.28,
+      "duration": 0.0,
+      "text": "Those are the six observables that we would<00:26:40.480><c> like</c><00:26:40.840><c> to</c><00:26:40.960><c> recover.</c><00:26:41.800><c> So,</c><00:26:41.960><c> we</c><00:26:42.080><c> start</c><00:26:42.880><c> with</c>"
+    },
+    {
+      "start": 1603.71,
+      "duration": 0.0,
+      "text": "would like to recover. So, we start with"
+    },
+    {
+      "start": 1603.72,
+      "duration": 0.0,
+      "text": "would like to recover. So, we start with a<00:26:43.800><c> parameter</c><00:26:44.320><c> guess.</c>"
+    },
+    {
+      "start": 1605.91,
+      "duration": 0.0,
+      "text": "a parameter guess."
+    },
+    {
+      "start": 1605.92,
+      "duration": 0.0,
+      "text": "a parameter guess. Um<00:26:46.760><c> we</c><00:26:47.400><c> use</c><00:26:47.760><c> this</c><00:26:47.960><c> parameter</c><00:26:48.440><c> guess</c><00:26:48.960><c> to</c>"
+    },
+    {
+      "start": 1609.15,
+      "duration": 0.0,
+      "text": "Um we use this parameter guess to"
+    },
+    {
+      "start": 1609.16,
+      "duration": 0.0,
+      "text": "Um we use this parameter guess to integrate<00:26:49.840><c> the</c><00:26:49.960><c> model</c><00:26:50.240><c> forward.</c>"
+    },
+    {
+      "start": 1611.83,
+      "duration": 0.0,
+      "text": "integrate the model forward."
+    },
+    {
+      "start": 1611.84,
+      "duration": 0.0,
+      "text": "integrate the model forward. We<00:26:52.000><c> use</c><00:26:52.960><c> this</c><00:26:53.320><c> data</c><00:26:53.679><c> set</c><00:26:54.640><c> to</c><00:26:55.040><c> estimate</c><00:26:55.720><c> the</c>"
+    },
+    {
+      "start": 1615.83,
+      "duration": 0.0,
+      "text": "We use this data set to estimate the"
+    },
+    {
+      "start": 1615.84,
+      "duration": 0.0,
+      "text": "We use this data set to estimate the parameter<00:26:56.320><c> sensitivities.</c><00:26:57.679><c> And</c><00:26:57.920><c> then</c><00:26:58.320><c> we</c><00:26:58.480><c> are</c>"
+    },
+    {
+      "start": 1618.83,
+      "duration": 0.0,
+      "text": "parameter sensitivities. And then we are"
+    },
+    {
+      "start": 1618.84,
+      "duration": 0.0,
+      "text": "parameter sensitivities. And then we are updating<00:26:59.679><c> the</c><00:26:59.800><c> parameters</c><00:27:00.720><c> using</c><00:27:01.320><c> the</c>"
+    },
+    {
+      "start": 1621.47,
+      "duration": 0.0,
+      "text": "updating the parameters using the"
+    },
+    {
+      "start": 1621.48,
+      "duration": 0.0,
+      "text": "updating the parameters using the knowledge<00:27:01.840><c> of</c><00:27:01.960><c> the</c><00:27:02.040><c> parameter</c>"
+    },
+    {
+      "start": 1622.51,
+      "duration": 0.0,
+      "text": "knowledge of the parameter"
+    },
+    {
+      "start": 1622.52,
+      "duration": 0.0,
+      "text": "knowledge of the parameter sensitivities."
+    },
+    {
+      "start": 1623.91,
+      "duration": 0.0,
+      "text": "sensitivities."
+    },
+    {
+      "start": 1623.92,
+      "duration": 0.0,
+      "text": "sensitivities. And<00:27:04.160><c> we</c><00:27:04.280><c> are</c><00:27:04.400><c> iterating</c><00:27:04.960><c> this</c><00:27:05.160><c> procedure</c>"
+    },
+    {
+      "start": 1625.99,
+      "duration": 0.0,
+      "text": "And we are iterating this procedure"
+    },
+    {
+      "start": 1626.0,
+      "duration": 0.0,
+      "text": "And we are iterating this procedure until"
+    },
+    {
+      "start": 1627.71,
+      "duration": 0.0,
+      "text": "until"
+    },
+    {
+      "start": 1627.72,
+      "duration": 0.0,
+      "text": "until the<00:27:07.880><c> statistics</c><00:27:08.720><c> of</c><00:27:08.960><c> our</c><00:27:09.280><c> predicted</c><00:27:09.880><c> system</c>"
+    },
+    {
+      "start": 1630.39,
+      "duration": 0.0,
+      "text": "the statistics of our predicted system"
+    },
+    {
+      "start": 1630.4,
+      "duration": 0.0,
+      "text": "the statistics of our predicted system will<00:27:10.600><c> converge</c>"
+    },
+    {
+      "start": 1632.03,
+      "duration": 0.0,
+      "text": "will converge"
+    },
+    {
+      "start": 1632.04,
+      "duration": 0.0,
+      "text": "will converge to<00:27:12.160><c> the</c><00:27:12.280><c> target</c><00:27:12.800><c> statistical</c><00:27:13.320><c> observables.</c>"
+    },
+    {
+      "start": 1634.83,
+      "duration": 0.0,
+      "text": "to the target statistical observables."
+    },
+    {
+      "start": 1634.84,
+      "duration": 0.0,
+      "text": "to the target statistical observables. We<00:27:15.040><c> are</c><00:27:15.200><c> doing</c><00:27:15.520><c> this</c><00:27:15.720><c> procedure</c><00:27:16.440><c> using</c><00:27:17.280><c> three</c>"
+    },
+    {
+      "start": 1637.43,
+      "duration": 0.0,
+      "text": "We are doing this procedure using three"
+    },
+    {
+      "start": 1637.44,
+      "duration": 0.0,
+      "text": "We are doing this procedure using three different<00:27:18.000><c> methods</c><00:27:18.960><c> to</c><00:27:19.120><c> evaluate</c><00:27:20.040><c> the</c>"
+    },
+    {
+      "start": 1640.19,
+      "duration": 0.0,
+      "text": "different methods to evaluate the"
+    },
+    {
+      "start": 1640.2,
+      "duration": 0.0,
+      "text": "different methods to evaluate the parameter<00:27:20.760><c> sensitivities.</c><00:27:22.160><c> We</c><00:27:22.320><c> are</c><00:27:22.520><c> first</c>"
+    },
+    {
+      "start": 1642.87,
+      "duration": 0.0,
+      "text": "parameter sensitivities. We are first"
+    },
+    {
+      "start": 1642.88,
+      "duration": 0.0,
+      "text": "parameter sensitivities. We are first using<00:27:23.280><c> finite</c><00:27:23.640><c> difference.</c><00:27:24.480><c> So,</c><00:27:24.679><c> we</c><00:27:24.760><c> are</c><00:27:24.880><c> just</c>"
+    },
+    {
+      "start": 1645.59,
+      "duration": 0.0,
+      "text": "using finite difference. So, we are just"
+    },
+    {
+      "start": 1645.6,
+      "duration": 0.0,
+      "text": "using finite difference. So, we are just integrating<00:27:26.320><c> many</c><00:27:26.600><c> time</c><00:27:27.200><c> the</c><00:27:27.480><c> trajectory</c>"
+    },
+    {
+      "start": 1648.03,
+      "duration": 0.0,
+      "text": "integrating many time the trajectory"
+    },
+    {
+      "start": 1648.04,
+      "duration": 0.0,
+      "text": "integrating many time the trajectory forward<00:27:28.840><c> one</c><00:27:29.200><c> for</c><00:27:29.360><c> each</c><00:27:29.640><c> parameter.</c><00:27:30.840><c> And</c><00:27:31.000><c> then</c>"
+    },
+    {
+      "start": 1651.35,
+      "duration": 0.0,
+      "text": "forward one for each parameter. And then"
+    },
+    {
+      "start": 1651.36,
+      "duration": 0.0,
+      "text": "forward one for each parameter. And then we<00:27:31.480><c> are</c><00:27:31.560><c> using</c><00:27:31.880><c> finite</c><00:27:32.160><c> difference</c><00:27:32.640><c> to</c>"
+    },
+    {
+      "start": 1652.71,
+      "duration": 0.0,
+      "text": "we are using finite difference to"
+    },
+    {
+      "start": 1652.72,
+      "duration": 0.0,
+      "text": "we are using finite difference to estimate<00:27:33.240><c> the</c><00:27:33.360><c> parameter</c><00:27:34.240><c> Jacobian.</c>"
+    },
+    {
+      "start": 1655.83,
+      "duration": 0.0,
+      "text": "estimate the parameter Jacobian."
+    },
+    {
+      "start": 1655.84,
+      "duration": 0.0,
+      "text": "estimate the parameter Jacobian. And<00:27:36.000><c> we</c><00:27:36.080><c> are</c><00:27:36.120><c> using</c><00:27:36.440><c> that</c><00:27:36.560><c> information</c><00:27:37.200><c> to</c>"
+    },
+    {
+      "start": 1657.27,
+      "duration": 0.0,
+      "text": "And we are using that information to"
+    },
+    {
+      "start": 1657.28,
+      "duration": 0.0,
+      "text": "And we are using that information to update<00:27:37.720><c> the</c><00:27:37.800><c> parameter</c><00:27:38.200><c> value.</c>"
+    },
+    {
+      "start": 1659.43,
+      "duration": 0.0,
+      "text": "update the parameter value."
+    },
+    {
+      "start": 1659.44,
+      "duration": 0.0,
+      "text": "update the parameter value. Then<00:27:40.040><c> we</c><00:27:40.160><c> are</c><00:27:40.240><c> using</c><00:27:40.800><c> the</c><00:27:40.920><c> generalized</c>"
+    },
+    {
+      "start": 1661.59,
+      "duration": 0.0,
+      "text": "Then we are using the generalized"
+    },
+    {
+      "start": 1661.6,
+      "duration": 0.0,
+      "text": "Then we are using the generalized fluctuation-dissipation<00:27:42.640><c> theorem</c><00:27:43.440><c> using</c>"
+    },
+    {
+      "start": 1664.03,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem using"
+    },
+    {
+      "start": 1664.04,
+      "duration": 0.0,
+      "text": "fluctuation-dissipation theorem using two<00:27:44.160><c> different</c><00:27:44.600><c> ways</c><00:27:45.400><c> to</c><00:27:45.960><c> estimate</c><00:27:46.520><c> the</c><00:27:46.640><c> score</c>"
+    },
+    {
+      "start": 1666.87,
+      "duration": 0.0,
+      "text": "two different ways to estimate the score"
+    },
+    {
+      "start": 1666.88,
+      "duration": 0.0,
+      "text": "two different ways to estimate the score function."
+    },
+    {
+      "start": 1668.15,
+      "duration": 0.0,
+      "text": "function."
+    },
+    {
+      "start": 1668.16,
+      "duration": 0.0,
+      "text": "function. We<00:27:48.360><c> first</c><00:27:48.760><c> used</c><00:27:49.440><c> the</c><00:27:49.679><c> quasi-Gaussian</c>"
+    },
+    {
+      "start": 1671.15,
+      "duration": 0.0,
+      "text": "We first used the quasi-Gaussian"
+    },
+    {
+      "start": 1671.16,
+      "duration": 0.0,
+      "text": "We first used the quasi-Gaussian approximation.<00:27:52.160><c> So,</c><00:27:52.360><c> we</c><00:27:52.480><c> just</c>"
+    },
+    {
+      "start": 1674.19,
+      "duration": 0.0,
+      "text": "approximation. So, we just"
+    },
+    {
+      "start": 1674.2,
+      "duration": 0.0,
+      "text": "approximation. So, we just wrote<00:27:54.600><c> the</c><00:27:54.679><c> score</c><00:27:54.960><c> function</c><00:27:55.600><c> in</c><00:27:55.760><c> terms</c><00:27:56.320><c> of</c><00:27:56.840><c> a</c>"
+    },
+    {
+      "start": 1676.91,
+      "duration": 0.0,
+      "text": "wrote the score function in terms of a"
+    },
+    {
+      "start": 1676.92,
+      "duration": 0.0,
+      "text": "wrote the score function in terms of a linear<00:27:57.280><c> function</c><00:27:57.960><c> that</c><00:27:58.160><c> depends</c><00:27:59.040><c> on</c><00:27:59.200><c> the</c>"
+    },
+    {
+      "start": 1679.27,
+      "duration": 0.0,
+      "text": "linear function that depends on the"
+    },
+    {
+      "start": 1679.28,
+      "duration": 0.0,
+      "text": "linear function that depends on the covariance<00:27:59.840><c> matrix</c><00:28:00.640><c> of</c><00:28:00.760><c> the</c><00:28:00.880><c> data.</c><00:28:01.800><c> And</c><00:28:02.000><c> then</c>"
+    },
+    {
+      "start": 1682.23,
+      "duration": 0.0,
+      "text": "covariance matrix of the data. And then"
+    },
+    {
+      "start": 1682.24,
+      "duration": 0.0,
+      "text": "covariance matrix of the data. And then we<00:28:02.360><c> are</c><00:28:02.520><c> using</c><00:28:03.840><c> the</c>"
+    },
+    {
+      "start": 1685.31,
+      "duration": 0.0,
+      "text": "we are using the"
+    },
+    {
+      "start": 1685.32,
+      "duration": 0.0,
+      "text": "we are using the denoising<00:28:05.920><c> score</c><00:28:06.200><c> matching</c><00:28:06.640><c> approach</c><00:28:07.240><c> to</c>"
+    },
+    {
+      "start": 1687.35,
+      "duration": 0.0,
+      "text": "denoising score matching approach to"
+    },
+    {
+      "start": 1687.36,
+      "duration": 0.0,
+      "text": "denoising score matching approach to estimate<00:28:07.960><c> the</c><00:28:08.160><c> score</c><00:28:08.440><c> function.</c>"
+    },
+    {
+      "start": 1689.669,
+      "duration": 0.0,
+      "text": "estimate the score function."
+    },
+    {
+      "start": 1689.679,
+      "duration": 0.0,
+      "text": "estimate the score function. And<00:28:09.800><c> we</c><00:28:09.920><c> will</c><00:28:10.240><c> be</c><00:28:10.600><c> comparing</c><00:28:11.280><c> those</c><00:28:11.520><c> three</c>"
+    },
+    {
+      "start": 1691.71,
+      "duration": 0.0,
+      "text": "And we will be comparing those three"
+    },
+    {
+      "start": 1691.72,
+      "duration": 0.0,
+      "text": "And we will be comparing those three approaches."
+    },
+    {
+      "start": 1693.11,
+      "duration": 0.0,
+      "text": "approaches."
+    },
+    {
+      "start": 1693.12,
+      "duration": 0.0,
+      "text": "approaches. These<00:28:13.320><c> are</c><00:28:13.440><c> the</c><00:28:13.560><c> results.</c>"
+    },
+    {
+      "start": 1694.95,
+      "duration": 0.0,
+      "text": "These are the results."
+    },
+    {
+      "start": 1694.96,
+      "duration": 0.0,
+      "text": "These are the results. So,<00:28:15.120><c> here</c><00:28:15.679><c> on</c><00:28:15.800><c> the</c><00:28:15.920><c> left</c><00:28:16.440><c> you</c><00:28:16.679><c> can</c><00:28:16.960><c> see</c><00:28:17.679><c> the</c><00:28:18.280><c> L2</c>"
+    },
+    {
+      "start": 1698.59,
+      "duration": 0.0,
+      "text": "So, here on the left you can see the L2"
+    },
+    {
+      "start": 1698.6,
+      "duration": 0.0,
+      "text": "So, here on the left you can see the L2 norm<00:28:19.280><c> between</c><00:28:19.880><c> predicted</c><00:28:20.800><c> versus</c>"
+    },
+    {
+      "start": 1702.31,
+      "duration": 0.0,
+      "text": "norm between predicted versus"
+    },
+    {
+      "start": 1702.32,
+      "duration": 0.0,
+      "text": "norm between predicted versus target<00:28:23.000><c> statistical</c><00:28:23.600><c> observables</c><00:28:24.919><c> as</c><00:28:25.080><c> a</c>"
+    },
+    {
+      "start": 1705.149,
+      "duration": 0.0,
+      "text": "target statistical observables as a"
+    },
+    {
+      "start": 1705.159,
+      "duration": 0.0,
+      "text": "target statistical observables as a function<00:28:25.800><c> of</c><00:28:26.400><c> the</c>"
+    },
+    {
+      "start": 1707.59,
+      "duration": 0.0,
+      "text": "function of the"
+    },
+    {
+      "start": 1707.6,
+      "duration": 0.0,
+      "text": "function of the algorithm<00:28:28.480><c> iteration.</c><00:28:29.480><c> So,</c><00:28:29.720><c> in</c><00:28:29.840><c> this</c><00:28:30.000><c> case</c>"
+    },
+    {
+      "start": 1711.149,
+      "duration": 0.0,
+      "text": "algorithm iteration. So, in this case"
+    },
+    {
+      "start": 1711.159,
+      "duration": 0.0,
+      "text": "algorithm iteration. So, in this case we<00:28:31.320><c> introduced</c><00:28:32.080><c> the</c><00:28:32.840><c> breaking</c><00:28:33.240><c> point</c><00:28:34.480><c> when</c>"
+    },
+    {
+      "start": 1715.07,
+      "duration": 0.0,
+      "text": "we introduced the breaking point when"
+    },
+    {
+      "start": 1715.08,
+      "duration": 0.0,
+      "text": "we introduced the breaking point when the"
+    },
+    {
+      "start": 1715.95,
+      "duration": 0.0,
+      "text": "the"
+    },
+    {
+      "start": 1715.96,
+      "duration": 0.0,
+      "text": "the L2<00:28:36.240><c> norm</c>"
+    },
+    {
+      "start": 1717.87,
+      "duration": 0.0,
+      "text": "L2 norm"
+    },
+    {
+      "start": 1717.88,
+      "duration": 0.0,
+      "text": "L2 norm was<00:28:38.080><c> falling</c><00:28:38.679><c> below</c><00:28:39.200><c> 10</c><00:28:39.520><c> at</c><00:28:39.600><c> the</c><00:28:39.679><c> minus</c><00:28:39.960><c> three.</c>"
+    },
+    {
+      "start": 1720.909,
+      "duration": 0.0,
+      "text": "was falling below 10 at the minus three."
+    },
+    {
+      "start": 1720.919,
+      "duration": 0.0,
+      "text": "was falling below 10 at the minus three. And<00:28:41.080><c> as</c><00:28:41.200><c> you</c><00:28:41.320><c> can</c><00:28:41.560><c> see</c><00:28:42.080><c> the</c><00:28:42.240><c> blue</c><00:28:42.480><c> curve</c><00:28:43.440><c> and</c>"
+    },
+    {
+      "start": 1723.59,
+      "duration": 0.0,
+      "text": "And as you can see the blue curve and"
+    },
+    {
+      "start": 1723.6,
+      "duration": 0.0,
+      "text": "And as you can see the blue curve and the<00:28:43.679><c> gray</c><00:28:44.159><c> and</c><00:28:44.320><c> the</c><00:28:44.400><c> gray</c><00:28:44.600><c> curve,</c><00:28:45.000><c> which</c>"
+    },
+    {
+      "start": 1725.23,
+      "duration": 0.0,
+      "text": "the gray and the gray curve, which"
+    },
+    {
+      "start": 1725.24,
+      "duration": 0.0,
+      "text": "the gray and the gray curve, which represent<00:28:46.679><c> the</c><00:28:47.080><c> calibration</c><00:28:48.120><c> algorithm</c>"
+    },
+    {
+      "start": 1728.909,
+      "duration": 0.0,
+      "text": "represent the calibration algorithm"
+    },
+    {
+      "start": 1728.919,
+      "duration": 0.0,
+      "text": "represent the calibration algorithm using<00:28:49.960><c> the</c><00:28:50.040><c> denoising</c><00:28:50.480><c> score</c><00:28:50.760><c> matching</c><00:28:51.360><c> score</c>"
+    },
+    {
+      "start": 1731.669,
+      "duration": 0.0,
+      "text": "using the denoising score matching score"
+    },
+    {
+      "start": 1731.679,
+      "duration": 0.0,
+      "text": "using the denoising score matching score function<00:28:52.400><c> plus</c><00:28:52.840><c> GFDT</c><00:28:53.919><c> and</c><00:28:54.480><c> the</c><00:28:54.880><c> naive</c>"
+    },
+    {
+      "start": 1737.35,
+      "duration": 0.0,
+      "text": "function plus GFDT and the naive"
+    },
+    {
+      "start": 1737.36,
+      "duration": 0.0,
+      "text": "function plus GFDT and the naive and<00:28:57.880><c> and</c><00:28:58.000><c> the</c><00:28:58.120><c> naive</c>"
+    },
+    {
+      "start": 1739.43,
+      "duration": 0.0,
+      "text": "and and the naive"
+    },
+    {
+      "start": 1739.44,
+      "duration": 0.0,
+      "text": "and and the naive finite<00:28:59.840><c> difference</c><00:29:00.840><c> estimation</c><00:29:01.760><c> for</c><00:29:01.919><c> the</c>"
+    },
+    {
+      "start": 1742.07,
+      "duration": 0.0,
+      "text": "finite difference estimation for the"
+    },
+    {
+      "start": 1742.08,
+      "duration": 0.0,
+      "text": "finite difference estimation for the parameter<00:29:02.560><c> Jacobians</c><00:29:03.560><c> in</c><00:29:04.120><c> just</c><00:29:04.560><c> five</c>"
+    },
+    {
+      "start": 1744.83,
+      "duration": 0.0,
+      "text": "parameter Jacobians in just five"
+    },
+    {
+      "start": 1744.84,
+      "duration": 0.0,
+      "text": "parameter Jacobians in just five iteration<00:29:05.720><c> we</c><00:29:05.800><c> are</c><00:29:05.960><c> falling</c><00:29:06.520><c> below</c><00:29:06.800><c> the</c>"
+    },
+    {
+      "start": 1746.95,
+      "duration": 0.0,
+      "text": "iteration we are falling below the"
+    },
+    {
+      "start": 1746.96,
+      "duration": 0.0,
+      "text": "iteration we are falling below the threshold."
+    },
+    {
+      "start": 1748.35,
+      "duration": 0.0,
+      "text": "threshold."
+    },
+    {
+      "start": 1748.36,
+      "duration": 0.0,
+      "text": "threshold. Which<00:29:08.480><c> essentially</c><00:29:09.000><c> means</c><00:29:09.440><c> in</c><00:29:09.679><c> five</c>"
+    },
+    {
+      "start": 1749.909,
+      "duration": 0.0,
+      "text": "Which essentially means in five"
+    },
+    {
+      "start": 1749.919,
+      "duration": 0.0,
+      "text": "Which essentially means in five iteration<00:29:10.679><c> we</c><00:29:10.800><c> were</c><00:29:11.080><c> able,</c><00:29:11.760><c> as</c><00:29:11.919><c> you</c><00:29:12.000><c> can</c><00:29:12.200><c> see</c>"
+    },
+    {
+      "start": 1752.39,
+      "duration": 0.0,
+      "text": "iteration we were able, as you can see"
+    },
+    {
+      "start": 1752.4,
+      "duration": 0.0,
+      "text": "iteration we were able, as you can see here<00:29:13.280><c> in</c><00:29:13.400><c> this</c><00:29:13.880><c> panel</c><00:29:14.320><c> showing</c><00:29:14.760><c> the</c><00:29:14.880><c> parameter</c>"
+    },
+    {
+      "start": 1755.31,
+      "duration": 0.0,
+      "text": "here in this panel showing the parameter"
+    },
+    {
+      "start": 1755.32,
+      "duration": 0.0,
+      "text": "here in this panel showing the parameter deviation,<00:29:16.480><c> to</c><00:29:16.800><c> precisely</c><00:29:17.800><c> recover</c><00:29:18.480><c> the</c>"
+    },
+    {
+      "start": 1758.63,
+      "duration": 0.0,
+      "text": "deviation, to precisely recover the"
+    },
+    {
+      "start": 1758.64,
+      "duration": 0.0,
+      "text": "deviation, to precisely recover the correct<00:29:19.000><c> parameters</c><00:29:19.560><c> of</c><00:29:19.679><c> the</c><00:29:19.800><c> model.</c>"
+    },
+    {
+      "start": 1761.35,
+      "duration": 0.0,
+      "text": "correct parameters of the model."
+    },
+    {
+      "start": 1761.36,
+      "duration": 0.0,
+      "text": "correct parameters of the model. And<00:29:22.159><c> instead</c><00:29:22.760><c> using</c><00:29:23.200><c> the</c><00:29:23.320><c> Gaussian</c>"
+    },
+    {
+      "start": 1763.669,
+      "duration": 0.0,
+      "text": "And instead using the Gaussian"
+    },
+    {
+      "start": 1763.679,
+      "duration": 0.0,
+      "text": "And instead using the Gaussian approximation<00:29:24.480><c> for</c><00:29:24.640><c> the</c><00:29:24.760><c> score,</c><00:29:25.560><c> so</c><00:29:25.800><c> like</c><00:29:26.080><c> a</c>"
+    },
+    {
+      "start": 1766.149,
+      "duration": 0.0,
+      "text": "approximation for the score, so like a"
+    },
+    {
+      "start": 1766.159,
+      "duration": 0.0,
+      "text": "approximation for the score, so like a more<00:29:27.120><c> so,</c><00:29:27.240><c> less</c><00:29:27.520><c> precise</c><00:29:28.760><c> estimation</c><00:29:29.480><c> of</c><00:29:29.600><c> the</c>"
+    },
+    {
+      "start": 1769.669,
+      "duration": 0.0,
+      "text": "more so, less precise estimation of the"
+    },
+    {
+      "start": 1769.679,
+      "duration": 0.0,
+      "text": "more so, less precise estimation of the score<00:29:29.960><c> function</c><00:29:30.600><c> was</c><00:29:30.840><c> very</c><00:29:31.040><c> difficult</c><00:29:31.520><c> to</c>"
+    },
+    {
+      "start": 1772.35,
+      "duration": 0.0,
+      "text": "score function was very difficult to"
+    },
+    {
+      "start": 1772.36,
+      "duration": 0.0,
+      "text": "score function was very difficult to have<00:29:32.840><c> the</c><00:29:32.960><c> algorithm</c>"
+    },
+    {
+      "start": 1774.31,
+      "duration": 0.0,
+      "text": "have the algorithm"
+    },
+    {
+      "start": 1774.32,
+      "duration": 0.0,
+      "text": "have the algorithm con"
+    },
+    {
+      "start": 1775.79,
+      "duration": 0.0,
+      "text": "con"
+    },
+    {
+      "start": 1775.8,
+      "duration": 0.0,
+      "text": "con converged"
+    },
+    {
+      "start": 1777.31,
+      "duration": 0.0,
+      "text": "converged"
+    },
+    {
+      "start": 1777.32,
+      "duration": 0.0,
+      "text": "converged to<00:29:37.440><c> the</c><00:29:37.560><c> correct</c><00:29:37.960><c> value.</c>"
+    },
+    {
+      "start": 1779.35,
+      "duration": 0.0,
+      "text": "to the correct value."
+    },
+    {
+      "start": 1779.36,
+      "duration": 0.0,
+      "text": "to the correct value. But<00:29:39.520><c> so,</c><00:29:39.600><c> here</c><00:29:39.840><c> what</c><00:29:40.000><c> we</c><00:29:40.280><c> can</c><00:29:40.480><c> see</c><00:29:40.600><c> that</c><00:29:40.760><c> using</c>"
+    },
+    {
+      "start": 1782.149,
+      "duration": 0.0,
+      "text": "But so, here what we can see that using"
+    },
+    {
+      "start": 1782.159,
+      "duration": 0.0,
+      "text": "But so, here what we can see that using the<00:29:42.520><c> generalized</c><00:29:43.000><c> fluctuation-dissipation</c>"
+    },
+    {
+      "start": 1783.95,
+      "duration": 0.0,
+      "text": "the generalized fluctuation-dissipation"
+    },
+    {
+      "start": 1783.96,
+      "duration": 0.0,
+      "text": "the generalized fluctuation-dissipation theorem<00:29:44.440><c> plus</c><00:29:45.000><c> the</c><00:29:45.120><c> denoising</c><00:29:45.520><c> score</c>"
+    },
+    {
+      "start": 1785.79,
+      "duration": 0.0,
+      "text": "theorem plus the denoising score"
+    },
+    {
+      "start": 1785.8,
+      "duration": 0.0,
+      "text": "theorem plus the denoising score matching<00:29:46.480><c> to</c><00:29:46.600><c> estimate</c><00:29:47.200><c> this</c><00:29:47.600><c> score</c>"
+    },
+    {
+      "start": 1787.79,
+      "duration": 0.0,
+      "text": "matching to estimate this score"
+    },
+    {
+      "start": 1787.8,
+      "duration": 0.0,
+      "text": "matching to estimate this score function,<00:29:48.480><c> we</c><00:29:48.640><c> were</c><00:29:48.880><c> able</c><00:29:49.520><c> to</c><00:29:49.640><c> have</c><00:29:50.000><c> very</c>"
+    },
+    {
+      "start": 1790.31,
+      "duration": 0.0,
+      "text": "function, we were able to have very"
+    },
+    {
+      "start": 1790.32,
+      "duration": 0.0,
+      "text": "function, we were able to have very similar<00:29:51.000><c> performances</c>"
+    },
+    {
+      "start": 1793.03,
+      "duration": 0.0,
+      "text": "similar performances"
+    },
+    {
+      "start": 1793.04,
+      "duration": 0.0,
+      "text": "similar performances with<00:29:53.400><c> respect</c><00:29:54.120><c> to</c><00:29:54.320><c> the</c><00:29:54.880><c> naive</c>"
+    },
+    {
+      "start": 1795.99,
+      "duration": 0.0,
+      "text": "with respect to the naive"
+    },
+    {
+      "start": 1796.0,
+      "duration": 0.0,
+      "text": "with respect to the naive finite<00:29:56.360><c> difference</c><00:29:57.200><c> method</c><00:29:58.480><c> at</c><00:29:58.640><c> a</c><00:29:58.720><c> fraction</c>"
+    },
+    {
+      "start": 1799.23,
+      "duration": 0.0,
+      "text": "finite difference method at a fraction"
+    },
+    {
+      "start": 1799.24,
+      "duration": 0.0,
+      "text": "finite difference method at a fraction of<00:29:59.360><c> the</c><00:29:59.480><c> computational</c><00:30:00.040><c> cost</c><00:30:00.480><c> because</c>"
+    },
+    {
+      "start": 1801.59,
+      "duration": 0.0,
+      "text": "of the computational cost because"
+    },
+    {
+      "start": 1801.6,
+      "duration": 0.0,
+      "text": "of the computational cost because we"
+    },
+    {
+      "start": 1803.03,
+      "duration": 0.0,
+      "text": "we"
+    },
+    {
+      "start": 1803.04,
+      "duration": 0.0,
+      "text": "we So,<00:30:03.200><c> for</c><00:30:03.400><c> every</c><00:30:03.640><c> iteration,</c><00:30:04.440><c> we</c><00:30:04.560><c> needed</c><00:30:05.200><c> to</c>"
+    },
+    {
+      "start": 1805.31,
+      "duration": 0.0,
+      "text": "So, for every iteration, we needed to"
+    },
+    {
+      "start": 1805.32,
+      "duration": 0.0,
+      "text": "So, for every iteration, we needed to integrate<00:30:06.040><c> the</c><00:30:06.160><c> system</c><00:30:06.920><c> forward</c><00:30:08.000><c> only</c><00:30:08.360><c> one</c>"
+    },
+    {
+      "start": 1808.59,
+      "duration": 0.0,
+      "text": "integrate the system forward only one"
+    },
+    {
+      "start": 1808.6,
+      "duration": 0.0,
+      "text": "integrate the system forward only one time"
+    },
+    {
+      "start": 1809.71,
+      "duration": 0.0,
+      "text": "time"
+    },
+    {
+      "start": 1809.72,
+      "duration": 0.0,
+      "text": "time instead<00:30:10.560><c> of</c><00:30:11.520><c> six</c><00:30:12.000><c> time</c>"
+    },
+    {
+      "start": 1813.47,
+      "duration": 0.0,
+      "text": "instead of six time"
+    },
+    {
+      "start": 1813.48,
+      "duration": 0.0,
+      "text": "instead of six time So,<00:30:13.680><c> the</c><00:30:13.800><c> number</c><00:30:14.160><c> of</c><00:30:14.320><c> the</c><00:30:14.720><c> parameters</c><00:30:15.600><c> that</c><00:30:15.800><c> we</c>"
+    },
+    {
+      "start": 1815.91,
+      "duration": 0.0,
+      "text": "So, the number of the parameters that we"
+    },
+    {
+      "start": 1815.92,
+      "duration": 0.0,
+      "text": "So, the number of the parameters that we want<00:30:16.240><c> to</c><00:30:16.320><c> calibrate</c><00:30:17.280><c> or</c><00:30:17.720><c> like</c><00:30:18.000><c> in</c><00:30:18.200><c> in</c><00:30:18.320><c> this</c>"
+    },
+    {
+      "start": 1818.47,
+      "duration": 0.0,
+      "text": "want to calibrate or like in in this"
+    },
+    {
+      "start": 1818.48,
+      "duration": 0.0,
+      "text": "want to calibrate or like in in this case<00:30:18.840><c> 12</c><00:30:19.360><c> because</c><00:30:19.720><c> we</c><00:30:19.880><c> use</c><00:30:20.160><c> the</c><00:30:20.240><c> center</c>"
+    },
+    {
+      "start": 1820.59,
+      "duration": 0.0,
+      "text": "case 12 because we use the center"
+    },
+    {
+      "start": 1820.6,
+      "duration": 0.0,
+      "text": "case 12 because we use the center difference<00:30:21.840><c> for</c>"
+    },
+    {
+      "start": 1823.03,
+      "duration": 0.0,
+      "text": "difference for"
+    },
+    {
+      "start": 1823.04,
+      "duration": 0.0,
+      "text": "difference for parameter<00:30:23.640><c> for</c><00:30:23.800><c> the</c>"
+    },
+    {
+      "start": 1824.75,
+      "duration": 0.0,
+      "text": "parameter for the"
+    },
+    {
+      "start": 1824.76,
+      "duration": 0.0,
+      "text": "parameter for the uh<00:30:24.800><c> um</c>"
+    },
+    {
+      "start": 1826.11,
+      "duration": 0.0,
+      "text": "uh um"
+    },
+    {
+      "start": 1826.12,
+      "duration": 0.0,
+      "text": "uh um parameter<00:30:26.600><c> Jacobian</c><00:30:27.320><c> estimation.</c><00:30:28.120><c> So,</c>"
+    },
+    {
+      "start": 1828.27,
+      "duration": 0.0,
+      "text": "parameter Jacobian estimation. So,"
+    },
+    {
+      "start": 1828.28,
+      "duration": 0.0,
+      "text": "parameter Jacobian estimation. So, essentially<00:30:28.800><c> here</c><00:30:29.200><c> we</c><00:30:29.320><c> have</c><00:30:29.520><c> an</c><00:30:29.640><c> algorithm</c>"
+    },
+    {
+      "start": 1830.75,
+      "duration": 0.0,
+      "text": "essentially here we have an algorithm"
+    },
+    {
+      "start": 1830.76,
+      "duration": 0.0,
+      "text": "essentially here we have an algorithm that's<00:30:31.520><c> doesn't</c><00:30:31.840><c> scale</c><00:30:32.960><c> So,</c><00:30:33.120><c> doesn't</c><00:30:34.040><c> So,</c><00:30:34.200><c> the</c>"
+    },
+    {
+      "start": 1834.87,
+      "duration": 0.0,
+      "text": "that's doesn't scale So, doesn't So, the"
+    },
+    {
+      "start": 1834.88,
+      "duration": 0.0,
+      "text": "that's doesn't scale So, doesn't So, the for<00:30:35.040><c> which</c><00:30:35.400><c> the</c><00:30:35.920><c> computational</c><00:30:36.520><c> time</c><00:30:37.000><c> doesn't</c>"
+    },
+    {
+      "start": 1837.31,
+      "duration": 0.0,
+      "text": "for which the computational time doesn't"
+    },
+    {
+      "start": 1837.32,
+      "duration": 0.0,
+      "text": "for which the computational time doesn't scale"
+    },
+    {
+      "start": 1838.67,
+      "duration": 0.0,
+      "text": "scale"
+    },
+    {
+      "start": 1838.68,
+      "duration": 0.0,
+      "text": "scale linearly<00:30:39.280><c> with</c><00:30:39.440><c> the</c><00:30:39.520><c> number</c><00:30:40.000><c> of</c><00:30:40.200><c> parameters,</c>"
+    },
+    {
+      "start": 1841.11,
+      "duration": 0.0,
+      "text": "linearly with the number of parameters,"
+    },
+    {
+      "start": 1841.12,
+      "duration": 0.0,
+      "text": "linearly with the number of parameters, but<00:30:41.320><c> is</c><00:30:41.440><c> constant</c><00:30:42.320><c> since</c><00:30:42.880><c> we</c><00:30:43.040><c> only</c><00:30:43.240><c> need</c><00:30:43.600><c> to</c>"
+    },
+    {
+      "start": 1843.75,
+      "duration": 0.0,
+      "text": "but is constant since we only need to"
+    },
+    {
+      "start": 1843.76,
+      "duration": 0.0,
+      "text": "but is constant since we only need to run<00:30:44.160><c> the</c><00:30:44.280><c> model</c><00:30:44.560><c> forward</c><00:30:45.280><c> one</c>"
+    },
+    {
+      "start": 1846.11,
+      "duration": 0.0,
+      "text": "run the model forward one"
+    },
+    {
+      "start": 1846.12,
+      "duration": 0.0,
+      "text": "run the model forward one one<00:30:46.320><c> single</c><00:30:46.600><c> time</c><00:30:47.160><c> for</c><00:30:47.360><c> every</c><00:30:47.640><c> iteration.</c>"
+    },
+    {
+      "start": 1849.35,
+      "duration": 0.0,
+      "text": "one single time for every iteration."
+    },
+    {
+      "start": 1849.36,
+      "duration": 0.0,
+      "text": "one single time for every iteration. Okay.<00:30:49.640><c> So,</c><00:30:50.000><c> in</c><00:30:50.120><c> this</c><00:30:50.240><c> case</c><00:30:50.560><c> we</c><00:30:50.720><c> considered</c><00:30:51.440><c> a</c>"
+    },
+    {
+      "start": 1851.51,
+      "duration": 0.0,
+      "text": "Okay. So, in this case we considered a"
+    },
+    {
+      "start": 1851.52,
+      "duration": 0.0,
+      "text": "Okay. So, in this case we considered a quite<00:30:51.920><c> a</c><00:30:51.960><c> low</c><00:30:52.200><c> dimensional</c><00:30:52.720><c> system.</c>"
+    },
+    {
+      "start": 1854.11,
+      "duration": 0.0,
+      "text": "quite a low dimensional system."
+    },
+    {
+      "start": 1854.12,
+      "duration": 0.0,
+      "text": "quite a low dimensional system. Next,<00:30:54.920><c> I</c><00:30:55.040><c> will</c><00:30:55.240><c> consider</c><00:30:56.080><c> this</c><00:30:56.520><c> coupled</c>"
+    },
+    {
+      "start": 1857.15,
+      "duration": 0.0,
+      "text": "Next, I will consider this coupled"
+    },
+    {
+      "start": 1857.16,
+      "duration": 0.0,
+      "text": "Next, I will consider this coupled Lorenz<00:30:58.040><c> '96</c>"
+    },
+    {
+      "start": 1859.55,
+      "duration": 0.0,
+      "text": "Lorenz '96"
+    },
+    {
+      "start": 1859.56,
+      "duration": 0.0,
+      "text": "Lorenz '96 system,<00:31:00.640><c> which</c><00:31:00.840><c> is</c><00:31:01.000><c> around</c><00:31:01.760><c> a</c><00:31:01.800><c> 400</c>"
+    },
+    {
+      "start": 1862.59,
+      "duration": 0.0,
+      "text": "system, which is around a 400"
+    },
+    {
+      "start": 1862.6,
+      "duration": 0.0,
+      "text": "system, which is around a 400 dimensional<00:31:03.240><c> system.</c><00:31:04.560><c> We</c><00:31:04.760><c> have</c><00:31:05.200><c> 36</c><00:31:06.160><c> slow</c><00:31:06.480><c> mode</c>"
+    },
+    {
+      "start": 1867.51,
+      "duration": 0.0,
+      "text": "dimensional system. We have 36 slow mode"
+    },
+    {
+      "start": 1867.52,
+      "duration": 0.0,
+      "text": "dimensional system. We have 36 slow mode and<00:31:07.720><c> we</c><00:31:07.840><c> have</c><00:31:08.120><c> a</c><00:31:08.160><c> for</c><00:31:08.400><c> each</c><00:31:08.640><c> slow</c><00:31:08.880><c> mode</c><00:31:09.400><c> we</c><00:31:09.560><c> have</c>"
+    },
+    {
+      "start": 1869.79,
+      "duration": 0.0,
+      "text": "and we have a for each slow mode we have"
+    },
+    {
+      "start": 1869.8,
+      "duration": 0.0,
+      "text": "and we have a for each slow mode we have 10<00:31:10.400><c> fast</c><00:31:10.720><c> modes.</c><00:31:11.640><c> Plus,</c><00:31:12.080><c> we</c><00:31:12.240><c> also</c><00:31:12.560><c> have</c><00:31:13.240><c> some</c>"
+    },
+    {
+      "start": 1873.67,
+      "duration": 0.0,
+      "text": "10 fast modes. Plus, we also have some"
+    },
+    {
+      "start": 1873.68,
+      "duration": 0.0,
+      "text": "10 fast modes. Plus, we also have some white<00:31:13.920><c> noise</c><00:31:14.680><c> in</c><00:31:14.800><c> each</c><00:31:15.000><c> of</c><00:31:15.440><c> um</c>"
+    },
+    {
+      "start": 1876.11,
+      "duration": 0.0,
+      "text": "white noise in each of um"
+    },
+    {
+      "start": 1876.12,
+      "duration": 0.0,
+      "text": "white noise in each of um those"
+    },
+    {
+      "start": 1877.11,
+      "duration": 0.0,
+      "text": "those"
+    },
+    {
+      "start": 1877.12,
+      "duration": 0.0,
+      "text": "those variables."
+    },
+    {
+      "start": 1878.55,
+      "duration": 0.0,
+      "text": "variables."
+    },
+    {
+      "start": 1878.56,
+      "duration": 0.0,
+      "text": "variables. And<00:31:18.840><c> what</c><00:31:19.000><c> we</c><00:31:19.120><c> want</c><00:31:19.320><c> to</c><00:31:19.400><c> do</c><00:31:19.560><c> now</c><00:31:19.920><c> is</c><00:31:20.040><c> to</c><00:31:20.160><c> do</c>"
+    },
+    {
+      "start": 1880.39,
+      "duration": 0.0,
+      "text": "And what we want to do now is to do"
+    },
+    {
+      "start": 1880.4,
+      "duration": 0.0,
+      "text": "And what we want to do now is to do something<00:31:20.800><c> different.</c><00:31:21.640><c> So,</c><00:31:22.440><c> we</c><00:31:22.600><c> would</c><00:31:22.800><c> like</c>"
+    },
+    {
+      "start": 1883.47,
+      "duration": 0.0,
+      "text": "something different. So, we would like"
+    },
+    {
+      "start": 1883.48,
+      "duration": 0.0,
+      "text": "something different. So, we would like to<00:31:23.640><c> build</c><00:31:24.520><c> a</c><00:31:24.640><c> stochastic</c><00:31:25.240><c> closure</c><00:31:26.520><c> for</c><00:31:27.200><c> the</c><00:31:27.520><c> X</c>"
+    },
+    {
+      "start": 1888.15,
+      "duration": 0.0,
+      "text": "to build a stochastic closure for the X"
+    },
+    {
+      "start": 1888.16,
+      "duration": 0.0,
+      "text": "to build a stochastic closure for the X So,<00:31:28.320><c> the</c><00:31:28.440><c> slow</c><00:31:28.720><c> variables.</c><00:31:29.720><c> So,</c><00:31:29.840><c> essentially</c>"
+    },
+    {
+      "start": 1890.43,
+      "duration": 0.0,
+      "text": "So, the slow variables. So, essentially"
+    },
+    {
+      "start": 1890.44,
+      "duration": 0.0,
+      "text": "So, the slow variables. So, essentially we<00:31:30.560><c> would</c><00:31:30.760><c> like</c><00:31:31.160><c> to</c><00:31:31.280><c> build</c><00:31:32.000><c> a</c><00:31:32.120><c> 36</c><00:31:32.720><c> dimensional</c>"
+    },
+    {
+      "start": 1893.27,
+      "duration": 0.0,
+      "text": "we would like to build a 36 dimensional"
+    },
+    {
+      "start": 1893.28,
+      "duration": 0.0,
+      "text": "we would like to build a 36 dimensional model"
+    },
+    {
+      "start": 1894.63,
+      "duration": 0.0,
+      "text": "model"
+    },
+    {
+      "start": 1894.64,
+      "duration": 0.0,
+      "text": "model instead<00:31:35.040><c> of</c><00:31:35.240><c> this</c><00:31:35.400><c> model</c><00:31:35.680><c> here</c><00:31:35.920><c> that</c><00:31:36.120><c> is</c>"
+    },
+    {
+      "start": 1896.71,
+      "duration": 0.0,
+      "text": "instead of this model here that is"
+    },
+    {
+      "start": 1896.72,
+      "duration": 0.0,
+      "text": "instead of this model here that is around<00:31:37.240><c> a</c><00:31:37.280><c> 400</c><00:31:37.840><c> dimensional</c>"
+    },
+    {
+      "start": 1899.19,
+      "duration": 0.0,
+      "text": "around a 400 dimensional"
+    },
+    {
+      "start": 1899.2,
+      "duration": 0.0,
+      "text": "around a 400 dimensional which<00:31:39.360><c> is</c><00:31:39.520><c> able</c><00:31:40.000><c> to</c><00:31:40.520><c> precisely</c><00:31:41.480><c> recover</c>"
+    },
+    {
+      "start": 1902.99,
+      "duration": 0.0,
+      "text": "which is able to precisely recover"
+    },
+    {
+      "start": 1903.0,
+      "duration": 0.0,
+      "text": "which is able to precisely recover the<00:31:43.440><c> target</c><00:31:43.920><c> statistical</c><00:31:44.520><c> observables</c>"
+    },
+    {
+      "start": 1906.31,
+      "duration": 0.0,
+      "text": "the target statistical observables"
+    },
+    {
+      "start": 1906.32,
+      "duration": 0.0,
+      "text": "the target statistical observables evaluated<00:31:47.160><c> from</c><00:31:47.840><c> the</c><00:31:48.000><c> high</c><00:31:48.160><c> dimensional</c>"
+    },
+    {
+      "start": 1908.67,
+      "duration": 0.0,
+      "text": "evaluated from the high dimensional"
+    },
+    {
+      "start": 1908.68,
+      "duration": 0.0,
+      "text": "evaluated from the high dimensional model.<00:31:49.320><c> So,</c><00:31:49.480><c> we</c><00:31:49.640><c> have</c><00:31:49.880><c> observations</c><00:31:50.800><c> for</c><00:31:51.120><c> X</c>"
+    },
+    {
+      "start": 1912.07,
+      "duration": 0.0,
+      "text": "model. So, we have observations for X"
+    },
+    {
+      "start": 1912.08,
+      "duration": 0.0,
+      "text": "model. So, we have observations for X which<00:31:52.760><c> have</c><00:31:53.040><c> been</c><00:31:53.280><c> generated</c><00:31:54.040><c> integrating</c>"
+    },
+    {
+      "start": 1914.669,
+      "duration": 0.0,
+      "text": "which have been generated integrating"
+    },
+    {
+      "start": 1914.679,
+      "duration": 0.0,
+      "text": "which have been generated integrating this<00:31:54.920><c> very</c><00:31:55.120><c> high</c><00:31:55.280><c> dimensional</c><00:31:56.040><c> system</c><00:31:57.200><c> and</c><00:31:57.400><c> we</c>"
+    },
+    {
+      "start": 1917.51,
+      "duration": 0.0,
+      "text": "this very high dimensional system and we"
+    },
+    {
+      "start": 1917.52,
+      "duration": 0.0,
+      "text": "this very high dimensional system and we would<00:31:57.720><c> like</c><00:31:58.360><c> to</c><00:31:58.520><c> build</c><00:31:59.440><c> this</c><00:31:59.880><c> reduced</c><00:32:00.320><c> order</c>"
+    },
+    {
+      "start": 1920.59,
+      "duration": 0.0,
+      "text": "would like to build this reduced order"
+    },
+    {
+      "start": 1920.6,
+      "duration": 0.0,
+      "text": "would like to build this reduced order model<00:32:01.400><c> which</c><00:32:01.600><c> is</c><00:32:01.760><c> only</c><00:32:02.040><c> 36</c><00:32:02.600><c> dimensional</c>"
+    },
+    {
+      "start": 1924.03,
+      "duration": 0.0,
+      "text": "model which is only 36 dimensional"
+    },
+    {
+      "start": 1924.04,
+      "duration": 0.0,
+      "text": "model which is only 36 dimensional with<00:32:04.360><c> the</c><00:32:04.480><c> correct</c><00:32:04.920><c> values</c><00:32:05.520><c> of</c><00:32:05.840><c> alphas</c><00:32:06.560><c> of</c><00:32:06.720><c> the</c>"
+    },
+    {
+      "start": 1926.83,
+      "duration": 0.0,
+      "text": "with the correct values of alphas of the"
+    },
+    {
+      "start": 1926.84,
+      "duration": 0.0,
+      "text": "with the correct values of alphas of the alpha<00:32:07.520><c> coefficients</c><00:32:08.400><c> and</c><00:32:08.560><c> the</c><00:32:08.640><c> sigma</c>"
+    },
+    {
+      "start": 1929.19,
+      "duration": 0.0,
+      "text": "alpha coefficients and the sigma"
+    },
+    {
+      "start": 1929.2,
+      "duration": 0.0,
+      "text": "alpha coefficients and the sigma coefficients<00:32:10.240><c> such</c><00:32:10.480><c> that</c><00:32:10.960><c> they</c><00:32:11.560><c> reproduce</c>"
+    },
+    {
+      "start": 1932.39,
+      "duration": 0.0,
+      "text": "coefficients such that they reproduce"
+    },
+    {
+      "start": 1932.4,
+      "duration": 0.0,
+      "text": "coefficients such that they reproduce this<00:32:12.760><c> set</c><00:32:13.200><c> of</c><00:32:13.400><c> target</c><00:32:13.920><c> statistical</c>"
+    },
+    {
+      "start": 1934.43,
+      "duration": 0.0,
+      "text": "this set of target statistical"
+    },
+    {
+      "start": 1934.44,
+      "duration": 0.0,
+      "text": "this set of target statistical observables<00:32:15.440><c> which</c><00:32:15.720><c> are</c><00:32:16.280><c> the</c><00:32:16.400><c> mean</c><00:32:17.640><c> uh</c><00:32:18.120><c> the</c>"
+    },
+    {
+      "start": 1938.27,
+      "duration": 0.0,
+      "text": "observables which are the mean uh the"
+    },
+    {
+      "start": 1938.28,
+      "duration": 0.0,
+      "text": "observables which are the mean uh the variance,<00:32:18.800><c> skewness,</c><00:32:19.880><c> excess</c><00:32:20.400><c> kurtosis,</c><00:32:21.640><c> and</c>"
+    },
+    {
+      "start": 1942.47,
+      "duration": 0.0,
+      "text": "variance, skewness, excess kurtosis, and"
+    },
+    {
+      "start": 1942.48,
+      "duration": 0.0,
+      "text": "variance, skewness, excess kurtosis, and uh<00:32:22.560><c> covariance</c><00:32:23.480><c> C1.</c>"
+    },
+    {
+      "start": 1945.03,
+      "duration": 0.0,
+      "text": "uh covariance C1."
+    },
+    {
+      "start": 1945.04,
+      "duration": 0.0,
+      "text": "uh covariance C1. We<00:32:25.240><c> have</c><00:32:25.600><c> in</c><00:32:25.720><c> total</c><00:32:26.280><c> five</c><00:32:26.880><c> parameters</c><00:32:27.600><c> that</c><00:32:27.800><c> we</c>"
+    },
+    {
+      "start": 1947.91,
+      "duration": 0.0,
+      "text": "We have in total five parameters that we"
+    },
+    {
+      "start": 1947.92,
+      "duration": 0.0,
+      "text": "We have in total five parameters that we want<00:32:28.080><c> to</c><00:32:28.200><c> calibrate</c>"
+    },
+    {
+      "start": 1949.292,
+      "duration": 0.0,
+      "text": "want to calibrate"
+    },
+    {
+      "start": 1949.302,
+      "duration": 0.0,
+      "text": "want to calibrate &gt;&gt; [gasps]"
+    },
+    {
+      "start": 1949.95,
+      "duration": 0.0,
+      "text": "&gt;&gt; [gasps]"
+    },
+    {
+      "start": 1949.96,
+      "duration": 0.0,
+      "text": "&gt;&gt; [gasps] &gt;&gt; uh<00:32:30.280><c> on</c><00:32:30.760><c> the</c>"
+    },
+    {
+      "start": 1951.75,
+      "duration": 0.0,
+      "text": "&gt;&gt; uh on the"
+    },
+    {
+      "start": 1951.76,
+      "duration": 0.0,
+      "text": "&gt;&gt; uh on the on<00:32:31.960><c> these</c><00:32:32.280><c> five</c><00:32:32.800><c> different</c><00:32:33.520><c> statistical</c>"
+    },
+    {
+      "start": 1954.03,
+      "duration": 0.0,
+      "text": "on these five different statistical"
+    },
+    {
+      "start": 1954.04,
+      "duration": 0.0,
+      "text": "on these five different statistical observables."
+    },
+    {
+      "start": 1955.43,
+      "duration": 0.0,
+      "text": "observables."
+    },
+    {
+      "start": 1955.44,
+      "duration": 0.0,
+      "text": "observables. And<00:32:36.160><c> so,</c><00:32:36.320><c> we</c><00:32:36.480><c> used</c><00:32:37.120><c> also</c><00:32:37.440><c> in</c><00:32:37.560><c> this</c><00:32:37.760><c> case</c><00:32:38.640><c> these</c>"
+    },
+    {
+      "start": 1959.07,
+      "duration": 0.0,
+      "text": "And so, we used also in this case these"
+    },
+    {
+      "start": 1959.08,
+      "duration": 0.0,
+      "text": "And so, we used also in this case these three<00:32:39.240><c> different</c><00:32:39.640><c> methods.</c>"
+    },
+    {
+      "start": 1960.99,
+      "duration": 0.0,
+      "text": "three different methods."
+    },
+    {
+      "start": 1961.0,
+      "duration": 0.0,
+      "text": "three different methods. We<00:32:41.160><c> have</c><00:32:41.440><c> in</c><00:32:41.600><c> orange</c><00:32:42.560><c> GFDT</c><00:32:43.120><c> plus</c><00:32:43.800><c> Gaussian</c>"
+    },
+    {
+      "start": 1965.71,
+      "duration": 0.0,
+      "text": "We have in orange GFDT plus Gaussian"
+    },
+    {
+      "start": 1965.72,
+      "duration": 0.0,
+      "text": "We have in orange GFDT plus Gaussian uh<00:32:46.360><c> estimation</c><00:32:47.040><c> of</c><00:32:47.160><c> the</c><00:32:47.240><c> score</c><00:32:47.480><c> function.</c><00:32:48.679><c> And</c>"
+    },
+    {
+      "start": 1968.83,
+      "duration": 0.0,
+      "text": "uh estimation of the score function. And"
+    },
+    {
+      "start": 1968.84,
+      "duration": 0.0,
+      "text": "uh estimation of the score function. And then<00:32:49.160><c> in</c><00:32:49.440><c> gray</c><00:32:50.080><c> and</c><00:32:50.240><c> in</c><00:32:50.360><c> blue,</c><00:32:50.920><c> we</c><00:32:51.080><c> have</c>"
+    },
+    {
+      "start": 1972.23,
+      "duration": 0.0,
+      "text": "then in gray and in blue, we have"
+    },
+    {
+      "start": 1972.24,
+      "duration": 0.0,
+      "text": "then in gray and in blue, we have the<00:32:52.360><c> finite</c><00:32:52.679><c> difference</c><00:32:53.120><c> method</c><00:32:53.920><c> and</c><00:32:54.160><c> then</c>"
+    },
+    {
+      "start": 1974.669,
+      "duration": 0.0,
+      "text": "the finite difference method and then"
+    },
+    {
+      "start": 1974.679,
+      "duration": 0.0,
+      "text": "the finite difference method and then the<00:32:54.880><c> GFDT</c><00:32:55.480><c> plus</c><00:32:55.840><c> the</c><00:32:55.960><c> noise</c><00:32:56.320><c> score</c><00:32:56.520><c> matching</c>"
+    },
+    {
+      "start": 1977.15,
+      "duration": 0.0,
+      "text": "the GFDT plus the noise score matching"
+    },
+    {
+      "start": 1977.16,
+      "duration": 0.0,
+      "text": "the GFDT plus the noise score matching for<00:32:57.280><c> the</c><00:32:57.400><c> score</c>"
+    },
+    {
+      "start": 1978.63,
+      "duration": 0.0,
+      "text": "for the score"
+    },
+    {
+      "start": 1978.64,
+      "duration": 0.0,
+      "text": "for the score estimation.<00:32:59.480><c> So,</c><00:32:59.600><c> essentially</c><00:32:59.960><c> in</c><00:33:00.240><c> in</c><00:33:00.320><c> this</c>"
+    },
+    {
+      "start": 1980.47,
+      "duration": 0.0,
+      "text": "estimation. So, essentially in in this"
+    },
+    {
+      "start": 1980.48,
+      "duration": 0.0,
+      "text": "estimation. So, essentially in in this case<00:33:00.920><c> we</c><00:33:01.040><c> can</c><00:33:01.280><c> see</c><00:33:01.520><c> like</c><00:33:02.240><c> yeah,</c><00:33:02.480><c> clear</c>"
+    },
+    {
+      "start": 1982.91,
+      "duration": 0.0,
+      "text": "case we can see like yeah, clear"
+    },
+    {
+      "start": 1982.92,
+      "duration": 0.0,
+      "text": "case we can see like yeah, clear advantage<00:33:03.840><c> in</c><00:33:04.040><c> using</c><00:33:04.960><c> the</c>"
+    },
+    {
+      "start": 1986.23,
+      "duration": 0.0,
+      "text": "advantage in using the"
+    },
+    {
+      "start": 1986.24,
+      "duration": 0.0,
+      "text": "advantage in using the the<00:33:06.400><c> noise</c><00:33:06.720><c> score</c><00:33:06.960><c> matching</c><00:33:07.480><c> to</c><00:33:07.640><c> build</c><00:33:08.280><c> the</c>"
+    },
+    {
+      "start": 1988.39,
+      "duration": 0.0,
+      "text": "the noise score matching to build the"
+    },
+    {
+      "start": 1988.4,
+      "duration": 0.0,
+      "text": "the noise score matching to build the score<00:33:08.640><c> function.</c><00:33:09.880><c> And</c><00:33:10.280><c> from</c><00:33:10.480><c> the</c><00:33:10.600><c> knowledge</c>"
+    },
+    {
+      "start": 1990.95,
+      "duration": 0.0,
+      "text": "score function. And from the knowledge"
+    },
+    {
+      "start": 1990.96,
+      "duration": 0.0,
+      "text": "score function. And from the knowledge of<00:33:11.040><c> the</c><00:33:11.160><c> score</c><00:33:11.400><c> function,</c><00:33:11.960><c> also</c><00:33:12.240><c> in</c><00:33:12.480><c> this</c><00:33:12.640><c> case</c>"
+    },
+    {
+      "start": 1993.07,
+      "duration": 0.0,
+      "text": "of the score function, also in this case"
+    },
+    {
+      "start": 1993.08,
+      "duration": 0.0,
+      "text": "of the score function, also in this case which<00:33:13.280><c> is</c><00:33:13.400><c> quite</c><00:33:13.720><c> high</c><00:33:13.840><c> dimensional,</c><00:33:14.920><c> we</c><00:33:15.040><c> can</c>"
+    },
+    {
+      "start": 1995.23,
+      "duration": 0.0,
+      "text": "which is quite high dimensional, we can"
+    },
+    {
+      "start": 1995.24,
+      "duration": 0.0,
+      "text": "which is quite high dimensional, we can observe"
+    },
+    {
+      "start": 1996.75,
+      "duration": 0.0,
+      "text": "observe"
+    },
+    {
+      "start": 1996.76,
+      "duration": 0.0,
+      "text": "observe how<00:33:17.440><c> we</c><00:33:17.640><c> get</c><00:33:18.080><c> quite</c><00:33:18.320><c> similar</c><00:33:18.960><c> performance</c>"
+    },
+    {
+      "start": 2000.51,
+      "duration": 0.0,
+      "text": "how we get quite similar performance"
+    },
+    {
+      "start": 2000.52,
+      "duration": 0.0,
+      "text": "how we get quite similar performance than<00:33:21.440><c> of</c><00:33:21.679><c> using</c><00:33:22.600><c> finite</c><00:33:22.920><c> difference</c><00:33:23.800><c> at</c><00:33:23.960><c> a</c>"
+    },
+    {
+      "start": 2004.03,
+      "duration": 0.0,
+      "text": "than of using finite difference at a"
+    },
+    {
+      "start": 2004.04,
+      "duration": 0.0,
+      "text": "than of using finite difference at a fraction<00:33:24.480><c> of</c><00:33:24.600><c> the</c><00:33:24.679><c> computational</c><00:33:25.200><c> cost.</c><00:33:25.560><c> And</c>"
+    },
+    {
+      "start": 2005.669,
+      "duration": 0.0,
+      "text": "fraction of the computational cost. And"
+    },
+    {
+      "start": 2005.679,
+      "duration": 0.0,
+      "text": "fraction of the computational cost. And so,<00:33:25.800><c> essentially</c><00:33:26.240><c> at</c><00:33:26.480><c> at</c><00:33:26.640><c> a</c><00:33:26.679><c> fraction</c><00:33:27.280><c> of</c><00:33:27.400><c> the</c>"
+    },
+    {
+      "start": 2007.51,
+      "duration": 0.0,
+      "text": "so, essentially at at a fraction of the"
+    },
+    {
+      "start": 2007.52,
+      "duration": 0.0,
+      "text": "so, essentially at at a fraction of the number<00:33:27.800><c> of</c><00:33:27.920><c> time</c><00:33:28.720><c> that</c><00:33:28.920><c> we</c><00:33:29.040><c> have</c><00:33:29.480><c> to</c><00:33:29.600><c> integrate</c>"
+    },
+    {
+      "start": 2010.43,
+      "duration": 0.0,
+      "text": "number of time that we have to integrate"
+    },
+    {
+      "start": 2010.44,
+      "duration": 0.0,
+      "text": "number of time that we have to integrate our<00:33:30.720><c> model</c><00:33:31.040><c> forward.</c>"
+    },
+    {
+      "start": 2013.64,
+      "duration": 0.0,
+      "text": "Okay.<00:33:34.000><c> So,</c><00:33:34.160><c> this</c>"
+    },
+    {
+      "start": 2015.31,
+      "duration": 0.0,
+      "text": "Okay. So, this"
+    },
+    {
+      "start": 2015.32,
+      "duration": 0.0,
+      "text": "Okay. So, this is<00:33:35.520><c> the</c><00:33:35.600><c> first</c><00:33:35.840><c> direction.</c><00:33:36.600><c> So,</c><00:33:36.800><c> start</c><00:33:37.440><c> from</c><00:33:37.640><c> a</c>"
+    },
+    {
+      "start": 2017.669,
+      "duration": 0.0,
+      "text": "is the first direction. So, start from a"
+    },
+    {
+      "start": 2017.679,
+      "duration": 0.0,
+      "text": "is the first direction. So, start from a model<00:33:38.040><c> answers</c><00:33:39.120><c> and</c><00:33:39.360><c> use</c><00:33:39.720><c> this</c><00:33:39.920><c> combination</c>"
+    },
+    {
+      "start": 2020.63,
+      "duration": 0.0,
+      "text": "model answers and use this combination"
+    },
+    {
+      "start": 2020.64,
+      "duration": 0.0,
+      "text": "model answers and use this combination between<00:33:41.160><c> the</c><00:33:41.280><c> generalized</c><00:33:41.960><c> fluctuation</c>"
+    },
+    {
+      "start": 2022.39,
+      "duration": 0.0,
+      "text": "between the generalized fluctuation"
+    },
+    {
+      "start": 2022.4,
+      "duration": 0.0,
+      "text": "between the generalized fluctuation dissipation<00:33:42.960><c> theorem</c><00:33:43.640><c> from</c><00:33:44.640><c> non-equilibrium</c>"
+    },
+    {
+      "start": 2025.91,
+      "duration": 0.0,
+      "text": "dissipation theorem from non-equilibrium"
+    },
+    {
+      "start": 2025.92,
+      "duration": 0.0,
+      "text": "dissipation theorem from non-equilibrium statistical<00:33:46.400><c> physics</c><00:33:47.080><c> with</c><00:33:47.600><c> the</c><00:33:47.720><c> noise</c><00:33:48.080><c> score</c>"
+    },
+    {
+      "start": 2028.31,
+      "duration": 0.0,
+      "text": "statistical physics with the noise score"
+    },
+    {
+      "start": 2028.32,
+      "duration": 0.0,
+      "text": "statistical physics with the noise score matching<00:33:49.120><c> from</c><00:33:49.679><c> generative</c><00:33:50.120><c> modeling</c>"
+    },
+    {
+      "start": 2031.75,
+      "duration": 0.0,
+      "text": "matching from generative modeling"
+    },
+    {
+      "start": 2031.76,
+      "duration": 0.0,
+      "text": "matching from generative modeling to<00:33:52.560><c> to</c><00:33:52.880><c> to</c><00:33:52.960><c> estimate</c>"
+    },
+    {
+      "start": 2034.99,
+      "duration": 0.0,
+      "text": "to to to estimate"
+    },
+    {
+      "start": 2035.0,
+      "duration": 0.0,
+      "text": "to to to estimate uh<00:33:55.080><c> the</c><00:33:55.760><c> parameter</c><00:33:56.280><c> sensitivities</c>"
+    },
+    {
+      "start": 2038.23,
+      "duration": 0.0,
+      "text": "uh the parameter sensitivities"
+    },
+    {
+      "start": 2038.24,
+      "duration": 0.0,
+      "text": "uh the parameter sensitivities with<00:33:58.640><c> a</c><00:33:58.720><c> very</c><00:33:59.040><c> limited</c><00:33:59.560><c> number</c><00:34:00.280><c> of</c><00:34:00.600><c> model</c>"
+    },
+    {
+      "start": 2040.99,
+      "duration": 0.0,
+      "text": "with a very limited number of model"
+    },
+    {
+      "start": 2041.0,
+      "duration": 0.0,
+      "text": "with a very limited number of model integrations."
+    },
+    {
+      "start": 2042.71,
+      "duration": 0.0,
+      "text": "integrations."
+    },
+    {
+      "start": 2042.72,
+      "duration": 0.0,
+      "text": "integrations. Now,<00:34:02.960><c> let's</c><00:34:03.280><c> see</c><00:34:03.440><c> the</c><00:34:03.640><c> second</c><00:34:03.920><c> direction.</c>"
+    },
+    {
+      "start": 2045.11,
+      "duration": 0.0,
+      "text": "Now, let's see the second direction."
+    },
+    {
+      "start": 2045.12,
+      "duration": 0.0,
+      "text": "Now, let's see the second direction. So,<00:34:05.240><c> in</c><00:34:05.320><c> this</c><00:34:05.480><c> case</c><00:34:06.520><c> we</c><00:34:06.679><c> don't</c><00:34:07.000><c> have</c><00:34:07.480><c> any</c><00:34:08.359><c> model</c>"
+    },
+    {
+      "start": 2048.869,
+      "duration": 0.0,
+      "text": "So, in this case we don't have any model"
+    },
+    {
+      "start": 2048.879,
+      "duration": 0.0,
+      "text": "So, in this case we don't have any model answers<00:34:09.960><c> for</c>"
+    },
+    {
+      "start": 2051.389,
+      "duration": 0.0,
+      "text": "answers for"
+    },
+    {
+      "start": 2051.399,
+      "duration": 0.0,
+      "text": "answers for the<00:34:11.600><c> functional</c><00:34:12.200><c> form</c><00:34:12.720><c> of</c><00:34:12.879><c> our</c><00:34:13.159><c> mathematical</c>"
+    },
+    {
+      "start": 2053.79,
+      "duration": 0.0,
+      "text": "the functional form of our mathematical"
+    },
+    {
+      "start": 2053.8,
+      "duration": 0.0,
+      "text": "the functional form of our mathematical model."
+    },
+    {
+      "start": 2055.31,
+      "duration": 0.0,
+      "text": "model."
+    },
+    {
+      "start": 2055.32,
+      "duration": 0.0,
+      "text": "model. We<00:34:15.560><c> have</c><00:34:16.200><c> a</c><00:34:16.320><c> set</c><00:34:17.120><c> of</c><00:34:18.240><c> statistical</c><00:34:18.919><c> and</c>"
+    },
+    {
+      "start": 2059.07,
+      "duration": 0.0,
+      "text": "We have a set of statistical and"
+    },
+    {
+      "start": 2059.08,
+      "duration": 0.0,
+      "text": "We have a set of statistical and dynamical<00:34:19.640><c> observables</c><00:34:20.600><c> that</c><00:34:20.840><c> we</c><00:34:21.000><c> want</c><00:34:21.440><c> our</c>"
+    },
+    {
+      "start": 2061.63,
+      "duration": 0.0,
+      "text": "dynamical observables that we want our"
+    },
+    {
+      "start": 2061.64,
+      "duration": 0.0,
+      "text": "dynamical observables that we want our model<00:34:22.000><c> to</c><00:34:22.120><c> reproduce,</c><00:34:23.000><c> which</c><00:34:23.240><c> in</c><00:34:23.320><c> this</c><00:34:23.520><c> case</c>"
+    },
+    {
+      "start": 2064.149,
+      "duration": 0.0,
+      "text": "model to reproduce, which in this case"
+    },
+    {
+      "start": 2064.159,
+      "duration": 0.0,
+      "text": "model to reproduce, which in this case are<00:34:24.720><c> the</c><00:34:24.879><c> full</c><00:34:25.320><c> steady</c><00:34:25.640><c> state</c><00:34:26.040><c> distribution</c>"
+    },
+    {
+      "start": 2067.59,
+      "duration": 0.0,
+      "text": "are the full steady state distribution"
+    },
+    {
+      "start": 2067.6,
+      "duration": 0.0,
+      "text": "are the full steady state distribution and<00:34:28.159><c> a</c><00:34:28.240><c> set</c><00:34:28.720><c> of</c><00:34:28.879><c> correlation</c><00:34:29.480><c> functions</c><00:34:30.440><c> where</c>"
+    },
+    {
+      "start": 2071.07,
+      "duration": 0.0,
+      "text": "and a set of correlation functions where"
+    },
+    {
+      "start": 2071.08,
+      "duration": 0.0,
+      "text": "and a set of correlation functions where phi<00:34:31.320><c> m</c><00:34:31.720><c> and</c><00:34:31.919><c> phi</c><00:34:32.080><c> n</c><00:34:32.720><c> are</c>"
+    },
+    {
+      "start": 2075.07,
+      "duration": 0.0,
+      "text": "phi m and phi n are"
+    },
+    {
+      "start": 2075.08,
+      "duration": 0.0,
+      "text": "phi m and phi n are So,<00:34:35.200><c> it's</c><00:34:35.320><c> a</c><00:34:35.359><c> set</c><00:34:35.679><c> of</c><00:34:35.840><c> observables</c><00:34:37.320><c> of</c><00:34:37.919><c> the</c><00:34:38.600><c> the</c>"
+    },
+    {
+      "start": 2078.75,
+      "duration": 0.0,
+      "text": "So, it's a set of observables of the the"
+    },
+    {
+      "start": 2078.76,
+      "duration": 0.0,
+      "text": "So, it's a set of observables of the the state<00:34:39.399><c> variable</c><00:34:39.960><c> of</c><00:34:40.080><c> the</c><00:34:40.200><c> system.</c>"
+    },
+    {
+      "start": 2081.79,
+      "duration": 0.0,
+      "text": "state variable of the system."
+    },
+    {
+      "start": 2081.8,
+      "duration": 0.0,
+      "text": "state variable of the system. So,<00:34:41.960><c> given</c><00:34:42.879><c> these</c><00:34:43.120><c> constraints,</c><00:34:44.399><c> we</c><00:34:44.560><c> would</c>"
+    },
+    {
+      "start": 2084.79,
+      "duration": 0.0,
+      "text": "So, given these constraints, we would"
+    },
+    {
+      "start": 2084.8,
+      "duration": 0.0,
+      "text": "So, given these constraints, we would like<00:34:45.840><c> to</c><00:34:46.000><c> build</c><00:34:46.840><c> a</c><00:34:46.879><c> mathematical</c><00:34:47.840><c> model</c><00:34:47.879><c> that</c>"
+    },
+    {
+      "start": 2088.07,
+      "duration": 0.0,
+      "text": "like to build a mathematical model that"
+    },
+    {
+      "start": 2088.08,
+      "duration": 0.0,
+      "text": "like to build a mathematical model that by<00:34:48.200><c> construction</c><00:34:49.159><c> reproduces</c><00:34:50.359><c> those</c>"
+    },
+    {
+      "start": 2090.55,
+      "duration": 0.0,
+      "text": "by construction reproduces those"
+    },
+    {
+      "start": 2090.56,
+      "duration": 0.0,
+      "text": "by construction reproduces those constraints<00:34:51.560><c> without</c><00:34:52.399><c> integrating</c><00:34:53.000><c> our</c>"
+    },
+    {
+      "start": 2093.19,
+      "duration": 0.0,
+      "text": "constraints without integrating our"
+    },
+    {
+      "start": 2093.2,
+      "duration": 0.0,
+      "text": "constraints without integrating our model<00:34:53.520><c> forward.</c>"
+    },
+    {
+      "start": 2095.59,
+      "duration": 0.0,
+      "text": "model forward."
+    },
+    {
+      "start": 2095.6,
+      "duration": 0.0,
+      "text": "model forward. And<00:34:55.840><c> again,</c><00:34:56.320><c> we</c><00:34:56.440><c> will</c><00:34:56.600><c> use</c><00:34:57.000><c> the</c><00:34:57.120><c> score</c>"
+    },
+    {
+      "start": 2097.39,
+      "duration": 0.0,
+      "text": "And again, we will use the score"
+    },
+    {
+      "start": 2097.4,
+      "duration": 0.0,
+      "text": "And again, we will use the score function<00:34:58.280><c> to</c><00:34:58.400><c> do</c><00:34:58.560><c> that.</c>"
+    },
+    {
+      "start": 2099.43,
+      "duration": 0.0,
+      "text": "function to do that."
+    },
+    {
+      "start": 2099.44,
+      "duration": 0.0,
+      "text": "function to do that. Specifically,<00:35:00.160><c> we</c><00:35:00.280><c> will</c><00:35:00.440><c> use</c><00:35:00.800><c> two</c><00:35:01.000><c> different</c>"
+    },
+    {
+      "start": 2102.19,
+      "duration": 0.0,
+      "text": "Specifically, we will use two different"
+    },
+    {
+      "start": 2102.2,
+      "duration": 0.0,
+      "text": "Specifically, we will use two different score<00:35:02.560><c> functions</c><00:35:03.400><c> in</c><00:35:03.520><c> this</c><00:35:03.720><c> case.</c><00:35:04.160><c> We</c><00:35:04.320><c> have</c>"
+    },
+    {
+      "start": 2104.55,
+      "duration": 0.0,
+      "text": "score functions in this case. We have"
+    },
+    {
+      "start": 2104.56,
+      "duration": 0.0,
+      "text": "score functions in this case. We have the<00:35:04.680><c> plain</c><00:35:05.080><c> score</c><00:35:05.320><c> function</c><00:35:05.840><c> that</c><00:35:06.000><c> we've</c><00:35:06.160><c> seen</c>"
+    },
+    {
+      "start": 2106.47,
+      "duration": 0.0,
+      "text": "the plain score function that we've seen"
+    },
+    {
+      "start": 2106.48,
+      "duration": 0.0,
+      "text": "the plain score function that we've seen before"
+    },
+    {
+      "start": 2107.67,
+      "duration": 0.0,
+      "text": "before"
+    },
+    {
+      "start": 2107.68,
+      "duration": 0.0,
+      "text": "before and<00:35:07.840><c> also</c><00:35:08.520><c> we</c><00:35:08.640><c> will</c><00:35:08.840><c> use</c><00:35:09.320><c> the</c><00:35:09.440><c> conditional</c>"
+    },
+    {
+      "start": 2109.99,
+      "duration": 0.0,
+      "text": "and also we will use the conditional"
+    },
+    {
+      "start": 2110.0,
+      "duration": 0.0,
+      "text": "and also we will use the conditional score<00:35:10.240><c> function.</c><00:35:11.080><c> This</c><00:35:11.560><c> essentially</c><00:35:12.200><c> is</c><00:35:12.440><c> the</c>"
+    },
+    {
+      "start": 2112.87,
+      "duration": 0.0,
+      "text": "score function. This essentially is the"
+    },
+    {
+      "start": 2112.88,
+      "duration": 0.0,
+      "text": "score function. This essentially is the gradient"
+    },
+    {
+      "start": 2114.43,
+      "duration": 0.0,
+      "text": "gradient"
+    },
+    {
+      "start": 2114.44,
+      "duration": 0.0,
+      "text": "gradient of<00:35:15.120><c> with</c><00:35:15.480><c> respect</c><00:35:16.359><c> to</c><00:35:16.520><c> X0</c><00:35:17.400><c> of</c><00:35:17.560><c> the</c><00:35:17.680><c> logarithm</c>"
+    },
+    {
+      "start": 2118.47,
+      "duration": 0.0,
+      "text": "of with respect to X0 of the logarithm"
+    },
+    {
+      "start": 2118.48,
+      "duration": 0.0,
+      "text": "of with respect to X0 of the logarithm of<00:35:18.640><c> the</c><00:35:18.720><c> conditional</c><00:35:19.400><c> probability</c><00:35:20.000><c> density</c>"
+    },
+    {
+      "start": 2120.39,
+      "duration": 0.0,
+      "text": "of the conditional probability density"
+    },
+    {
+      "start": 2120.4,
+      "duration": 0.0,
+      "text": "of the conditional probability density function.<00:35:21.160><c> So,</c><00:35:21.480><c> the</c><00:35:22.120><c> probability</c><00:35:22.680><c> density</c>"
+    },
+    {
+      "start": 2123.11,
+      "duration": 0.0,
+      "text": "function. So, the probability density"
+    },
+    {
+      "start": 2123.12,
+      "duration": 0.0,
+      "text": "function. So, the probability density function<00:35:23.800><c> of</c><00:35:24.040><c> X</c><00:35:24.600><c> at</c><00:35:24.760><c> time</c><00:35:25.080><c> T</c><00:35:25.560><c> conditioned</c><00:35:26.400><c> on</c>"
+    },
+    {
+      "start": 2126.59,
+      "duration": 0.0,
+      "text": "function of X at time T conditioned on"
+    },
+    {
+      "start": 2126.6,
+      "duration": 0.0,
+      "text": "function of X at time T conditioned on X0."
+    },
+    {
+      "start": 2128.59,
+      "duration": 0.0,
+      "text": "X0."
+    },
+    {
+      "start": 2128.6,
+      "duration": 0.0,
+      "text": "X0. The<00:35:28.760><c> conditional</c><00:35:29.200><c> score</c><00:35:29.400><c> function</c><00:35:29.960><c> can</c><00:35:30.240><c> be</c>"
+    },
+    {
+      "start": 2131.349,
+      "duration": 0.0,
+      "text": "The conditional score function can be"
+    },
+    {
+      "start": 2131.359,
+      "duration": 0.0,
+      "text": "The conditional score function can be constructed<00:35:32.280><c> using</c><00:35:32.640><c> the</c><00:35:32.720><c> noise</c><00:35:33.080><c> score</c>"
+    },
+    {
+      "start": 2133.349,
+      "duration": 0.0,
+      "text": "constructed using the noise score"
+    },
+    {
+      "start": 2133.359,
+      "duration": 0.0,
+      "text": "constructed using the noise score matching<00:35:33.960><c> precisely</c><00:35:34.920><c> as</c><00:35:35.280><c> we</c><00:35:35.400><c> did</c><00:35:35.640><c> it</c><00:35:36.280><c> for</c><00:35:36.560><c> the</c>"
+    },
+    {
+      "start": 2136.71,
+      "duration": 0.0,
+      "text": "matching precisely as we did it for the"
+    },
+    {
+      "start": 2136.72,
+      "duration": 0.0,
+      "text": "matching precisely as we did it for the plain<00:35:37.080><c> score</c><00:35:37.320><c> function.</c><00:35:37.960><c> In</c><00:35:38.120><c> fact,</c><00:35:38.440><c> we</c><00:35:38.560><c> can</c>"
+    },
+    {
+      "start": 2138.79,
+      "duration": 0.0,
+      "text": "plain score function. In fact, we can"
+    },
+    {
+      "start": 2138.8,
+      "duration": 0.0,
+      "text": "plain score function. In fact, we can write<00:35:39.320><c> the</c><00:35:39.480><c> conditional</c><00:35:40.040><c> score</c><00:35:40.280><c> function</c><00:35:41.000><c> in</c>"
+    },
+    {
+      "start": 2141.19,
+      "duration": 0.0,
+      "text": "write the conditional score function in"
+    },
+    {
+      "start": 2141.2,
+      "duration": 0.0,
+      "text": "write the conditional score function in terms<00:35:42.120><c> of</c><00:35:42.880><c> the</c><00:35:43.080><c> joint</c><00:35:43.960><c> score</c><00:35:44.320><c> function</c><00:35:45.160><c> and</c>"
+    },
+    {
+      "start": 2145.51,
+      "duration": 0.0,
+      "text": "terms of the joint score function and"
+    },
+    {
+      "start": 2145.52,
+      "duration": 0.0,
+      "text": "terms of the joint score function and the<00:35:45.760><c> plain</c><00:35:46.160><c> score</c><00:35:46.400><c> function.</c>"
+    },
+    {
+      "start": 2147.95,
+      "duration": 0.0,
+      "text": "the plain score function."
+    },
+    {
+      "start": 2147.96,
+      "duration": 0.0,
+      "text": "the plain score function. For<00:35:48.200><c> the</c><00:35:48.320><c> joint</c><00:35:48.760><c> score</c><00:35:49.000><c> function,</c><00:35:49.640><c> we</c><00:35:49.760><c> just</c>"
+    },
+    {
+      "start": 2150.43,
+      "duration": 0.0,
+      "text": "For the joint score function, we just"
+    },
+    {
+      "start": 2150.44,
+      "duration": 0.0,
+      "text": "For the joint score function, we just take<00:35:50.920><c> our</c><00:35:51.160><c> data</c><00:35:51.440><c> set.</c><00:35:52.000><c> We</c><00:35:52.160><c> use</c><00:35:52.720><c> a</c><00:35:52.840><c> delay</c>"
+    },
+    {
+      "start": 2153.11,
+      "duration": 0.0,
+      "text": "take our data set. We use a delay"
+    },
+    {
+      "start": 2153.12,
+      "duration": 0.0,
+      "text": "take our data set. We use a delay embedding"
+    },
+    {
+      "start": 2154.91,
+      "duration": 0.0,
+      "text": "embedding"
+    },
+    {
+      "start": 2154.92,
+      "duration": 0.0,
+      "text": "embedding in<00:35:55.120><c> order</c><00:35:55.600><c> to</c><00:35:56.440><c> build</c>"
+    },
+    {
+      "start": 2158.27,
+      "duration": 0.0,
+      "text": "in order to build"
+    },
+    {
+      "start": 2158.28,
+      "duration": 0.0,
+      "text": "in order to build a<00:35:59.080><c> time</c><00:35:59.400><c> series</c><00:36:00.400><c> of</c><00:36:00.720><c> X0</c><00:36:01.600><c> and</c><00:36:01.840><c> XT.</c>"
+    },
+    {
+      "start": 2162.91,
+      "duration": 0.0,
+      "text": "a time series of X0 and XT."
+    },
+    {
+      "start": 2162.92,
+      "duration": 0.0,
+      "text": "a time series of X0 and XT. We<00:36:03.080><c> do</c><00:36:03.280><c> it</c><00:36:03.560><c> for</c><00:36:03.800><c> different</c><00:36:04.320><c> value</c><00:36:04.880><c> of</c><00:36:05.000><c> the</c><00:36:05.160><c> time</c>"
+    },
+    {
+      "start": 2165.43,
+      "duration": 0.0,
+      "text": "We do it for different value of the time"
+    },
+    {
+      "start": 2165.44,
+      "duration": 0.0,
+      "text": "We do it for different value of the time delay<00:36:06.240><c> and</c><00:36:06.400><c> in</c><00:36:06.480><c> this</c><00:36:06.680><c> way</c><00:36:07.160><c> we</c><00:36:07.400><c> estimate</c><00:36:08.080><c> the</c>"
+    },
+    {
+      "start": 2168.19,
+      "duration": 0.0,
+      "text": "delay and in this way we estimate the"
+    },
+    {
+      "start": 2168.2,
+      "duration": 0.0,
+      "text": "delay and in this way we estimate the score<00:36:08.440><c> function</c><00:36:08.920><c> the</c><00:36:09.160><c> joint</c><00:36:09.880><c> score</c><00:36:10.120><c> function.</c>"
+    },
+    {
+      "start": 2171.39,
+      "duration": 0.0,
+      "text": "score function the joint score function."
+    },
+    {
+      "start": 2171.4,
+      "duration": 0.0,
+      "text": "score function the joint score function. And<00:36:11.560><c> then</c><00:36:11.920><c> we</c><00:36:12.040><c> can</c><00:36:12.240><c> combine</c><00:36:12.760><c> it</c><00:36:13.160><c> with</c><00:36:13.840><c> us</c><00:36:14.640><c> and</c>"
+    },
+    {
+      "start": 2174.79,
+      "duration": 0.0,
+      "text": "And then we can combine it with us and"
+    },
+    {
+      "start": 2174.8,
+      "duration": 0.0,
+      "text": "And then we can combine it with us and using<00:36:15.120><c> the</c><00:36:15.240><c> same</c><00:36:15.680><c> the</c><00:36:15.800><c> noise</c><00:36:16.160><c> score</c><00:36:16.359><c> matching</c>"
+    },
+    {
+      "start": 2176.75,
+      "duration": 0.0,
+      "text": "using the same the noise score matching"
+    },
+    {
+      "start": 2176.76,
+      "duration": 0.0,
+      "text": "using the same the noise score matching machinery<00:36:17.520><c> we</c><00:36:17.640><c> have</c><00:36:17.800><c> seen</c><00:36:17.960><c> before,</c><00:36:19.000><c> we</c><00:36:19.120><c> can</c>"
+    },
+    {
+      "start": 2179.31,
+      "duration": 0.0,
+      "text": "machinery we have seen before, we can"
+    },
+    {
+      "start": 2179.32,
+      "duration": 0.0,
+      "text": "machinery we have seen before, we can estimate<00:36:19.880><c> both</c>"
+    },
+    {
+      "start": 2181.19,
+      "duration": 0.0,
+      "text": "estimate both"
+    },
+    {
+      "start": 2181.2,
+      "duration": 0.0,
+      "text": "estimate both the<00:36:21.359><c> conditional</c><00:36:21.880><c> score</c><00:36:22.680><c> and</c><00:36:22.880><c> the</c><00:36:22.960><c> score</c>"
+    },
+    {
+      "start": 2183.19,
+      "duration": 0.0,
+      "text": "the conditional score and the score"
+    },
+    {
+      "start": 2183.2,
+      "duration": 0.0,
+      "text": "the conditional score and the score function<00:36:23.600><c> from</c><00:36:23.760><c> data.</c>"
+    },
+    {
+      "start": 2184.75,
+      "duration": 0.0,
+      "text": "function from data."
+    },
+    {
+      "start": 2184.76,
+      "duration": 0.0,
+      "text": "function from data. And<00:36:24.920><c> as</c><00:36:25.040><c> we</c><00:36:25.080><c> have</c><00:36:25.200><c> seen</c><00:36:25.359><c> before,</c><00:36:26.080><c> both</c><00:36:26.280><c> these</c>"
+    },
+    {
+      "start": 2186.47,
+      "duration": 0.0,
+      "text": "And as we have seen before, both these"
+    },
+    {
+      "start": 2186.48,
+      "duration": 0.0,
+      "text": "And as we have seen before, both these algorithms<00:36:27.440><c> scale</c><00:36:28.080><c> quite</c><00:36:28.280><c> well</c><00:36:28.600><c> with</c><00:36:28.720><c> the</c>"
+    },
+    {
+      "start": 2188.79,
+      "duration": 0.0,
+      "text": "algorithms scale quite well with the"
+    },
+    {
+      "start": 2188.8,
+      "duration": 0.0,
+      "text": "algorithms scale quite well with the dimension<00:36:29.240><c> of</c><00:36:29.359><c> the</c><00:36:29.440><c> system.</c>"
+    },
+    {
+      "start": 2190.83,
+      "duration": 0.0,
+      "text": "dimension of the system."
+    },
+    {
+      "start": 2190.84,
+      "duration": 0.0,
+      "text": "dimension of the system. So,<00:36:31.040><c> the</c><00:36:31.160><c> idea</c><00:36:31.440><c> here</c><00:36:32.359><c> is</c><00:36:32.560><c> then</c><00:36:32.960><c> to</c><00:36:33.120><c> use</c><00:36:33.560><c> those</c>"
+    },
+    {
+      "start": 2193.83,
+      "duration": 0.0,
+      "text": "So, the idea here is then to use those"
+    },
+    {
+      "start": 2193.84,
+      "duration": 0.0,
+      "text": "So, the idea here is then to use those two<00:36:33.960><c> quantities</c>"
+    },
+    {
+      "start": 2195.75,
+      "duration": 0.0,
+      "text": "two quantities"
+    },
+    {
+      "start": 2195.76,
+      "duration": 0.0,
+      "text": "two quantities where<00:36:36.520><c> the</c><00:36:36.800><c> first</c><00:36:37.120><c> quantity</c><00:36:37.600><c> essentially</c>"
+    },
+    {
+      "start": 2198.67,
+      "duration": 0.0,
+      "text": "where the first quantity essentially"
+    },
+    {
+      "start": 2198.68,
+      "duration": 0.0,
+      "text": "where the first quantity essentially takes<00:36:38.960><c> into</c><00:36:39.160><c> account</c><00:36:40.480><c> the</c><00:36:40.680><c> geometry</c><00:36:41.280><c> of</c><00:36:41.400><c> the</c>"
+    },
+    {
+      "start": 2201.47,
+      "duration": 0.0,
+      "text": "takes into account the geometry of the"
+    },
+    {
+      "start": 2201.48,
+      "duration": 0.0,
+      "text": "takes into account the geometry of the steady<00:36:41.760><c> state</c><00:36:42.120><c> distribution.</c>"
+    },
+    {
+      "start": 2203.83,
+      "duration": 0.0,
+      "text": "steady state distribution."
+    },
+    {
+      "start": 2203.84,
+      "duration": 0.0,
+      "text": "steady state distribution. Instead,<00:36:44.720><c> the</c><00:36:44.920><c> second</c><00:36:45.280><c> quantity</c><00:36:45.800><c> essentially</c>"
+    },
+    {
+      "start": 2206.59,
+      "duration": 0.0,
+      "text": "Instead, the second quantity essentially"
+    },
+    {
+      "start": 2206.6,
+      "duration": 0.0,
+      "text": "Instead, the second quantity essentially takes<00:36:46.880><c> into</c><00:36:47.080><c> account</c><00:36:48.160><c> how</c><00:36:48.640><c> the</c><00:36:48.840><c> system</c>"
+    },
+    {
+      "start": 2209.349,
+      "duration": 0.0,
+      "text": "takes into account how the system"
+    },
+    {
+      "start": 2209.359,
+      "duration": 0.0,
+      "text": "takes into account how the system relaxes<00:36:50.080><c> towards</c><00:36:50.640><c> the</c><00:36:50.760><c> steady</c><00:36:51.040><c> state</c>"
+    },
+    {
+      "start": 2211.39,
+      "duration": 0.0,
+      "text": "relaxes towards the steady state"
+    },
+    {
+      "start": 2211.4,
+      "duration": 0.0,
+      "text": "relaxes towards the steady state distribution.<00:36:52.240><c> So,</c><00:36:52.400><c> it's</c><00:36:52.520><c> carrying</c>"
+    },
+    {
+      "start": 2212.99,
+      "duration": 0.0,
+      "text": "distribution. So, it's carrying"
+    },
+    {
+      "start": 2213.0,
+      "duration": 0.0,
+      "text": "distribution. So, it's carrying information<00:36:53.680><c> also</c><00:36:53.920><c> about</c><00:36:54.400><c> the</c>"
+    },
+    {
+      "start": 2215.03,
+      "duration": 0.0,
+      "text": "information also about the"
+    },
+    {
+      "start": 2215.04,
+      "duration": 0.0,
+      "text": "information also about the the<00:36:55.160><c> dynamics</c><00:36:55.720><c> of</c><00:36:55.800><c> the</c><00:36:55.920><c> system</c><00:36:56.720><c> and</c><00:36:56.920><c> not</c><00:36:57.160><c> only</c>"
+    },
+    {
+      "start": 2217.349,
+      "duration": 0.0,
+      "text": "the dynamics of the system and not only"
+    },
+    {
+      "start": 2217.359,
+      "duration": 0.0,
+      "text": "the dynamics of the system and not only about<00:36:57.760><c> the</c><00:36:57.880><c> statistics.</c>"
+    },
+    {
+      "start": 2220.72,
+      "duration": 0.0,
+      "text": "So,<00:37:00.920><c> this</c><00:37:01.120><c> is</c><00:37:01.280><c> the</c><00:37:01.440><c> intuition.</c><00:37:02.240><c> So,</c><00:37:03.160><c> try</c><00:37:03.400><c> to</c>"
+    },
+    {
+      "start": 2223.55,
+      "duration": 0.0,
+      "text": "So, this is the intuition. So, try to"
+    },
+    {
+      "start": 2223.56,
+      "duration": 0.0,
+      "text": "So, this is the intuition. So, try to use<00:37:03.840><c> those</c><00:37:04.200><c> two</c><00:37:04.960><c> quantities</c><00:37:05.840><c> that</c><00:37:06.280><c> can</c><00:37:06.480><c> be</c>"
+    },
+    {
+      "start": 2226.59,
+      "duration": 0.0,
+      "text": "use those two quantities that can be"
+    },
+    {
+      "start": 2226.6,
+      "duration": 0.0,
+      "text": "use those two quantities that can be evaluated<00:37:07.600><c> quite</c><00:37:07.880><c> well</c><00:37:08.120><c> also</c><00:37:08.400><c> for</c><00:37:08.560><c> very</c><00:37:08.800><c> high</c>"
+    },
+    {
+      "start": 2228.95,
+      "duration": 0.0,
+      "text": "evaluated quite well also for very high"
+    },
+    {
+      "start": 2228.96,
+      "duration": 0.0,
+      "text": "evaluated quite well also for very high dimensional<00:37:09.480><c> systems</c>"
+    },
+    {
+      "start": 2230.95,
+      "duration": 0.0,
+      "text": "dimensional systems"
+    },
+    {
+      "start": 2230.96,
+      "duration": 0.0,
+      "text": "dimensional systems to<00:37:11.280><c> build</c><00:37:11.760><c> our</c>"
+    },
+    {
+      "start": 2233.03,
+      "duration": 0.0,
+      "text": "to build our"
+    },
+    {
+      "start": 2233.04,
+      "duration": 0.0,
+      "text": "to build our stochastic<00:37:13.560><c> modeling</c><00:37:14.520><c> approach.</c>"
+    },
+    {
+      "start": 2236.15,
+      "duration": 0.0,
+      "text": "stochastic modeling approach."
+    },
+    {
+      "start": 2236.16,
+      "duration": 0.0,
+      "text": "stochastic modeling approach. So,<00:37:16.280><c> let's</c><00:37:16.480><c> start</c><00:37:17.000><c> from</c><00:37:17.560><c> our</c>"
+    },
+    {
+      "start": 2239.15,
+      "duration": 0.0,
+      "text": "So, let's start from our"
+    },
+    {
+      "start": 2239.16,
+      "duration": 0.0,
+      "text": "So, let's start from our Langevin<00:37:19.640><c> equation.</c><00:37:20.280><c> So,</c><00:37:20.480><c> this</c><00:37:20.640><c> is</c><00:37:21.000><c> the</c><00:37:21.120><c> same</c>"
+    },
+    {
+      "start": 2241.43,
+      "duration": 0.0,
+      "text": "Langevin equation. So, this is the same"
+    },
+    {
+      "start": 2241.44,
+      "duration": 0.0,
+      "text": "Langevin equation. So, this is the same Langevin<00:37:21.840><c> equation</c><00:37:22.400><c> I</c><00:37:22.480><c> wrote</c><00:37:22.760><c> at</c><00:37:22.840><c> the</c>"
+    },
+    {
+      "start": 2242.91,
+      "duration": 0.0,
+      "text": "Langevin equation I wrote at the"
+    },
+    {
+      "start": 2242.92,
+      "duration": 0.0,
+      "text": "Langevin equation I wrote at the beginning.<00:37:23.880><c> Yeah,</c><00:37:24.200><c> just</c><00:37:24.480><c> have</c><00:37:24.760><c> here</c><00:37:25.280><c> a</c><00:37:25.359><c> factor</c>"
+    },
+    {
+      "start": 2245.71,
+      "duration": 0.0,
+      "text": "beginning. Yeah, just have here a factor"
+    },
+    {
+      "start": 2245.72,
+      "duration": 0.0,
+      "text": "beginning. Yeah, just have here a factor square<00:37:25.920><c> root</c><00:37:26.120><c> of</c><00:37:26.240><c> two.</c>"
+    },
+    {
+      "start": 2248.07,
+      "duration": 0.0,
+      "text": "square root of two."
+    },
+    {
+      "start": 2248.08,
+      "duration": 0.0,
+      "text": "square root of two. And<00:37:28.520><c> then</c><00:37:28.600><c> let's</c><00:37:28.840><c> first</c><00:37:29.520><c> impose</c>"
+    },
+    {
+      "start": 2250.75,
+      "duration": 0.0,
+      "text": "And then let's first impose"
+    },
+    {
+      "start": 2250.76,
+      "duration": 0.0,
+      "text": "And then let's first impose stationarity.<00:37:31.880><c> So,</c><00:37:32.040><c> we</c><00:37:32.200><c> want</c><00:37:33.240><c> for</c><00:37:33.440><c> a</c><00:37:33.560><c> given</c>"
+    },
+    {
+      "start": 2254.83,
+      "duration": 0.0,
+      "text": "stationarity. So, we want for a given"
+    },
+    {
+      "start": 2254.84,
+      "duration": 0.0,
+      "text": "stationarity. So, we want for a given sigma<00:37:35.480><c> X</c><00:37:36.080><c> to</c><00:37:36.240><c> find</c><00:37:37.200><c> our</c><00:37:37.520><c> drift</c><00:37:37.880><c> term</c><00:37:38.359><c> F</c>"
+    },
+    {
+      "start": 2259.43,
+      "duration": 0.0,
+      "text": "sigma X to find our drift term F"
+    },
+    {
+      "start": 2259.44,
+      "duration": 0.0,
+      "text": "sigma X to find our drift term F such<00:37:39.720><c> that</c><00:37:40.200><c> by</c><00:37:40.440><c> construction</c>"
+    },
+    {
+      "start": 2262.23,
+      "duration": 0.0,
+      "text": "such that by construction"
+    },
+    {
+      "start": 2262.24,
+      "duration": 0.0,
+      "text": "such that by construction reproduces<00:37:43.280><c> the</c><00:37:43.440><c> steady</c><00:37:43.720><c> state</c>"
+    },
+    {
+      "start": 2264.07,
+      "duration": 0.0,
+      "text": "reproduces the steady state"
+    },
+    {
+      "start": 2264.08,
+      "duration": 0.0,
+      "text": "reproduces the steady state distribution."
+    },
+    {
+      "start": 2265.47,
+      "duration": 0.0,
+      "text": "distribution."
+    },
+    {
+      "start": 2265.48,
+      "duration": 0.0,
+      "text": "distribution. And<00:37:45.720><c> to</c><00:37:45.800><c> do</c><00:37:45.960><c> that,</c><00:37:46.560><c> we</c><00:37:46.720><c> can</c><00:37:47.080><c> write</c><00:37:47.520><c> the</c>"
+    },
+    {
+      "start": 2267.63,
+      "duration": 0.0,
+      "text": "And to do that, we can write the"
+    },
+    {
+      "start": 2267.64,
+      "duration": 0.0,
+      "text": "And to do that, we can write the Fokker-Planck<00:37:48.200><c> equation</c><00:37:49.000><c> relative</c><00:37:49.520><c> to</c><00:37:49.640><c> the</c>"
+    },
+    {
+      "start": 2270.55,
+      "duration": 0.0,
+      "text": "Fokker-Planck equation relative to the"
+    },
+    {
+      "start": 2270.56,
+      "duration": 0.0,
+      "text": "Fokker-Planck equation relative to the Langevin<00:37:50.911><c> [clears throat]</c><00:37:51.000><c> equation,</c>"
+    },
+    {
+      "start": 2271.83,
+      "duration": 0.0,
+      "text": "Langevin [clears throat] equation,"
+    },
+    {
+      "start": 2271.84,
+      "duration": 0.0,
+      "text": "Langevin [clears throat] equation, impose<00:37:52.359><c> the</c><00:37:52.720><c> stationarity</c>"
+    },
+    {
+      "start": 2274.15,
+      "duration": 0.0,
+      "text": "impose the stationarity"
+    },
+    {
+      "start": 2274.16,
+      "duration": 0.0,
+      "text": "impose the stationarity and<00:37:54.359><c> we</c><00:37:54.480><c> can</c><00:37:54.680><c> show</c><00:37:55.080><c> that</c><00:37:55.800><c> without</c><00:37:56.440><c> losing</c><00:37:56.800><c> any</c>"
+    },
+    {
+      "start": 2276.91,
+      "duration": 0.0,
+      "text": "and we can show that without losing any"
+    },
+    {
+      "start": 2276.92,
+      "duration": 0.0,
+      "text": "and we can show that without losing any generality"
+    },
+    {
+      "start": 2278.27,
+      "duration": 0.0,
+      "text": "generality"
+    },
+    {
+      "start": 2278.28,
+      "duration": 0.0,
+      "text": "generality we<00:37:58.520><c> can</c><00:37:59.200><c> write</c>"
+    },
+    {
+      "start": 2280.47,
+      "duration": 0.0,
+      "text": "we can write"
+    },
+    {
+      "start": 2280.48,
+      "duration": 0.0,
+      "text": "we can write the<00:38:01.120><c> drift</c><00:38:01.480><c> term</c>"
+    },
+    {
+      "start": 2282.55,
+      "duration": 0.0,
+      "text": "the drift term"
+    },
+    {
+      "start": 2282.56,
+      "duration": 0.0,
+      "text": "the drift term in<00:38:03.200><c> this</c><00:38:03.440><c> way.</c><00:38:04.280><c> So,</c><00:38:04.960><c> in</c><00:38:05.160><c> terms</c><00:38:05.920><c> of</c><00:38:06.520><c> the</c><00:38:06.680><c> score</c>"
+    },
+    {
+      "start": 2286.91,
+      "duration": 0.0,
+      "text": "in this way. So, in terms of the score"
+    },
+    {
+      "start": 2286.92,
+      "duration": 0.0,
+      "text": "in this way. So, in terms of the score function<00:38:07.600><c> that</c><00:38:07.760><c> we</c><00:38:07.840><c> defined</c><00:38:08.280><c> before</c>"
+    },
+    {
+      "start": 2289.55,
+      "duration": 0.0,
+      "text": "function that we defined before"
+    },
+    {
+      "start": 2289.56,
+      "duration": 0.0,
+      "text": "function that we defined before and<00:38:10.600><c> the</c><00:38:10.920><c> diffusion</c><00:38:11.440><c> matrix.</c><00:38:12.040><c> So,</c><00:38:12.280><c> this</c>"
+    },
+    {
+      "start": 2292.99,
+      "duration": 0.0,
+      "text": "and the diffusion matrix. So, this"
+    },
+    {
+      "start": 2293.0,
+      "duration": 0.0,
+      "text": "and the diffusion matrix. So, this symmetric<00:38:13.680><c> matrix</c><00:38:14.520><c> D</c><00:38:14.680><c> of</c><00:38:14.880><c> X</c>"
+    },
+    {
+      "start": 2295.87,
+      "duration": 0.0,
+      "text": "symmetric matrix D of X"
+    },
+    {
+      "start": 2295.88,
+      "duration": 0.0,
+      "text": "symmetric matrix D of X and<00:38:16.880><c> another</c><00:38:17.760><c> anti-symmetric</c><00:38:18.600><c> matrix</c><00:38:19.400><c> R</c><00:38:19.640><c> of</c>"
+    },
+    {
+      "start": 2299.75,
+      "duration": 0.0,
+      "text": "and another anti-symmetric matrix R of"
+    },
+    {
+      "start": 2299.76,
+      "duration": 0.0,
+      "text": "and another anti-symmetric matrix R of X."
+    },
+    {
+      "start": 2301.15,
+      "duration": 0.0,
+      "text": "X."
+    },
+    {
+      "start": 2301.16,
+      "duration": 0.0,
+      "text": "X. So,<00:38:21.359><c> this</c><00:38:21.520><c> is</c><00:38:21.640><c> a</c><00:38:21.680><c> very</c><00:38:21.920><c> general</c>"
+    },
+    {
+      "start": 2303.63,
+      "duration": 0.0,
+      "text": "So, this is a very general"
+    },
+    {
+      "start": 2303.64,
+      "duration": 0.0,
+      "text": "So, this is a very general expression.<00:38:24.400><c> We're</c><00:38:24.560><c> not</c><00:38:24.720><c> doing</c><00:38:24.960><c> any</c>"
+    },
+    {
+      "start": 2305.67,
+      "duration": 0.0,
+      "text": "expression. We're not doing any"
+    },
+    {
+      "start": 2305.68,
+      "duration": 0.0,
+      "text": "expression. We're not doing any approximation<00:38:26.560><c> here.</c><00:38:27.480><c> We</c><00:38:27.640><c> are</c><00:38:28.280><c> We</c><00:38:28.400><c> are</c><00:38:28.520><c> just</c>"
+    },
+    {
+      "start": 2308.95,
+      "duration": 0.0,
+      "text": "approximation here. We are We are just"
+    },
+    {
+      "start": 2308.96,
+      "duration": 0.0,
+      "text": "approximation here. We are We are just finding<00:38:29.480><c> the</c><00:38:29.600><c> most</c><00:38:29.840><c> general</c><00:38:30.200><c> way</c><00:38:30.680><c> to</c><00:38:30.760><c> express</c>"
+    },
+    {
+      "start": 2311.349,
+      "duration": 0.0,
+      "text": "finding the most general way to express"
+    },
+    {
+      "start": 2311.359,
+      "duration": 0.0,
+      "text": "finding the most general way to express the<00:38:31.520><c> drift</c><00:38:32.120><c> for</c><00:38:32.320><c> a</c><00:38:32.400><c> given</c><00:38:33.359><c> the</c><00:38:33.640><c> diffusion</c><00:38:34.920><c> in</c>"
+    },
+    {
+      "start": 2315.03,
+      "duration": 0.0,
+      "text": "the drift for a given the diffusion in"
+    },
+    {
+      "start": 2315.04,
+      "duration": 0.0,
+      "text": "the drift for a given the diffusion in such<00:38:35.280><c> a</c><00:38:35.320><c> way</c><00:38:35.720><c> that</c><00:38:35.960><c> it</c><00:38:36.120><c> reproduces</c><00:38:37.120><c> the</c><00:38:37.280><c> steady</c>"
+    },
+    {
+      "start": 2317.59,
+      "duration": 0.0,
+      "text": "such a way that it reproduces the steady"
+    },
+    {
+      "start": 2317.6,
+      "duration": 0.0,
+      "text": "such a way that it reproduces the steady state<00:38:38.000><c> distribution</c><00:38:38.800><c> by</c><00:38:38.920><c> construction,</c>"
+    },
+    {
+      "start": 2319.91,
+      "duration": 0.0,
+      "text": "state distribution by construction,"
+    },
+    {
+      "start": 2319.92,
+      "duration": 0.0,
+      "text": "state distribution by construction, which<00:38:40.120><c> essentially</c><00:38:40.600><c> means</c><00:38:41.560><c> in</c><00:38:41.720><c> such</c><00:38:41.880><c> a</c><00:38:41.960><c> way</c>"
+    },
+    {
+      "start": 2322.47,
+      "duration": 0.0,
+      "text": "which essentially means in such a way"
+    },
+    {
+      "start": 2322.48,
+      "duration": 0.0,
+      "text": "which essentially means in such a way that<00:38:42.800><c> F</c><00:38:43.480><c> So,</c><00:38:43.640><c> this</c><00:38:43.800><c> specific</c><00:38:44.280><c> shape</c><00:38:45.000><c> of</c><00:38:45.240><c> F</c>"
+    },
+    {
+      "start": 2326.31,
+      "duration": 0.0,
+      "text": "that F So, this specific shape of F"
+    },
+    {
+      "start": 2326.32,
+      "duration": 0.0,
+      "text": "that F So, this specific shape of F solve<00:38:46.920><c> the</c><00:38:47.040><c> stationary</c><00:38:48.080><c> Fokker-Planck</c>"
+    },
+    {
+      "start": 2328.63,
+      "duration": 0.0,
+      "text": "solve the stationary Fokker-Planck"
+    },
+    {
+      "start": 2328.64,
+      "duration": 0.0,
+      "text": "solve the stationary Fokker-Planck equation."
+    },
+    {
+      "start": 2330.67,
+      "duration": 0.0,
+      "text": "equation."
+    },
+    {
+      "start": 2330.68,
+      "duration": 0.0,
+      "text": "equation. Now,<00:38:51.359><c> we</c><00:38:51.520><c> can</c><00:38:52.320><c> So,</c><00:38:52.480><c> we</c><00:38:52.600><c> can</c><00:38:52.960><c> see</c><00:38:53.240><c> that</c><00:38:53.760><c> we</c><00:38:53.920><c> have</c>"
+    },
+    {
+      "start": 2334.43,
+      "duration": 0.0,
+      "text": "Now, we can So, we can see that we have"
+    },
+    {
+      "start": 2334.44,
+      "duration": 0.0,
+      "text": "Now, we can So, we can see that we have two<00:38:54.640><c> different</c><00:38:55.680><c> tensors</c><00:38:56.640><c> D</c><00:38:56.920><c> of</c><00:38:57.160><c> X</c><00:38:57.840><c> and</c><00:38:58.040><c> R</c><00:38:58.160><c> of</c><00:38:58.320><c> X.</c>"
+    },
+    {
+      "start": 2338.87,
+      "duration": 0.0,
+      "text": "two different tensors D of X and R of X."
+    },
+    {
+      "start": 2338.88,
+      "duration": 0.0,
+      "text": "two different tensors D of X and R of X. D<00:38:59.000><c> of</c><00:38:59.160><c> X</c><00:38:59.720><c> is</c><00:38:59.880><c> symmetric</c><00:39:01.200><c> and</c><00:39:01.680><c> represent</c><00:39:02.600><c> the</c>"
+    },
+    {
+      "start": 2342.71,
+      "duration": 0.0,
+      "text": "D of X is symmetric and represent the"
+    },
+    {
+      "start": 2342.72,
+      "duration": 0.0,
+      "text": "D of X is symmetric and represent the diffusion<00:39:03.160><c> tensor.</c>"
+    },
+    {
+      "start": 2344.71,
+      "duration": 0.0,
+      "text": "diffusion tensor."
+    },
+    {
+      "start": 2344.72,
+      "duration": 0.0,
+      "text": "diffusion tensor. Instead,<00:39:05.359><c> R</c><00:39:05.840><c> which</c><00:39:06.040><c> is</c><00:39:06.160><c> the</c><00:39:06.280><c> anti-symmetric</c>"
+    },
+    {
+      "start": 2347.07,
+      "duration": 0.0,
+      "text": "Instead, R which is the anti-symmetric"
+    },
+    {
+      "start": 2347.08,
+      "duration": 0.0,
+      "text": "Instead, R which is the anti-symmetric part<00:39:07.560><c> can</c><00:39:07.720><c> be</c><00:39:07.840><c> interpreted</c><00:39:09.040><c> as</c><00:39:09.520><c> the</c><00:39:09.680><c> term</c><00:39:10.480><c> that</c>"
+    },
+    {
+      "start": 2350.91,
+      "duration": 0.0,
+      "text": "part can be interpreted as the term that"
+    },
+    {
+      "start": 2350.92,
+      "duration": 0.0,
+      "text": "part can be interpreted as the term that breaks<00:39:11.520><c> the</c><00:39:11.640><c> tail</c><00:39:11.880><c> balance</c><00:39:12.920><c> and</c><00:39:13.560><c> that</c>"
+    },
+    {
+      "start": 2353.71,
+      "duration": 0.0,
+      "text": "breaks the tail balance and that"
+    },
+    {
+      "start": 2353.72,
+      "duration": 0.0,
+      "text": "breaks the tail balance and that introduces<00:39:14.760><c> some</c><00:39:15.120><c> rotational</c><00:39:15.760><c> component</c><00:39:17.120><c> to</c>"
+    },
+    {
+      "start": 2357.27,
+      "duration": 0.0,
+      "text": "introduces some rotational component to"
+    },
+    {
+      "start": 2357.28,
+      "duration": 0.0,
+      "text": "introduces some rotational component to our<00:39:17.520><c> system</c><00:39:18.120><c> without</c><00:39:18.600><c> changing</c><00:39:19.280><c> the</c><00:39:19.400><c> steady</c>"
+    },
+    {
+      "start": 2359.63,
+      "duration": 0.0,
+      "text": "our system without changing the steady"
+    },
+    {
+      "start": 2359.64,
+      "duration": 0.0,
+      "text": "our system without changing the steady state<00:39:20.320><c> distribution.</c><00:39:21.240><c> So,</c><00:39:21.440><c> this</c><00:39:22.080><c> can</c><00:39:22.240><c> be</c>"
+    },
+    {
+      "start": 2362.349,
+      "duration": 0.0,
+      "text": "state distribution. So, this can be"
+    },
+    {
+      "start": 2362.359,
+      "duration": 0.0,
+      "text": "state distribution. So, this can be related<00:39:23.280><c> to</c><00:39:23.440><c> an</c><00:39:23.640><c> Helmholtz</c><00:39:24.160><c> decomposition</c><00:39:25.400><c> of</c>"
+    },
+    {
+      "start": 2365.83,
+      "duration": 0.0,
+      "text": "related to an Helmholtz decomposition of"
+    },
+    {
+      "start": 2365.84,
+      "duration": 0.0,
+      "text": "related to an Helmholtz decomposition of the<00:39:26.720><c> drift</c><00:39:27.120><c> term.</c><00:39:27.680><c> So,</c><00:39:27.800><c> we</c><00:39:27.920><c> have</c><00:39:28.480><c> a</c>"
+    },
+    {
+      "start": 2369.19,
+      "duration": 0.0,
+      "text": "the drift term. So, we have a"
+    },
+    {
+      "start": 2369.2,
+      "duration": 0.0,
+      "text": "the drift term. So, we have a a<00:39:29.320><c> term</c><00:39:29.920><c> a</c><00:39:29.960><c> symmetric</c><00:39:30.480><c> term</c><00:39:31.120><c> which</c>"
+    },
+    {
+      "start": 2371.27,
+      "duration": 0.0,
+      "text": "a term a symmetric term which"
+    },
+    {
+      "start": 2371.28,
+      "duration": 0.0,
+      "text": "a term a symmetric term which essentially"
+    },
+    {
+      "start": 2373.15,
+      "duration": 0.0,
+      "text": "essentially"
+    },
+    {
+      "start": 2373.16,
+      "duration": 0.0,
+      "text": "essentially satisfies<00:39:33.960><c> the</c><00:39:34.040><c> detail</c><00:39:34.359><c> balance</c><00:39:34.920><c> and</c><00:39:35.120><c> give</c><00:39:35.359><c> us</c>"
+    },
+    {
+      "start": 2375.51,
+      "duration": 0.0,
+      "text": "satisfies the detail balance and give us"
+    },
+    {
+      "start": 2375.52,
+      "duration": 0.0,
+      "text": "satisfies the detail balance and give us a<00:39:35.600><c> system</c><00:39:36.400><c> which</c><00:39:36.600><c> is</c><00:39:36.760><c> just</c><00:39:37.080><c> a</c><00:39:37.120><c> Brownian</c><00:39:37.520><c> motion</c>"
+    },
+    {
+      "start": 2377.99,
+      "duration": 0.0,
+      "text": "a system which is just a Brownian motion"
+    },
+    {
+      "start": 2378.0,
+      "duration": 0.0,
+      "text": "a system which is just a Brownian motion inside<00:39:38.960><c> a</c><00:39:39.040><c> potential.</c><00:39:40.200><c> And</c><00:39:40.359><c> then</c><00:39:40.560><c> we</c><00:39:40.680><c> have</c>"
+    },
+    {
+      "start": 2381.11,
+      "duration": 0.0,
+      "text": "inside a potential. And then we have"
+    },
+    {
+      "start": 2381.12,
+      "duration": 0.0,
+      "text": "inside a potential. And then we have this<00:39:41.400><c> other</c><00:39:42.360><c> circulatory</c><00:39:43.800><c> term</c><00:39:44.480><c> which</c>"
+    },
+    {
+      "start": 2384.67,
+      "duration": 0.0,
+      "text": "this other circulatory term which"
+    },
+    {
+      "start": 2384.68,
+      "duration": 0.0,
+      "text": "this other circulatory term which introduces<00:39:45.520><c> some</c><00:39:45.800><c> rotational</c><00:39:46.440><c> component</c>"
+    },
+    {
+      "start": 2387.15,
+      "duration": 0.0,
+      "text": "introduces some rotational component"
+    },
+    {
+      "start": 2387.16,
+      "duration": 0.0,
+      "text": "introduces some rotational component that<00:39:47.360><c> breaks</c><00:39:48.240><c> detailed</c><00:39:48.600><c> balance.</c>"
+    },
+    {
+      "start": 2389.75,
+      "duration": 0.0,
+      "text": "that breaks detailed balance."
+    },
+    {
+      "start": 2389.76,
+      "duration": 0.0,
+      "text": "that breaks detailed balance. Okay,<00:39:50.160><c> so</c><00:39:50.360><c> now</c><00:39:51.000><c> by</c><00:39:51.200><c> using</c><00:39:51.720><c> this</c><00:39:52.160><c> expression</c>"
+    },
+    {
+      "start": 2392.75,
+      "duration": 0.0,
+      "text": "Okay, so now by using this expression"
+    },
+    {
+      "start": 2392.76,
+      "duration": 0.0,
+      "text": "Okay, so now by using this expression here<00:39:53.480><c> for</c><00:39:54.320><c> the</c><00:39:54.920><c> drift</c><00:39:55.280><c> term,</c><00:39:56.280><c> we</c><00:39:56.480><c> are</c>"
+    },
+    {
+      "start": 2396.75,
+      "duration": 0.0,
+      "text": "here for the drift term, we are"
+    },
+    {
+      "start": 2396.76,
+      "duration": 0.0,
+      "text": "here for the drift term, we are guaranteed"
+    },
+    {
+      "start": 2398.31,
+      "duration": 0.0,
+      "text": "guaranteed"
+    },
+    {
+      "start": 2398.32,
+      "duration": 0.0,
+      "text": "guaranteed to<00:39:58.800><c> recover</c><00:39:59.400><c> the</c><00:39:59.520><c> steady</c><00:39:59.760><c> state</c>"
+    },
+    {
+      "start": 2400.11,
+      "duration": 0.0,
+      "text": "to recover the steady state"
+    },
+    {
+      "start": 2400.12,
+      "duration": 0.0,
+      "text": "to recover the steady state distribution.<00:40:01.360><c> So</c><00:40:01.640><c> we</c>"
+    },
+    {
+      "start": 2402.63,
+      "duration": 0.0,
+      "text": "distribution. So we"
+    },
+    {
+      "start": 2402.64,
+      "duration": 0.0,
+      "text": "distribution. So we achieved<00:40:03.360><c> the</c><00:40:03.520><c> first</c><00:40:03.840><c> goal</c><00:40:04.560><c> of</c><00:40:04.800><c> our</c><00:40:05.000><c> modeling</c>"
+    },
+    {
+      "start": 2405.39,
+      "duration": 0.0,
+      "text": "achieved the first goal of our modeling"
+    },
+    {
+      "start": 2405.4,
+      "duration": 0.0,
+      "text": "achieved the first goal of our modeling strategy<00:40:05.920><c> which</c><00:40:06.160><c> is</c><00:40:06.760><c> to</c><00:40:06.920><c> build</c><00:40:07.160><c> a</c><00:40:07.200><c> stochastic</c>"
+    },
+    {
+      "start": 2407.71,
+      "duration": 0.0,
+      "text": "strategy which is to build a stochastic"
+    },
+    {
+      "start": 2407.72,
+      "duration": 0.0,
+      "text": "strategy which is to build a stochastic model<00:40:08.200><c> that</c><00:40:08.440><c> by</c><00:40:08.560><c> construction</c><00:40:09.720><c> reproduces</c>"
+    },
+    {
+      "start": 2410.63,
+      "duration": 0.0,
+      "text": "model that by construction reproduces"
+    },
+    {
+      "start": 2410.64,
+      "duration": 0.0,
+      "text": "model that by construction reproduces the<00:40:10.760><c> observed</c><00:40:11.360><c> the</c><00:40:11.440><c> steady</c><00:40:11.640><c> state</c>"
+    },
+    {
+      "start": 2412.47,
+      "duration": 0.0,
+      "text": "the observed the steady state"
+    },
+    {
+      "start": 2412.48,
+      "duration": 0.0,
+      "text": "the observed the steady state distribution<00:40:13.160><c> of</c><00:40:13.240><c> the</c><00:40:13.360><c> data</c><00:40:13.640><c> set.</c><00:40:14.480><c> Now</c><00:40:14.800><c> let's</c>"
+    },
+    {
+      "start": 2415.15,
+      "duration": 0.0,
+      "text": "distribution of the data set. Now let's"
+    },
+    {
+      "start": 2415.16,
+      "duration": 0.0,
+      "text": "distribution of the data set. Now let's try<00:40:15.520><c> to</c><00:40:15.680><c> impose</c><00:40:16.400><c> also</c><00:40:16.760><c> the</c><00:40:16.880><c> second</c><00:40:17.840><c> constraint</c>"
+    },
+    {
+      "start": 2418.47,
+      "duration": 0.0,
+      "text": "try to impose also the second constraint"
+    },
+    {
+      "start": 2418.48,
+      "duration": 0.0,
+      "text": "try to impose also the second constraint which<00:40:18.720><c> is</c>"
+    },
+    {
+      "start": 2419.75,
+      "duration": 0.0,
+      "text": "which is"
+    },
+    {
+      "start": 2419.76,
+      "duration": 0.0,
+      "text": "which is we<00:40:19.880><c> want</c><00:40:20.360><c> to</c><00:40:20.560><c> reproduce</c><00:40:21.160><c> also</c><00:40:21.600><c> the</c><00:40:22.120><c> the</c><00:40:22.200><c> time</c>"
+    },
+    {
+      "start": 2422.47,
+      "duration": 0.0,
+      "text": "we want to reproduce also the the time"
+    },
+    {
+      "start": 2422.48,
+      "duration": 0.0,
+      "text": "we want to reproduce also the the time correlations."
+    },
+    {
+      "start": 2425.07,
+      "duration": 0.0,
+      "text": "correlations."
+    },
+    {
+      "start": 2425.08,
+      "duration": 0.0,
+      "text": "correlations. So<00:40:25.320><c> we</c><00:40:25.440><c> would</c><00:40:25.720><c> like</c><00:40:26.280><c> to</c>"
+    },
+    {
+      "start": 2427.35,
+      "duration": 0.0,
+      "text": "So we would like to"
+    },
+    {
+      "start": 2427.36,
+      "duration": 0.0,
+      "text": "So we would like to reproduce<00:40:28.520><c> this</c><00:40:29.200><c> time</c><00:40:29.520><c> correlations</c><00:40:30.440><c> for</c><00:40:30.920><c> a</c>"
+    },
+    {
+      "start": 2430.99,
+      "duration": 0.0,
+      "text": "reproduce this time correlations for a"
+    },
+    {
+      "start": 2431.0,
+      "duration": 0.0,
+      "text": "reproduce this time correlations for a set<00:40:31.600><c> of</c><00:40:31.800><c> observables</c><00:40:32.760><c> phi</c><00:40:33.000><c> n.</c>"
+    },
+    {
+      "start": 2434.23,
+      "duration": 0.0,
+      "text": "set of observables phi n."
+    },
+    {
+      "start": 2434.24,
+      "duration": 0.0,
+      "text": "set of observables phi n. So<00:40:34.480><c> without</c><00:40:35.160><c> going</c><00:40:35.480><c> into</c><00:40:35.720><c> the</c><00:40:35.800><c> mathematical</c>"
+    },
+    {
+      "start": 2436.31,
+      "duration": 0.0,
+      "text": "So without going into the mathematical"
+    },
+    {
+      "start": 2436.32,
+      "duration": 0.0,
+      "text": "So without going into the mathematical details<00:40:37.080><c> of</c><00:40:37.200><c> this</c><00:40:37.360><c> derivation,</c><00:40:38.120><c> we</c><00:40:38.240><c> can</c><00:40:38.480><c> show</c>"
+    },
+    {
+      "start": 2439.07,
+      "duration": 0.0,
+      "text": "details of this derivation, we can show"
+    },
+    {
+      "start": 2439.08,
+      "duration": 0.0,
+      "text": "details of this derivation, we can show that"
+    },
+    {
+      "start": 2440.23,
+      "duration": 0.0,
+      "text": "that"
+    },
+    {
+      "start": 2440.24,
+      "duration": 0.0,
+      "text": "that the<00:40:40.520><c> time</c><00:40:40.840><c> derivative</c><00:40:42.160><c> of</c><00:40:42.520><c> this</c><00:40:42.760><c> correlation</c>"
+    },
+    {
+      "start": 2443.31,
+      "duration": 0.0,
+      "text": "the time derivative of this correlation"
+    },
+    {
+      "start": 2443.32,
+      "duration": 0.0,
+      "text": "the time derivative of this correlation function<00:40:44.120><c> for</c><00:40:44.760><c> this</c><00:40:45.440><c> specific</c><00:40:46.080><c> model,</c><00:40:47.000><c> so</c><00:40:47.200><c> for</c>"
+    },
+    {
+      "start": 2447.51,
+      "duration": 0.0,
+      "text": "function for this specific model, so for"
+    },
+    {
+      "start": 2447.52,
+      "duration": 0.0,
+      "text": "function for this specific model, so for this<00:40:47.720><c> specific</c><00:40:48.240><c> Langevin</c><00:40:48.680><c> equation</c><00:40:49.440><c> with</c>"
+    },
+    {
+      "start": 2449.87,
+      "duration": 0.0,
+      "text": "this specific Langevin equation with"
+    },
+    {
+      "start": 2449.88,
+      "duration": 0.0,
+      "text": "this specific Langevin equation with drift<00:40:50.200><c> term</c><00:40:51.120><c> given</c><00:40:51.600><c> by</c><00:40:51.960><c> this</c><00:40:52.200><c> expression</c><00:40:52.760><c> over</c>"
+    },
+    {
+      "start": 2452.87,
+      "duration": 0.0,
+      "text": "drift term given by this expression over"
+    },
+    {
+      "start": 2452.88,
+      "duration": 0.0,
+      "text": "drift term given by this expression over here,"
+    },
+    {
+      "start": 2454.19,
+      "duration": 0.0,
+      "text": "here,"
+    },
+    {
+      "start": 2454.2,
+      "duration": 0.0,
+      "text": "here, can<00:40:54.440><c> be</c><00:40:54.560><c> written</c><00:40:55.200><c> in</c><00:40:55.320><c> this</c><00:40:55.480><c> way.</c>"
+    },
+    {
+      "start": 2456.99,
+      "duration": 0.0,
+      "text": "can be written in this way."
+    },
+    {
+      "start": 2457.0,
+      "duration": 0.0,
+      "text": "can be written in this way. So<00:40:57.200><c> essentially</c><00:40:57.600><c> we</c><00:40:57.720><c> can</c><00:40:57.920><c> relate</c><00:40:58.680><c> the</c><00:40:58.800><c> time</c>"
+    },
+    {
+      "start": 2459.07,
+      "duration": 0.0,
+      "text": "So essentially we can relate the time"
+    },
+    {
+      "start": 2459.08,
+      "duration": 0.0,
+      "text": "So essentially we can relate the time derivative<00:41:00.240><c> of</c><00:41:00.440><c> this</c><00:41:00.640><c> correlation</c><00:41:01.200><c> function</c>"
+    },
+    {
+      "start": 2462.31,
+      "duration": 0.0,
+      "text": "derivative of this correlation function"
+    },
+    {
+      "start": 2462.32,
+      "duration": 0.0,
+      "text": "derivative of this correlation function with<00:41:03.200><c> this</c>"
+    },
+    {
+      "start": 2464.43,
+      "duration": 0.0,
+      "text": "with this"
+    },
+    {
+      "start": 2464.44,
+      "duration": 0.0,
+      "text": "with this expression<00:41:05.000><c> here</c><00:41:05.880><c> which</c><00:41:06.160><c> contains</c><00:41:07.120><c> the</c><00:41:07.320><c> two</c>"
+    },
+    {
+      "start": 2467.63,
+      "duration": 0.0,
+      "text": "expression here which contains the two"
+    },
+    {
+      "start": 2467.64,
+      "duration": 0.0,
+      "text": "expression here which contains the two phi,<00:41:08.080><c> so</c><00:41:08.200><c> the</c><00:41:08.320><c> two</c><00:41:08.440><c> observable</c><00:41:09.160><c> phi</c><00:41:09.360><c> m</c><00:41:09.600><c> and</c><00:41:09.760><c> phi</c>"
+    },
+    {
+      "start": 2469.91,
+      "duration": 0.0,
+      "text": "phi, so the two observable phi m and phi"
+    },
+    {
+      "start": 2469.92,
+      "duration": 0.0,
+      "text": "phi, so the two observable phi m and phi n,<00:41:10.800><c> the</c><00:41:10.960><c> conditional</c><00:41:11.600><c> score</c><00:41:11.920><c> function,</c><00:41:13.080><c> and</c>"
+    },
+    {
+      "start": 2473.91,
+      "duration": 0.0,
+      "text": "n, the conditional score function, and"
+    },
+    {
+      "start": 2473.92,
+      "duration": 0.0,
+      "text": "n, the conditional score function, and the<00:41:14.080><c> matrix,</c><00:41:14.800><c> so</c><00:41:14.960><c> the</c><00:41:15.240><c> tensor</c><00:41:15.960><c> m.</c>"
+    },
+    {
+      "start": 2476.95,
+      "duration": 0.0,
+      "text": "the matrix, so the tensor m."
+    },
+    {
+      "start": 2476.96,
+      "duration": 0.0,
+      "text": "the matrix, so the tensor m. And<00:41:17.200><c> the</c><00:41:17.280><c> tensor</c><00:41:17.640><c> m</c><00:41:18.120><c> is</c><00:41:18.240><c> the</c><00:41:18.360><c> only</c><00:41:18.520><c> term</c><00:41:18.840><c> here</c>"
+    },
+    {
+      "start": 2479.31,
+      "duration": 0.0,
+      "text": "And the tensor m is the only term here"
+    },
+    {
+      "start": 2479.32,
+      "duration": 0.0,
+      "text": "And the tensor m is the only term here that<00:41:19.480><c> we</c><00:41:19.560><c> don't</c><00:41:19.720><c> know</c><00:41:19.960><c> because</c><00:41:20.320><c> we</c><00:41:20.440><c> can</c>"
+    },
+    {
+      "start": 2480.83,
+      "duration": 0.0,
+      "text": "that we don't know because we can"
+    },
+    {
+      "start": 2480.84,
+      "duration": 0.0,
+      "text": "that we don't know because we can estimate<00:41:21.960><c> this</c><00:41:22.240><c> quantity</c><00:41:22.680><c> here</c><00:41:23.000><c> from</c><00:41:23.200><c> data.</c>"
+    },
+    {
+      "start": 2483.87,
+      "duration": 0.0,
+      "text": "estimate this quantity here from data."
+    },
+    {
+      "start": 2483.88,
+      "duration": 0.0,
+      "text": "estimate this quantity here from data. We<00:41:24.360><c> just</c><00:41:24.600><c> evaluate</c>"
+    },
+    {
+      "start": 2486.15,
+      "duration": 0.0,
+      "text": "We just evaluate"
+    },
+    {
+      "start": 2486.16,
+      "duration": 0.0,
+      "text": "We just evaluate the<00:41:26.480><c> correlation</c><00:41:27.000><c> function</c><00:41:27.680><c> and</c><00:41:27.840><c> then</c><00:41:28.360><c> we</c>"
+    },
+    {
+      "start": 2488.67,
+      "duration": 0.0,
+      "text": "the correlation function and then we"
+    },
+    {
+      "start": 2488.68,
+      "duration": 0.0,
+      "text": "the correlation function and then we estimate<00:41:29.280><c> the</c><00:41:29.320><c> derivative.</c>"
+    },
+    {
+      "start": 2490.95,
+      "duration": 0.0,
+      "text": "estimate the derivative."
+    },
+    {
+      "start": 2490.96,
+      "duration": 0.0,
+      "text": "estimate the derivative. We<00:41:31.360><c> can</c><00:41:32.120><c> estimate</c><00:41:33.400><c> the</c><00:41:33.760><c> conditional</c><00:41:34.320><c> score.</c>"
+    },
+    {
+      "start": 2495.15,
+      "duration": 0.0,
+      "text": "We can estimate the conditional score."
+    },
+    {
+      "start": 2495.16,
+      "duration": 0.0,
+      "text": "We can estimate the conditional score. We<00:41:35.360><c> know</c><00:41:35.600><c> the</c><00:41:35.720><c> analytical</c><00:41:36.200><c> expression</c><00:41:36.920><c> for</c>"
+    },
+    {
+      "start": 2497.11,
+      "duration": 0.0,
+      "text": "We know the analytical expression for"
+    },
+    {
+      "start": 2497.12,
+      "duration": 0.0,
+      "text": "We know the analytical expression for both<00:41:37.480><c> phi</c><00:41:38.000><c> m</c><00:41:38.440><c> and</c><00:41:38.680><c> phi</c><00:41:38.880><c> n</c><00:41:39.360><c> because</c><00:41:39.960><c> this</c><00:41:40.200><c> is</c><00:41:40.360><c> the</c>"
+    },
+    {
+      "start": 2500.63,
+      "duration": 0.0,
+      "text": "both phi m and phi n because this is the"
+    },
+    {
+      "start": 2500.64,
+      "duration": 0.0,
+      "text": "both phi m and phi n because this is the libraries<00:41:41.240><c> of</c><00:41:41.360><c> observable</c><00:41:42.240><c> that</c><00:41:42.440><c> we</c><00:41:42.520><c> are</c>"
+    },
+    {
+      "start": 2502.67,
+      "duration": 0.0,
+      "text": "libraries of observable that we are"
+    },
+    {
+      "start": 2502.68,
+      "duration": 0.0,
+      "text": "libraries of observable that we are considering."
+    },
+    {
+      "start": 2504.07,
+      "duration": 0.0,
+      "text": "considering."
+    },
+    {
+      "start": 2504.08,
+      "duration": 0.0,
+      "text": "considering. The<00:41:44.400><c> only</c><00:41:44.560><c> term</c><00:41:44.920><c> that</c><00:41:45.080><c> we</c><00:41:45.160><c> don't</c><00:41:45.320><c> know</c><00:41:45.880><c> is</c><00:41:46.320><c> this</c>"
+    },
+    {
+      "start": 2506.55,
+      "duration": 0.0,
+      "text": "The only term that we don't know is this"
+    },
+    {
+      "start": 2506.56,
+      "duration": 0.0,
+      "text": "The only term that we don't know is this matrix"
+    },
+    {
+      "start": 2507.67,
+      "duration": 0.0,
+      "text": "matrix"
+    },
+    {
+      "start": 2507.68,
+      "duration": 0.0,
+      "text": "matrix m<00:41:47.720><c> x</c><00:41:47.960><c> of</c><00:41:48.080><c> 0.</c><00:41:48.760><c> So</c><00:41:48.920><c> now</c><00:41:49.120><c> let's</c><00:41:49.360><c> see</c><00:41:49.520><c> how</c><00:41:49.720><c> we</c><00:41:49.840><c> can</c>"
+    },
+    {
+      "start": 2509.99,
+      "duration": 0.0,
+      "text": "m x of 0. So now let's see how we can"
+    },
+    {
+      "start": 2510.0,
+      "duration": 0.0,
+      "text": "m x of 0. So now let's see how we can derive<00:41:50.840><c> this</c><00:41:51.160><c> matrix</c><00:41:52.080><c> m</c><00:41:52.720><c> x</c><00:41:52.960><c> of</c><00:41:53.080><c> 0.</c>"
+    },
+    {
+      "start": 2514.15,
+      "duration": 0.0,
+      "text": "derive this matrix m x of 0."
+    },
+    {
+      "start": 2514.16,
+      "duration": 0.0,
+      "text": "derive this matrix m x of 0. So<00:41:54.440><c> first</c>"
+    },
+    {
+      "start": 2515.55,
+      "duration": 0.0,
+      "text": "So first"
+    },
+    {
+      "start": 2515.56,
+      "duration": 0.0,
+      "text": "So first let's<00:41:55.960><c> do</c><00:41:56.160><c> this</c><00:41:56.400><c> decomposition.</c><00:41:57.440><c> So</c><00:41:57.600><c> let's</c>"
+    },
+    {
+      "start": 2517.99,
+      "duration": 0.0,
+      "text": "let's do this decomposition. So let's"
+    },
+    {
+      "start": 2518.0,
+      "duration": 0.0,
+      "text": "let's do this decomposition. So let's decompose<00:41:59.240><c> m</c><00:41:59.600><c> x</c><00:42:00.280><c> in</c><00:42:00.480><c> terms</c><00:42:00.920><c> of</c><00:42:01.120><c> a</c><00:42:01.160><c> constant</c>"
+    },
+    {
+      "start": 2521.71,
+      "duration": 0.0,
+      "text": "decompose m x in terms of a constant"
+    },
+    {
+      "start": 2521.72,
+      "duration": 0.0,
+      "text": "decompose m x in terms of a constant term<00:42:02.240><c> plus</c><00:42:02.480><c> a</c><00:42:02.560><c> fluctuation.</c>"
+    },
+    {
+      "start": 2524.31,
+      "duration": 0.0,
+      "text": "term plus a fluctuation."
+    },
+    {
+      "start": 2524.32,
+      "duration": 0.0,
+      "text": "term plus a fluctuation. This<00:42:04.640><c> fluctuation</c><00:42:06.120><c> is</c><00:42:06.640><c> so</c><00:42:06.840><c> the</c><00:42:07.000><c> average</c><00:42:07.320><c> value</c>"
+    },
+    {
+      "start": 2528.11,
+      "duration": 0.0,
+      "text": "This fluctuation is so the average value"
+    },
+    {
+      "start": 2528.12,
+      "duration": 0.0,
+      "text": "This fluctuation is so the average value over<00:42:08.600><c> the</c><00:42:08.760><c> stationary</c><00:42:09.240><c> density</c><00:42:09.760><c> of</c><00:42:09.880><c> this</c>"
+    },
+    {
+      "start": 2530.07,
+      "duration": 0.0,
+      "text": "over the stationary density of this"
+    },
+    {
+      "start": 2530.08,
+      "duration": 0.0,
+      "text": "over the stationary density of this fluctuation<00:42:11.200><c> must</c><00:42:11.400><c> be</c><00:42:11.480><c> equal</c><00:42:11.640><c> to</c><00:42:11.720><c> 0.</c><00:42:12.120><c> So</c>"
+    },
+    {
+      "start": 2532.23,
+      "duration": 0.0,
+      "text": "fluctuation must be equal to 0. So"
+    },
+    {
+      "start": 2532.24,
+      "duration": 0.0,
+      "text": "fluctuation must be equal to 0. So essentially<00:42:12.640><c> here</c><00:42:13.000><c> we</c><00:42:13.160><c> have</c>"
+    },
+    {
+      "start": 2534.55,
+      "duration": 0.0,
+      "text": "essentially here we have"
+    },
+    {
+      "start": 2534.56,
+      "duration": 0.0,
+      "text": "essentially here we have a<00:42:15.000><c> yeah,</c><00:42:15.240><c> a</c><00:42:15.320><c> constant</c><00:42:15.680><c> term</c><00:42:16.040><c> plus</c>"
+    },
+    {
+      "start": 2537.55,
+      "duration": 0.0,
+      "text": "a yeah, a constant term plus"
+    },
+    {
+      "start": 2537.56,
+      "duration": 0.0,
+      "text": "a yeah, a constant term plus zero<00:42:17.840><c> mean</c><00:42:18.480><c> fluctuation</c><00:42:19.120><c> term</c><00:42:19.640><c> delta</c><00:42:19.960><c> m.</c><00:42:20.520><c> So</c>"
+    },
+    {
+      "start": 2540.67,
+      "duration": 0.0,
+      "text": "zero mean fluctuation term delta m. So"
+    },
+    {
+      "start": 2540.68,
+      "duration": 0.0,
+      "text": "zero mean fluctuation term delta m. So let's<00:42:20.920><c> use</c><00:42:21.760><c> this</c><00:42:22.440><c> expression</c><00:42:23.040><c> here</c><00:42:23.280><c> for</c><00:42:23.520><c> m</c>"
+    },
+    {
+      "start": 2544.23,
+      "duration": 0.0,
+      "text": "let's use this expression here for m"
+    },
+    {
+      "start": 2544.24,
+      "duration": 0.0,
+      "text": "let's use this expression here for m inside<00:42:25.400><c> the</c><00:42:25.840><c> this</c><00:42:26.520><c> equation</c><00:42:27.000><c> over</c><00:42:27.200><c> there</c>"
+    },
+    {
+      "start": 2548.31,
+      "duration": 0.0,
+      "text": "inside the this equation over there"
+    },
+    {
+      "start": 2548.32,
+      "duration": 0.0,
+      "text": "inside the this equation over there and<00:42:28.680><c> we</c><00:42:28.840><c> can</c><00:42:29.120><c> then</c><00:42:29.480><c> rewrite</c><00:42:30.480><c> c</c><00:42:30.640><c> dot</c><00:42:31.720><c> in</c><00:42:31.920><c> terms</c>"
+    },
+    {
+      "start": 2552.39,
+      "duration": 0.0,
+      "text": "and we can then rewrite c dot in terms"
+    },
+    {
+      "start": 2552.4,
+      "duration": 0.0,
+      "text": "and we can then rewrite c dot in terms of<00:42:32.600><c> two</c><00:42:32.760><c> terms.</c><00:42:33.600><c> This</c><00:42:33.880><c> first</c><00:42:34.240><c> term</c><00:42:35.000><c> which</c>"
+    },
+    {
+      "start": 2555.35,
+      "duration": 0.0,
+      "text": "of two terms. This first term which"
+    },
+    {
+      "start": 2555.36,
+      "duration": 0.0,
+      "text": "of two terms. This first term which depends<00:42:36.040><c> on</c><00:42:36.440><c> phi</c>"
+    },
+    {
+      "start": 2557.83,
+      "duration": 0.0,
+      "text": "depends on phi"
+    },
+    {
+      "start": 2557.84,
+      "duration": 0.0,
+      "text": "depends on phi does<00:42:38.240><c> not</c><00:42:38.640><c> depend</c><00:42:39.480><c> on</c><00:42:39.680><c> the</c><00:42:39.800><c> conditional</c>"
+    },
+    {
+      "start": 2560.27,
+      "duration": 0.0,
+      "text": "does not depend on the conditional"
+    },
+    {
+      "start": 2560.28,
+      "duration": 0.0,
+      "text": "does not depend on the conditional score,<00:42:40.800><c> depends</c><00:42:41.360><c> only</c><00:42:41.680><c> on</c><00:42:41.800><c> the</c><00:42:41.880><c> stationary</c>"
+    },
+    {
+      "start": 2562.31,
+      "duration": 0.0,
+      "text": "score, depends only on the stationary"
+    },
+    {
+      "start": 2562.32,
+      "duration": 0.0,
+      "text": "score, depends only on the stationary score."
+    },
+    {
+      "start": 2563.27,
+      "duration": 0.0,
+      "text": "score."
+    },
+    {
+      "start": 2563.28,
+      "duration": 0.0,
+      "text": "score. Okay,<00:42:43.520><c> so</c><00:42:43.680><c> we</c><00:42:43.800><c> have</c><00:42:44.080><c> the</c><00:42:44.240><c> first</c><00:42:44.560><c> term</c><00:42:44.800><c> here</c>"
+    },
+    {
+      "start": 2565.99,
+      "duration": 0.0,
+      "text": "Okay, so we have the first term here"
+    },
+    {
+      "start": 2566.0,
+      "duration": 0.0,
+      "text": "Okay, so we have the first term here which<00:42:46.320><c> is</c><00:42:46.560><c> much</c><00:42:46.880><c> easier</c><00:42:47.880><c> to</c><00:42:48.520><c> evaluate</c><00:42:49.240><c> because</c>"
+    },
+    {
+      "start": 2569.67,
+      "duration": 0.0,
+      "text": "which is much easier to evaluate because"
+    },
+    {
+      "start": 2569.68,
+      "duration": 0.0,
+      "text": "which is much easier to evaluate because yeah,<00:42:49.920><c> we</c><00:42:50.080><c> don't</c><00:42:50.360><c> need</c><00:42:50.720><c> to</c><00:42:50.840><c> estimate</c><00:42:51.360><c> the</c>"
+    },
+    {
+      "start": 2571.43,
+      "duration": 0.0,
+      "text": "yeah, we don't need to estimate the"
+    },
+    {
+      "start": 2571.44,
+      "duration": 0.0,
+      "text": "yeah, we don't need to estimate the conditional<00:42:51.920><c> score</c>"
+    },
+    {
+      "start": 2573.11,
+      "duration": 0.0,
+      "text": "conditional score"
+    },
+    {
+      "start": 2573.12,
+      "duration": 0.0,
+      "text": "conditional score and<00:42:53.760><c> it</c><00:42:53.920><c> only</c><00:42:54.080><c> depends</c><00:42:54.520><c> on</c><00:42:54.640><c> this</c><00:42:54.800><c> constant</c>"
+    },
+    {
+      "start": 2575.27,
+      "duration": 0.0,
+      "text": "and it only depends on this constant"
+    },
+    {
+      "start": 2575.28,
+      "duration": 0.0,
+      "text": "and it only depends on this constant matrix<00:42:55.800><c> phi</c><00:42:56.280><c> and</c><00:42:56.520><c> this</c>"
+    },
+    {
+      "start": 2577.23,
+      "duration": 0.0,
+      "text": "matrix phi and this"
+    },
+    {
+      "start": 2577.24,
+      "duration": 0.0,
+      "text": "matrix phi and this plain<00:42:57.520><c> score</c><00:42:57.760><c> itself</c>"
+    },
+    {
+      "start": 2578.95,
+      "duration": 0.0,
+      "text": "plain score itself"
+    },
+    {
+      "start": 2578.96,
+      "duration": 0.0,
+      "text": "plain score itself minus<00:42:59.640><c> this</c><00:43:00.280><c> additional</c><00:43:00.960><c> so</c><00:43:01.560><c> this</c><00:43:01.840><c> additional</c>"
+    },
+    {
+      "start": 2582.31,
+      "duration": 0.0,
+      "text": "minus this additional so this additional"
+    },
+    {
+      "start": 2582.32,
+      "duration": 0.0,
+      "text": "minus this additional so this additional term<00:43:02.800><c> which</c><00:43:02.960><c> is</c><00:43:03.080><c> nothing</c><00:43:03.600><c> but</c><00:43:04.320><c> this</c><00:43:04.600><c> one</c><00:43:04.760><c> over</c>"
+    },
+    {
+      "start": 2584.95,
+      "duration": 0.0,
+      "text": "term which is nothing but this one over"
+    },
+    {
+      "start": 2584.96,
+      "duration": 0.0,
+      "text": "term which is nothing but this one over here<00:43:05.960><c> written</c><00:43:06.320><c> in</c><00:43:06.480><c> terms</c><00:43:06.920><c> of</c><00:43:07.080><c> delta</c><00:43:07.440><c> m</c><00:43:08.000><c> instead</c>"
+    },
+    {
+      "start": 2588.31,
+      "duration": 0.0,
+      "text": "here written in terms of delta m instead"
+    },
+    {
+      "start": 2588.32,
+      "duration": 0.0,
+      "text": "here written in terms of delta m instead of<00:43:08.440><c> m.</c>"
+    },
+    {
+      "start": 2590.84,
+      "duration": 0.0,
+      "text": "So<00:43:11.120><c> the</c><00:43:11.320><c> key</c><00:43:11.480><c> idea</c><00:43:11.880><c> here</c><00:43:12.720><c> is</c><00:43:12.920><c> that</c><00:43:13.480><c> if</c><00:43:13.760><c> we</c><00:43:13.920><c> have</c>"
+    },
+    {
+      "start": 2594.59,
+      "duration": 0.0,
+      "text": "So the key idea here is that if we have"
+    },
+    {
+      "start": 2594.6,
+      "duration": 0.0,
+      "text": "So the key idea here is that if we have a<00:43:14.680><c> library</c><00:43:15.560><c> of</c><00:43:15.760><c> observable</c><00:43:16.600><c> which</c><00:43:16.760><c> is</c><00:43:16.880><c> rich</c>"
+    },
+    {
+      "start": 2597.19,
+      "duration": 0.0,
+      "text": "a library of observable which is rich"
+    },
+    {
+      "start": 2597.2,
+      "duration": 0.0,
+      "text": "a library of observable which is rich enough<00:43:17.960><c> such</c>"
+    },
+    {
+      "start": 2599.03,
+      "duration": 0.0,
+      "text": "enough such"
+    },
+    {
+      "start": 2599.04,
+      "duration": 0.0,
+      "text": "enough such such<00:43:19.280><c> that</c><00:43:19.880><c> m</c><00:43:20.120><c> of</c><00:43:20.360><c> x</c><00:43:21.040><c> is</c><00:43:21.360><c> uniquely</c><00:43:21.920><c> determined,</c>"
+    },
+    {
+      "start": 2603.31,
+      "duration": 0.0,
+      "text": "such that m of x is uniquely determined,"
+    },
+    {
+      "start": 2603.32,
+      "duration": 0.0,
+      "text": "such that m of x is uniquely determined, then<00:43:23.520><c> we</c><00:43:23.640><c> can</c><00:43:24.200><c> estimate</c><00:43:25.120><c> m</c><00:43:25.800><c> m</c><00:43:26.000><c> of</c><00:43:26.200><c> x,</c><00:43:26.680><c> so</c><00:43:26.880><c> this</c>"
+    },
+    {
+      "start": 2608.07,
+      "duration": 0.0,
+      "text": "then we can estimate m m of x, so this"
+    },
+    {
+      "start": 2608.08,
+      "duration": 0.0,
+      "text": "then we can estimate m m of x, so this tensor<00:43:28.840><c> m</c><00:43:29.000><c> of</c><00:43:29.200><c> x</c><00:43:29.520><c> which</c><00:43:29.800><c> is</c><00:43:30.160><c> the</c><00:43:30.320><c> missing</c>"
+    },
+    {
+      "start": 2610.87,
+      "duration": 0.0,
+      "text": "tensor m of x which is the missing"
+    },
+    {
+      "start": 2610.88,
+      "duration": 0.0,
+      "text": "tensor m of x which is the missing element<00:43:32.040><c> for</c><00:43:33.000><c> so</c><00:43:33.120><c> in</c><00:43:33.280><c> our</c><00:43:33.600><c> stochastic</c><00:43:34.200><c> model</c>"
+    },
+    {
+      "start": 2615.43,
+      "duration": 0.0,
+      "text": "element for so in our stochastic model"
+    },
+    {
+      "start": 2615.44,
+      "duration": 0.0,
+      "text": "element for so in our stochastic model by<00:43:35.800><c> essentially</c><00:43:36.440><c> using</c><00:43:36.840><c> this</c><00:43:37.080><c> relationship</c>"
+    },
+    {
+      "start": 2617.75,
+      "duration": 0.0,
+      "text": "by essentially using this relationship"
+    },
+    {
+      "start": 2617.76,
+      "duration": 0.0,
+      "text": "by essentially using this relationship over<00:43:37.920><c> here.</c>"
+    },
+    {
+      "start": 2618.99,
+      "duration": 0.0,
+      "text": "over here."
+    },
+    {
+      "start": 2619.0,
+      "duration": 0.0,
+      "text": "over here. So<00:43:39.280><c> using</c><00:43:39.600><c> this</c><00:43:39.760><c> relationship</c><00:43:40.400><c> here</c>"
+    },
+    {
+      "start": 2621.55,
+      "duration": 0.0,
+      "text": "So using this relationship here"
+    },
+    {
+      "start": 2621.56,
+      "duration": 0.0,
+      "text": "So using this relationship here and<00:43:42.040><c> a</c><00:43:42.080><c> library</c><00:43:42.720><c> of</c><00:43:42.880><c> observable</c><00:43:43.680><c> which</c><00:43:43.880><c> is</c>"
+    },
+    {
+      "start": 2623.99,
+      "duration": 0.0,
+      "text": "and a library of observable which is"
+    },
+    {
+      "start": 2624.0,
+      "duration": 0.0,
+      "text": "and a library of observable which is rich<00:43:44.280><c> enough,</c><00:43:44.840><c> we</c><00:43:45.000><c> can</c><00:43:45.200><c> estimate</c><00:43:45.840><c> both</c><00:43:46.280><c> phi</c>"
+    },
+    {
+      "start": 2627.19,
+      "duration": 0.0,
+      "text": "rich enough, we can estimate both phi"
+    },
+    {
+      "start": 2627.2,
+      "duration": 0.0,
+      "text": "rich enough, we can estimate both phi and<00:43:47.880><c> delta</c><00:43:48.200><c> m.</c>"
+    },
+    {
+      "start": 2629.35,
+      "duration": 0.0,
+      "text": "and delta m."
+    },
+    {
+      "start": 2629.36,
+      "duration": 0.0,
+      "text": "and delta m. So<00:43:49.560><c> let's</c><00:43:49.720><c> see</c><00:43:49.800><c> now</c><00:43:50.160><c> how</c><00:43:50.400><c> we</c><00:43:50.520><c> can</c><00:43:50.680><c> estimate</c><00:43:51.280><c> phi</c>"
+    },
+    {
+      "start": 2632.19,
+      "duration": 0.0,
+      "text": "So let's see now how we can estimate phi"
+    },
+    {
+      "start": 2632.2,
+      "duration": 0.0,
+      "text": "So let's see now how we can estimate phi first."
+    },
+    {
+      "start": 2633.55,
+      "duration": 0.0,
+      "text": "first."
+    },
+    {
+      "start": 2633.56,
+      "duration": 0.0,
+      "text": "first. So<00:43:53.800><c> to</c><00:43:53.920><c> do</c><00:43:54.040><c> that,</c><00:43:54.480><c> let's</c><00:43:54.760><c> consider</c>"
+    },
+    {
+      "start": 2636.71,
+      "duration": 0.0,
+      "text": "So to do that, let's consider"
+    },
+    {
+      "start": 2636.72,
+      "duration": 0.0,
+      "text": "So to do that, let's consider just<00:43:57.240><c> the</c><00:43:57.440><c> coordinate</c><00:43:58.000><c> observable.</c><00:43:58.960><c> Okay,</c><00:43:59.240><c> so</c>"
+    },
+    {
+      "start": 2639.39,
+      "duration": 0.0,
+      "text": "just the coordinate observable. Okay, so"
+    },
+    {
+      "start": 2639.4,
+      "duration": 0.0,
+      "text": "just the coordinate observable. Okay, so we<00:43:59.520><c> have</c><00:43:59.760><c> here</c><00:44:00.240><c> in</c><00:44:00.400><c> theory</c><00:44:00.840><c> like</c><00:44:01.080><c> a</c><00:44:01.120><c> very</c><00:44:01.359><c> large</c>"
+    },
+    {
+      "start": 2641.71,
+      "duration": 0.0,
+      "text": "we have here in theory like a very large"
+    },
+    {
+      "start": 2641.72,
+      "duration": 0.0,
+      "text": "we have here in theory like a very large libraries<00:44:02.320><c> of</c><00:44:02.480><c> observable.</c><00:44:03.600><c> Now</c><00:44:03.880><c> let's</c><00:44:04.120><c> focus</c>"
+    },
+    {
+      "start": 2644.59,
+      "duration": 0.0,
+      "text": "libraries of observable. Now let's focus"
+    },
+    {
+      "start": 2644.6,
+      "duration": 0.0,
+      "text": "libraries of observable. Now let's focus on<00:44:04.920><c> a</c><00:44:05.000><c> few</c><00:44:05.200><c> of</c><00:44:05.320><c> them</c><00:44:05.920><c> and</c><00:44:06.160><c> few</c><00:44:06.359><c> of</c><00:44:06.440><c> them</c><00:44:07.400><c> of</c><00:44:07.600><c> x</c>"
+    },
+    {
+      "start": 2648.11,
+      "duration": 0.0,
+      "text": "on a few of them and few of them of x"
+    },
+    {
+      "start": 2648.12,
+      "duration": 0.0,
+      "text": "on a few of them and few of them of x equal<00:44:08.480><c> to</c><00:44:08.640><c> x.</c><00:44:09.080><c> So</c><00:44:09.240><c> we're</c><00:44:09.400><c> just</c><00:44:09.800><c> considering</c>"
+    },
+    {
+      "start": 2650.79,
+      "duration": 0.0,
+      "text": "equal to x. So we're just considering"
+    },
+    {
+      "start": 2650.8,
+      "duration": 0.0,
+      "text": "equal to x. So we're just considering the<00:44:11.440><c> observable</c><00:44:12.520><c> coordinate.</c>"
+    },
+    {
+      "start": 2654.51,
+      "duration": 0.0,
+      "text": "the observable coordinate."
+    },
+    {
+      "start": 2654.52,
+      "duration": 0.0,
+      "text": "the observable coordinate. By<00:44:14.920><c> doing</c><00:44:15.200><c> this</c><00:44:15.440><c> replacement,</c><00:44:16.480><c> the</c><00:44:16.680><c> first</c>"
+    },
+    {
+      "start": 2657.03,
+      "duration": 0.0,
+      "text": "By doing this replacement, the first"
+    },
+    {
+      "start": 2657.04,
+      "duration": 0.0,
+      "text": "By doing this replacement, the first term<00:44:18.000><c> becomes</c><00:44:18.640><c> this</c><00:44:19.320><c> expected</c><00:44:19.880><c> value</c>"
+    },
+    {
+      "start": 2660.23,
+      "duration": 0.0,
+      "text": "term becomes this expected value"
+    },
+    {
+      "start": 2660.24,
+      "duration": 0.0,
+      "text": "term becomes this expected value multiplied<00:44:20.760><c> by</c><00:44:20.920><c> phi.</c>"
+    },
+    {
+      "start": 2661.91,
+      "duration": 0.0,
+      "text": "multiplied by phi."
+    },
+    {
+      "start": 2661.92,
+      "duration": 0.0,
+      "text": "multiplied by phi. Now<00:44:22.760><c> if</c><00:44:22.920><c> we</c><00:44:23.040><c> consider</c><00:44:24.000><c> t</c><00:44:24.600><c> equal</c><00:44:24.880><c> to</c><00:44:25.000><c> 0,</c>"
+    },
+    {
+      "start": 2666.31,
+      "duration": 0.0,
+      "text": "Now if we consider t equal to 0,"
+    },
+    {
+      "start": 2666.32,
+      "duration": 0.0,
+      "text": "Now if we consider t equal to 0, then<00:44:27.359><c> this</c>"
+    },
+    {
+      "start": 2668.59,
+      "duration": 0.0,
+      "text": "then this"
+    },
+    {
+      "start": 2668.6,
+      "duration": 0.0,
+      "text": "then this expected<00:44:29.240><c> value</c><00:44:29.680><c> becomes</c><00:44:30.320><c> minus</c><00:44:30.880><c> the</c>"
+    },
+    {
+      "start": 2670.99,
+      "duration": 0.0,
+      "text": "expected value becomes minus the"
+    },
+    {
+      "start": 2671.0,
+      "duration": 0.0,
+      "text": "expected value becomes minus the identity<00:44:31.760><c> because</c><00:44:32.120><c> of</c><00:44:32.400><c> the</c><00:44:32.760><c> Stein</c><00:44:33.520><c> identity.</c>"
+    },
+    {
+      "start": 2675.55,
+      "duration": 0.0,
+      "text": "identity because of the Stein identity."
+    },
+    {
+      "start": 2675.56,
+      "duration": 0.0,
+      "text": "identity because of the Stein identity. This<00:44:35.960><c> term</c><00:44:36.200><c> here</c><00:44:37.160><c> becomes</c><00:44:37.640><c> equal</c><00:44:37.840><c> to</c><00:44:37.960><c> 0</c>"
+    },
+    {
+      "start": 2678.63,
+      "duration": 0.0,
+      "text": "This term here becomes equal to 0"
+    },
+    {
+      "start": 2678.64,
+      "duration": 0.0,
+      "text": "This term here becomes equal to 0 because<00:44:39.480><c> we</c><00:44:39.600><c> have</c><00:44:40.320><c> so</c><00:44:40.480><c> you</c><00:44:40.600><c> can</c><00:44:40.800><c> just</c>"
+    },
+    {
+      "start": 2681.27,
+      "duration": 0.0,
+      "text": "because we have so you can just"
+    },
+    {
+      "start": 2681.28,
+      "duration": 0.0,
+      "text": "because we have so you can just integrate<00:44:42.240><c> the</c><00:44:42.640><c> conditional</c><00:44:43.440><c> score</c><00:44:43.880><c> term</c><00:44:44.720><c> at</c>"
+    },
+    {
+      "start": 2684.95,
+      "duration": 0.0,
+      "text": "integrate the conditional score term at"
+    },
+    {
+      "start": 2684.96,
+      "duration": 0.0,
+      "text": "integrate the conditional score term at t<00:44:45.000><c> equal</c><00:44:45.200><c> to</c><00:44:45.280><c> 0</c><00:44:45.520><c> and</c><00:44:45.640><c> you</c><00:44:45.720><c> will</c><00:44:45.840><c> get</c><00:44:46.000><c> 0.</c>"
+    },
+    {
+      "start": 2687.11,
+      "duration": 0.0,
+      "text": "t equal to 0 and you will get 0."
+    },
+    {
+      "start": 2687.12,
+      "duration": 0.0,
+      "text": "t equal to 0 and you will get 0. So<00:44:47.320><c> this</c><00:44:47.480><c> essentially</c><00:44:47.960><c> means</c><00:44:48.320><c> that</c><00:44:48.520><c> if</c><00:44:48.840><c> if</c><00:44:48.920><c> you</c>"
+    },
+    {
+      "start": 2688.99,
+      "duration": 0.0,
+      "text": "So this essentially means that if if you"
+    },
+    {
+      "start": 2689.0,
+      "duration": 0.0,
+      "text": "So this essentially means that if if you consider<00:44:49.560><c> the</c><00:44:49.680><c> coordinate</c><00:44:50.120><c> observable</c><00:44:51.120><c> and</c><00:44:51.760><c> t</c>"
+    },
+    {
+      "start": 2691.99,
+      "duration": 0.0,
+      "text": "consider the coordinate observable and t"
+    },
+    {
+      "start": 2692.0,
+      "duration": 0.0,
+      "text": "consider the coordinate observable and t equal<00:44:52.240><c> to</c><00:44:52.280><c> 0,</c><00:44:53.240><c> we</c><00:44:53.400><c> are</c><00:44:53.560><c> able</c><00:44:54.000><c> to</c><00:44:54.120><c> derive</c><00:44:55.000><c> a</c>"
+    },
+    {
+      "start": 2695.11,
+      "duration": 0.0,
+      "text": "equal to 0, we are able to derive a"
+    },
+    {
+      "start": 2695.12,
+      "duration": 0.0,
+      "text": "equal to 0, we are able to derive a relationship<00:44:56.560><c> for</c><00:44:57.160><c> phi.</c>"
+    },
+    {
+      "start": 2698.03,
+      "duration": 0.0,
+      "text": "relationship for phi."
+    },
+    {
+      "start": 2698.04,
+      "duration": 0.0,
+      "text": "relationship for phi. So<00:44:58.320><c> we</c><00:44:58.480><c> can</c><00:44:58.760><c> essentially</c><00:44:59.240><c> fix</c><00:44:59.760><c> the</c><00:45:00.000><c> average</c>"
+    },
+    {
+      "start": 2700.47,
+      "duration": 0.0,
+      "text": "So we can essentially fix the average"
+    },
+    {
+      "start": 2700.48,
+      "duration": 0.0,
+      "text": "So we can essentially fix the average value<00:45:01.120><c> of</c><00:45:01.640><c> the</c><00:45:01.760><c> matrix</c><00:45:02.320><c> m</c>"
+    },
+    {
+      "start": 2704.43,
+      "duration": 0.0,
+      "text": "value of the matrix m"
+    },
+    {
+      "start": 2704.44,
+      "duration": 0.0,
+      "text": "value of the matrix m and<00:45:04.760><c> we</c><00:45:04.840><c> can</c><00:45:05.040><c> write</c><00:45:05.359><c> it</c><00:45:05.640><c> in</c><00:45:05.840><c> terms</c><00:45:06.520><c> of</c><00:45:07.480><c> the</c>"
+    },
+    {
+      "start": 2708.47,
+      "duration": 0.0,
+      "text": "and we can write it in terms of the"
+    },
+    {
+      "start": 2708.48,
+      "duration": 0.0,
+      "text": "and we can write it in terms of the coordinate<00:45:09.280><c> the</c><00:45:09.480><c> time</c><00:45:09.720><c> derivative</c><00:45:10.520><c> of</c><00:45:10.680><c> the</c>"
+    },
+    {
+      "start": 2710.79,
+      "duration": 0.0,
+      "text": "coordinate the time derivative of the"
+    },
+    {
+      "start": 2710.8,
+      "duration": 0.0,
+      "text": "coordinate the time derivative of the coordinate<00:45:11.560><c> correlation</c><00:45:12.600><c> at</c><00:45:12.840><c> t</c><00:45:13.120><c> equal</c><00:45:13.359><c> to</c><00:45:13.480><c> 0.</c>"
+    },
+    {
+      "start": 2715.79,
+      "duration": 0.0,
+      "text": "coordinate correlation at t equal to 0."
+    },
+    {
+      "start": 2715.8,
+      "duration": 0.0,
+      "text": "coordinate correlation at t equal to 0. Now<00:45:16.160><c> let's</c><00:45:16.400><c> see</c><00:45:16.520><c> what</c><00:45:17.240><c> this</c><00:45:17.480><c> implies.</c><00:45:18.240><c> So</c><00:45:18.680><c> we</c>"
+    },
+    {
+      "start": 2718.83,
+      "duration": 0.0,
+      "text": "Now let's see what this implies. So we"
+    },
+    {
+      "start": 2718.84,
+      "duration": 0.0,
+      "text": "Now let's see what this implies. So we have<00:45:19.359><c> then</c>"
+    },
+    {
+      "start": 2720.51,
+      "duration": 0.0,
+      "text": "have then"
+    },
+    {
+      "start": 2720.52,
+      "duration": 0.0,
+      "text": "have then fixed<00:45:20.920><c> the</c><00:45:21.040><c> phi.</c><00:45:21.760><c> We</c><00:45:21.920><c> have</c><00:45:22.359><c> this</c><00:45:22.800><c> additional</c>"
+    },
+    {
+      "start": 2723.43,
+      "duration": 0.0,
+      "text": "fixed the phi. We have this additional"
+    },
+    {
+      "start": 2723.44,
+      "duration": 0.0,
+      "text": "fixed the phi. We have this additional term,<00:45:24.200><c> this</c><00:45:24.400><c> correction</c><00:45:25.000><c> term</c><00:45:26.080><c> e</c><00:45:26.359><c> of</c><00:45:26.600><c> x.</c>"
+    },
+    {
+      "start": 2727.91,
+      "duration": 0.0,
+      "text": "term, this correction term e of x."
+    },
+    {
+      "start": 2727.92,
+      "duration": 0.0,
+      "text": "term, this correction term e of x. When<00:45:28.560><c> we</c><00:45:28.720><c> consider</c>"
+    },
+    {
+      "start": 2730.27,
+      "duration": 0.0,
+      "text": "When we consider"
+    },
+    {
+      "start": 2730.28,
+      "duration": 0.0,
+      "text": "When we consider the"
+    },
+    {
+      "start": 2731.43,
+      "duration": 0.0,
+      "text": "the"
+    },
+    {
+      "start": 2731.44,
+      "duration": 0.0,
+      "text": "the phi<00:45:32.000><c> phi</c><00:45:32.520><c> phi</c><00:45:32.840><c> phi</c><00:45:32.920><c> 1</c><00:45:33.160><c> of</c><00:45:33.359><c> x</c><00:45:33.800><c> equal</c><00:45:34.120><c> to</c><00:45:34.240><c> x,</c><00:45:34.600><c> we</c>"
+    },
+    {
+      "start": 2734.75,
+      "duration": 0.0,
+      "text": "phi phi phi phi 1 of x equal to x, we"
+    },
+    {
+      "start": 2734.76,
+      "duration": 0.0,
+      "text": "phi phi phi phi 1 of x equal to x, we then<00:45:35.000><c> have</c><00:45:35.440><c> this</c><00:45:36.320><c> coordinate</c><00:45:36.920><c> term</c><00:45:37.320><c> at</c><00:45:37.400><c> the</c>"
+    },
+    {
+      "start": 2737.47,
+      "duration": 0.0,
+      "text": "then have this coordinate term at the"
+    },
+    {
+      "start": 2737.48,
+      "duration": 0.0,
+      "text": "then have this coordinate term at the beginning."
+    },
+    {
+      "start": 2738.95,
+      "duration": 0.0,
+      "text": "beginning."
+    },
+    {
+      "start": 2738.96,
+      "duration": 0.0,
+      "text": "beginning. We<00:45:39.200><c> can</c><00:45:39.480><c> rewrite</c><00:45:40.080><c> this</c><00:45:40.320><c> expected</c><00:45:40.880><c> value</c><00:45:41.480><c> in</c>"
+    },
+    {
+      "start": 2741.67,
+      "duration": 0.0,
+      "text": "We can rewrite this expected value in"
+    },
+    {
+      "start": 2741.68,
+      "duration": 0.0,
+      "text": "We can rewrite this expected value in terms<00:45:42.120><c> of</c><00:45:42.280><c> the</c><00:45:42.400><c> gradient</c><00:45:43.160><c> with</c><00:45:43.320><c> respect</c><00:45:43.720><c> to</c><00:45:43.840><c> x</c>"
+    },
+    {
+      "start": 2743.91,
+      "duration": 0.0,
+      "text": "terms of the gradient with respect to x"
+    },
+    {
+      "start": 2743.92,
+      "duration": 0.0,
+      "text": "terms of the gradient with respect to x 0<00:45:44.840><c> of</c><00:45:45.240><c> the</c><00:45:45.359><c> expected</c><00:45:45.920><c> value</c><00:45:46.400><c> of</c><00:45:46.640><c> x</c><00:45:46.840><c> of</c><00:45:47.040><c> t</c>"
+    },
+    {
+      "start": 2747.43,
+      "duration": 0.0,
+      "text": "0 of the expected value of x of t"
+    },
+    {
+      "start": 2747.44,
+      "duration": 0.0,
+      "text": "0 of the expected value of x of t conditioned<00:45:48.160><c> on</c><00:45:48.320><c> x</c><00:45:48.480><c> 0</c><00:45:49.320><c> multiplied</c><00:45:50.080><c> by</c><00:45:50.280><c> delta</c>"
+    },
+    {
+      "start": 2750.59,
+      "duration": 0.0,
+      "text": "conditioned on x 0 multiplied by delta"
+    },
+    {
+      "start": 2750.6,
+      "duration": 0.0,
+      "text": "conditioned on x 0 multiplied by delta m."
+    },
+    {
+      "start": 2752.31,
+      "duration": 0.0,
+      "text": "m."
+    },
+    {
+      "start": 2752.32,
+      "duration": 0.0,
+      "text": "m. So<00:45:52.880><c> by</c><00:45:53.800><c> just</c><00:45:54.040><c> considering</c><00:45:54.640><c> the</c><00:45:54.760><c> coordinate</c>"
+    },
+    {
+      "start": 2755.23,
+      "duration": 0.0,
+      "text": "So by just considering the coordinate"
+    },
+    {
+      "start": 2755.24,
+      "duration": 0.0,
+      "text": "So by just considering the coordinate observable<00:45:55.800><c> case,</c><00:45:56.480><c> we</c><00:45:56.640><c> can</c><00:45:56.880><c> derive</c><00:45:57.880><c> phi</c>"
+    },
+    {
+      "start": 2759.27,
+      "duration": 0.0,
+      "text": "observable case, we can derive phi"
+    },
+    {
+      "start": 2759.28,
+      "duration": 0.0,
+      "text": "observable case, we can derive phi and<00:45:59.600><c> then</c><00:46:00.040><c> we</c><00:46:00.200><c> can</c><00:46:00.400><c> write</c><00:46:00.800><c> this</c><00:46:01.040><c> relationship</c>"
+    },
+    {
+      "start": 2761.83,
+      "duration": 0.0,
+      "text": "and then we can write this relationship"
+    },
+    {
+      "start": 2761.84,
+      "duration": 0.0,
+      "text": "and then we can write this relationship for<00:46:02.000><c> the</c><00:46:02.120><c> correction</c><00:46:02.640><c> term.</c><00:46:03.560><c> But</c><00:46:03.960><c> at</c><00:46:04.120><c> this</c>"
+    },
+    {
+      "start": 2764.349,
+      "duration": 0.0,
+      "text": "for the correction term. But at this"
+    },
+    {
+      "start": 2764.359,
+      "duration": 0.0,
+      "text": "for the correction term. But at this point<00:46:04.760><c> we</c><00:46:04.880><c> can</c><00:46:05.040><c> notice</c><00:46:05.760><c> that</c><00:46:06.120><c> if</c>"
+    },
+    {
+      "start": 2767.43,
+      "duration": 0.0,
+      "text": "point we can notice that if"
+    },
+    {
+      "start": 2767.44,
+      "duration": 0.0,
+      "text": "point we can notice that if m<00:46:07.800><c> of</c><00:46:08.000><c> x,</c><00:46:08.440><c> so</c><00:46:08.640><c> if</c><00:46:09.000><c> the</c><00:46:09.120><c> expected</c><00:46:09.720><c> value</c><00:46:10.280><c> of</c><00:46:10.520><c> x</c><00:46:10.720><c> of</c>"
+    },
+    {
+      "start": 2770.91,
+      "duration": 0.0,
+      "text": "m of x, so if the expected value of x of"
+    },
+    {
+      "start": 2770.92,
+      "duration": 0.0,
+      "text": "m of x, so if the expected value of x of t<00:46:11.480><c> conditioned</c><00:46:12.359><c> on</c><00:46:12.520><c> x</c><00:46:12.680><c> 0</c><00:46:13.760><c> is</c><00:46:14.200><c> approximately</c>"
+    },
+    {
+      "start": 2775.95,
+      "duration": 0.0,
+      "text": "t conditioned on x 0 is approximately"
+    },
+    {
+      "start": 2775.96,
+      "duration": 0.0,
+      "text": "t conditioned on x 0 is approximately affine<00:46:16.720><c> which</c><00:46:16.880><c> essentially</c><00:46:17.359><c> means</c><00:46:17.960><c> if</c><00:46:18.200><c> we</c><00:46:18.320><c> can</c>"
+    },
+    {
+      "start": 2778.59,
+      "duration": 0.0,
+      "text": "affine which essentially means if we can"
+    },
+    {
+      "start": 2778.6,
+      "duration": 0.0,
+      "text": "affine which essentially means if we can write<00:46:19.280><c> the</c><00:46:19.400><c> expected</c><00:46:19.920><c> value</c><00:46:20.600><c> of</c><00:46:20.800><c> x</c><00:46:21.040><c> t</c>"
+    },
+    {
+      "start": 2781.23,
+      "duration": 0.0,
+      "text": "write the expected value of x t"
+    },
+    {
+      "start": 2781.24,
+      "duration": 0.0,
+      "text": "write the expected value of x t conditioned<00:46:22.040><c> on</c><00:46:22.240><c> x</c><00:46:22.400><c> 0</c>"
+    },
+    {
+      "start": 2783.55,
+      "duration": 0.0,
+      "text": "conditioned on x 0"
+    },
+    {
+      "start": 2783.56,
+      "duration": 0.0,
+      "text": "conditioned on x 0 in<00:46:23.760><c> terms</c><00:46:24.320><c> of</c><00:46:24.520><c> a</c><00:46:24.600><c> linear</c><00:46:24.960><c> function</c><00:46:25.880><c> of</c><00:46:26.080><c> x</c><00:46:26.280><c> 0,</c>"
+    },
+    {
+      "start": 2787.349,
+      "duration": 0.0,
+      "text": "in terms of a linear function of x 0,"
+    },
+    {
+      "start": 2787.359,
+      "duration": 0.0,
+      "text": "in terms of a linear function of x 0, then<00:46:27.840><c> when</c><00:46:28.040><c> we</c><00:46:28.160><c> take</c><00:46:28.400><c> the</c><00:46:28.480><c> gradient,</c><00:46:29.240><c> we</c><00:46:29.359><c> will</c>"
+    },
+    {
+      "start": 2789.51,
+      "duration": 0.0,
+      "text": "then when we take the gradient, we will"
+    },
+    {
+      "start": 2789.52,
+      "duration": 0.0,
+      "text": "then when we take the gradient, we will get<00:46:29.760><c> a</c><00:46:29.840><c> constant</c><00:46:30.440><c> term</c><00:46:30.920><c> with</c><00:46:31.080><c> respect</c><00:46:31.520><c> to</c><00:46:31.640><c> x</c>"
+    },
+    {
+      "start": 2792.59,
+      "duration": 0.0,
+      "text": "get a constant term with respect to x"
+    },
+    {
+      "start": 2792.6,
+      "duration": 0.0,
+      "text": "get a constant term with respect to x and<00:46:32.800><c> then</c><00:46:33.440><c> by</c><00:46:33.640><c> construction</c><00:46:34.560><c> the</c><00:46:34.760><c> average</c>"
+    },
+    {
+      "start": 2795.19,
+      "duration": 0.0,
+      "text": "and then by construction the average"
+    },
+    {
+      "start": 2795.2,
+      "duration": 0.0,
+      "text": "and then by construction the average value<00:46:35.800><c> of</c><00:46:36.040><c> x</c><00:46:36.680><c> of</c><00:46:36.880><c> delta</c><00:46:37.320><c> m</c><00:46:37.960><c> is</c><00:46:38.160><c> equal</c><00:46:38.359><c> to</c><00:46:38.480><c> 0</c>"
+    },
+    {
+      "start": 2799.51,
+      "duration": 0.0,
+      "text": "value of x of delta m is equal to 0"
+    },
+    {
+      "start": 2799.52,
+      "duration": 0.0,
+      "text": "value of x of delta m is equal to 0 which<00:46:39.800><c> essentially</c><00:46:40.320><c> means</c><00:46:40.840><c> that</c><00:46:41.240><c> if</c><00:46:42.280><c> the</c>"
+    },
+    {
+      "start": 2802.43,
+      "duration": 0.0,
+      "text": "which essentially means that if the"
+    },
+    {
+      "start": 2802.44,
+      "duration": 0.0,
+      "text": "which essentially means that if the conditional<00:46:43.040><c> mean</c><00:46:44.120><c> is</c><00:46:44.480><c> approximately</c><00:46:45.520><c> affine</c>"
+    },
+    {
+      "start": 2806.15,
+      "duration": 0.0,
+      "text": "conditional mean is approximately affine"
+    },
+    {
+      "start": 2806.16,
+      "duration": 0.0,
+      "text": "conditional mean is approximately affine which<00:46:46.359><c> essentially</c>"
+    },
+    {
+      "start": 2807.79,
+      "duration": 0.0,
+      "text": "which essentially"
+    },
+    {
+      "start": 2807.8,
+      "duration": 0.0,
+      "text": "which essentially is<00:46:48.080><c> is</c><00:46:48.280><c> is</c><00:46:48.680><c> the</c><00:46:48.800><c> case</c><00:46:49.520><c> if</c>"
+    },
+    {
+      "start": 2810.95,
+      "duration": 0.0,
+      "text": "is is is the case if"
+    },
+    {
+      "start": 2810.96,
+      "duration": 0.0,
+      "text": "is is is the case if the<00:46:51.280><c> joint</c><00:46:51.880><c> probability</c><00:46:52.560><c> density</c><00:46:53.000><c> function</c>"
+    },
+    {
+      "start": 2814.43,
+      "duration": 0.0,
+      "text": "the joint probability density function"
+    },
+    {
+      "start": 2814.44,
+      "duration": 0.0,
+      "text": "the joint probability density function of<00:46:55.200><c> x</c><00:46:55.880><c> x</c><00:46:56.160><c> t</c><00:46:56.800><c> is</c><00:46:57.400><c> a</c><00:46:57.520><c> Gaussian,</c>"
+    },
+    {
+      "start": 2818.91,
+      "duration": 0.0,
+      "text": "of x x t is a Gaussian,"
+    },
+    {
+      "start": 2818.92,
+      "duration": 0.0,
+      "text": "of x x t is a Gaussian, we<00:46:59.280><c> can</c><00:47:00.000><c> then</c><00:47:00.680><c> use</c><00:47:01.359><c> so</c><00:47:01.480><c> we</c><00:47:01.640><c> can</c><00:47:02.200><c> then</c><00:47:02.640><c> replace</c>"
+    },
+    {
+      "start": 2823.67,
+      "duration": 0.0,
+      "text": "we can then use so we can then replace"
+    },
+    {
+      "start": 2823.68,
+      "duration": 0.0,
+      "text": "we can then use so we can then replace our<00:47:04.080><c> matrix</c><00:47:05.040><c> m</c><00:47:05.240><c> of</c><00:47:05.440><c> x</c><00:47:05.840><c> which</c><00:47:06.040><c> is</c><00:47:06.440><c> state</c>"
+    },
+    {
+      "start": 2827.15,
+      "duration": 0.0,
+      "text": "our matrix m of x which is state"
+    },
+    {
+      "start": 2827.16,
+      "duration": 0.0,
+      "text": "our matrix m of x which is state dependent<00:47:08.120><c> with</c><00:47:08.680><c> just</c><00:47:09.000><c> the</c><00:47:09.120><c> matrix</c><00:47:09.680><c> phi</c>"
+    },
+    {
+      "start": 2830.75,
+      "duration": 0.0,
+      "text": "dependent with just the matrix phi"
+    },
+    {
+      "start": 2830.76,
+      "duration": 0.0,
+      "text": "dependent with just the matrix phi and<00:47:11.120><c> we</c><00:47:11.280><c> have</c><00:47:11.640><c> a</c><00:47:11.720><c> model</c><00:47:12.400><c> that</c><00:47:12.640><c> by</c><00:47:12.800><c> construction</c>"
+    },
+    {
+      "start": 2833.59,
+      "duration": 0.0,
+      "text": "and we have a model that by construction"
+    },
+    {
+      "start": 2833.6,
+      "duration": 0.0,
+      "text": "and we have a model that by construction reproduces<00:47:14.720><c> both</c><00:47:15.200><c> the</c><00:47:15.320><c> temporal</c>"
+    },
+    {
+      "start": 2835.87,
+      "duration": 0.0,
+      "text": "reproduces both the temporal"
+    },
+    {
+      "start": 2835.88,
+      "duration": 0.0,
+      "text": "reproduces both the temporal correlations<00:47:17.200><c> and</c><00:47:17.400><c> the</c><00:47:17.480><c> steady</c><00:47:17.720><c> state</c>"
+    },
+    {
+      "start": 2838.03,
+      "duration": 0.0,
+      "text": "correlations and the steady state"
+    },
+    {
+      "start": 2838.04,
+      "duration": 0.0,
+      "text": "correlations and the steady state distribution."
+    },
+    {
+      "start": 2839.91,
+      "duration": 0.0,
+      "text": "distribution."
+    },
+    {
+      "start": 2839.92,
+      "duration": 0.0,
+      "text": "distribution. Okay,<00:47:20.320><c> so</c><00:47:20.520><c> if</c>"
+    },
+    {
+      "start": 2841.79,
+      "duration": 0.0,
+      "text": "Okay, so if"
+    },
+    {
+      "start": 2841.8,
+      "duration": 0.0,
+      "text": "Okay, so if this<00:47:22.120><c> term</c><00:47:22.880><c> so</c><00:47:23.080><c> if</c><00:47:23.400><c> m</c><00:47:23.600><c> of</c><00:47:23.800><c> t</c><00:47:24.359><c> is</c><00:47:24.600><c> linear</c><00:47:25.400><c> in</c><00:47:25.560><c> x</c><00:47:25.720><c> 0</c>"
+    },
+    {
+      "start": 2846.59,
+      "duration": 0.0,
+      "text": "this term so if m of t is linear in x 0"
+    },
+    {
+      "start": 2846.6,
+      "duration": 0.0,
+      "text": "this term so if m of t is linear in x 0 which<00:47:26.800><c> is</c><00:47:26.920><c> often</c><00:47:27.320><c> the</c><00:47:27.680><c> case</c><00:47:28.120><c> because</c><00:47:28.840><c> if</c><00:47:29.359><c> the</c>"
+    },
+    {
+      "start": 2849.47,
+      "duration": 0.0,
+      "text": "which is often the case because if the"
+    },
+    {
+      "start": 2849.48,
+      "duration": 0.0,
+      "text": "which is often the case because if the conditional<00:47:30.240><c> probability</c><00:47:31.000><c> density</c><00:47:32.280><c> so</c><00:47:32.440><c> if</c>"
+    },
+    {
+      "start": 2852.59,
+      "duration": 0.0,
+      "text": "conditional probability density so if"
+    },
+    {
+      "start": 2852.6,
+      "duration": 0.0,
+      "text": "conditional probability density so if the<00:47:32.720><c> joint</c><00:47:33.160><c> probability</c><00:47:33.800><c> density</c><00:47:34.440><c> of</c><00:47:34.680><c> x</c><00:47:34.840><c> 0</c><00:47:35.320><c> and</c>"
+    },
+    {
+      "start": 2855.67,
+      "duration": 0.0,
+      "text": "the joint probability density of x 0 and"
+    },
+    {
+      "start": 2855.68,
+      "duration": 0.0,
+      "text": "the joint probability density of x 0 and and<00:47:35.840><c> x</c><00:47:36.000><c> t</c><00:47:36.200><c> can</c><00:47:36.400><c> be</c><00:47:36.480><c> approximated</c><00:47:37.440><c> with</c><00:47:37.600><c> a</c>"
+    },
+    {
+      "start": 2857.63,
+      "duration": 0.0,
+      "text": "and x t can be approximated with a"
+    },
+    {
+      "start": 2857.64,
+      "duration": 0.0,
+      "text": "and x t can be approximated with a Gaussian<00:47:38.080><c> distribution,</c><00:47:39.080><c> then</c><00:47:40.120><c> m</c><00:47:40.480><c> of</c><00:47:40.680><c> t</c>"
+    },
+    {
+      "start": 2860.87,
+      "duration": 0.0,
+      "text": "Gaussian distribution, then m of t"
+    },
+    {
+      "start": 2860.88,
+      "duration": 0.0,
+      "text": "Gaussian distribution, then m of t depends<00:47:41.400><c> linearly</c><00:47:42.200><c> on</c><00:47:42.400><c> x</c><00:47:42.520><c> 0.</c>"
+    },
+    {
+      "start": 2863.83,
+      "duration": 0.0,
+      "text": "depends linearly on x 0."
+    },
+    {
+      "start": 2863.84,
+      "duration": 0.0,
+      "text": "depends linearly on x 0. So<00:47:44.200><c> if</c><00:47:44.440><c> this</c><00:47:44.680><c> term</c><00:47:45.400><c> is</c><00:47:45.600><c> negligible,</c>"
+    },
+    {
+      "start": 2867.47,
+      "duration": 0.0,
+      "text": "So if this term is negligible,"
+    },
+    {
+      "start": 2867.48,
+      "duration": 0.0,
+      "text": "So if this term is negligible, then<00:47:48.080><c> we</c><00:47:48.200><c> can</c>"
+    },
+    {
+      "start": 2869.349,
+      "duration": 0.0,
+      "text": "then we can"
+    },
+    {
+      "start": 2869.359,
+      "duration": 0.0,
+      "text": "then we can reproduce<00:47:50.359><c> the</c><00:47:50.640><c> time</c><00:47:50.960><c> correlations</c><00:47:51.840><c> of</c><00:47:52.040><c> the</c>"
+    },
+    {
+      "start": 2872.11,
+      "duration": 0.0,
+      "text": "reproduce the time correlations of the"
+    },
+    {
+      "start": 2872.12,
+      "duration": 0.0,
+      "text": "reproduce the time correlations of the observed<00:47:52.560><c> data</c><00:47:53.240><c> just</c><00:47:53.680><c> using</c><00:47:54.640><c> phi,</c><00:47:55.120><c> so</c><00:47:55.240><c> this</c>"
+    },
+    {
+      "start": 2875.43,
+      "duration": 0.0,
+      "text": "observed data just using phi, so this"
+    },
+    {
+      "start": 2875.44,
+      "duration": 0.0,
+      "text": "observed data just using phi, so this constant<00:47:55.880><c> matrix</c><00:47:56.280><c> phi</c><00:47:56.440><c> that</c><00:47:56.640><c> we</c><00:47:56.760><c> can</c><00:47:56.920><c> easily</c>"
+    },
+    {
+      "start": 2877.19,
+      "duration": 0.0,
+      "text": "constant matrix phi that we can easily"
+    },
+    {
+      "start": 2877.2,
+      "duration": 0.0,
+      "text": "constant matrix phi that we can easily determine<00:47:57.880><c> from</c><00:47:58.080><c> the</c><00:47:58.160><c> correlation</c><00:47:58.640><c> function</c>"
+    },
+    {
+      "start": 2879.71,
+      "duration": 0.0,
+      "text": "determine from the correlation function"
+    },
+    {
+      "start": 2879.72,
+      "duration": 0.0,
+      "text": "determine from the correlation function instead<00:48:00.440><c> of</c><00:48:00.880><c> the</c><00:48:01.000><c> state</c><00:48:01.240><c> dependent</c><00:48:01.840><c> matrix</c><00:48:02.280><c> m.</c>"
+    },
+    {
+      "start": 2883.07,
+      "duration": 0.0,
+      "text": "instead of the state dependent matrix m."
+    },
+    {
+      "start": 2883.08,
+      "duration": 0.0,
+      "text": "instead of the state dependent matrix m. And<00:48:03.240><c> then</c><00:48:03.440><c> we</c><00:48:03.600><c> have</c><00:48:03.840><c> built</c><00:48:04.160><c> a</c><00:48:04.359><c> Langevin</c>"
+    },
+    {
+      "start": 2884.75,
+      "duration": 0.0,
+      "text": "And then we have built a Langevin"
+    },
+    {
+      "start": 2884.76,
+      "duration": 0.0,
+      "text": "And then we have built a Langevin equation<00:48:05.520><c> that</c><00:48:05.760><c> by</c><00:48:05.880><c> construction</c><00:48:06.640><c> reproduces</c>"
+    },
+    {
+      "start": 2887.31,
+      "duration": 0.0,
+      "text": "equation that by construction reproduces"
+    },
+    {
+      "start": 2887.32,
+      "duration": 0.0,
+      "text": "equation that by construction reproduces both<00:48:07.760><c> the</c><00:48:07.880><c> steady</c><00:48:08.200><c> state</c><00:48:08.560><c> distribution</c><00:48:09.680><c> and</c>"
+    },
+    {
+      "start": 2889.83,
+      "duration": 0.0,
+      "text": "both the steady state distribution and"
+    },
+    {
+      "start": 2889.84,
+      "duration": 0.0,
+      "text": "both the steady state distribution and the<00:48:09.920><c> time</c><00:48:10.160><c> correlations.</c>"
+    },
+    {
+      "start": 2891.63,
+      "duration": 0.0,
+      "text": "the time correlations."
+    },
+    {
+      "start": 2891.64,
+      "duration": 0.0,
+      "text": "the time correlations. If<00:48:11.920><c> we</c><00:48:12.040><c> want</c><00:48:12.400><c> instead</c><00:48:13.160><c> to</c><00:48:13.359><c> add</c><00:48:14.080><c> more</c>"
+    },
+    {
+      "start": 2895.59,
+      "duration": 0.0,
+      "text": "If we want instead to add more"
+    },
+    {
+      "start": 2895.6,
+      "duration": 0.0,
+      "text": "If we want instead to add more constraint<00:48:16.560><c> on</c><00:48:16.840><c> the</c><00:48:16.960><c> correlations,</c><00:48:17.920><c> so</c><00:48:18.160><c> we</c>"
+    },
+    {
+      "start": 2898.27,
+      "duration": 0.0,
+      "text": "constraint on the correlations, so we"
+    },
+    {
+      "start": 2898.28,
+      "duration": 0.0,
+      "text": "constraint on the correlations, so we want<00:48:18.520><c> to</c><00:48:18.640><c> add</c><00:48:18.840><c> more</c><00:48:19.040><c> constraints</c><00:48:19.640><c> on</c><00:48:19.720><c> the</c>"
+    },
+    {
+      "start": 2899.83,
+      "duration": 0.0,
+      "text": "want to add more constraints on the"
+    },
+    {
+      "start": 2899.84,
+      "duration": 0.0,
+      "text": "want to add more constraints on the dynamics<00:48:20.440><c> adding</c><00:48:20.760><c> more</c><00:48:21.000><c> correlations,</c><00:48:22.200><c> then</c>"
+    },
+    {
+      "start": 2903.31,
+      "duration": 0.0,
+      "text": "dynamics adding more correlations, then"
+    },
+    {
+      "start": 2903.32,
+      "duration": 0.0,
+      "text": "dynamics adding more correlations, then we<00:48:23.480><c> have</c><00:48:24.240><c> to</c><00:48:24.680><c> obtain</c>"
+    },
+    {
+      "start": 2906.31,
+      "duration": 0.0,
+      "text": "we have to obtain"
+    },
+    {
+      "start": 2906.32,
+      "duration": 0.0,
+      "text": "we have to obtain also<00:48:27.440><c> the</c><00:48:27.840><c> matrix</c><00:48:28.520><c> delta</c><00:48:28.880><c> m</c><00:48:29.440><c> that</c><00:48:29.640><c> we</c><00:48:29.720><c> can</c>"
+    },
+    {
+      "start": 2909.99,
+      "duration": 0.0,
+      "text": "also the matrix delta m that we can"
+    },
+    {
+      "start": 2910.0,
+      "duration": 0.0,
+      "text": "also the matrix delta m that we can parameterize<00:48:30.760><c> with</c><00:48:30.960><c> a</c><00:48:31.000><c> neural</c><00:48:31.240><c> network.</c>"
+    },
+    {
+      "start": 2912.63,
+      "duration": 0.0,
+      "text": "parameterize with a neural network."
+    },
+    {
+      "start": 2912.64,
+      "duration": 0.0,
+      "text": "parameterize with a neural network. So<00:48:32.880><c> in</c><00:48:32.960><c> this</c><00:48:33.080><c> specific</c><00:48:33.480><c> case,</c><00:48:34.120><c> we</c>"
+    },
+    {
+      "start": 2914.27,
+      "duration": 0.0,
+      "text": "So in this specific case, we"
+    },
+    {
+      "start": 2914.28,
+      "duration": 0.0,
+      "text": "So in this specific case, we parameterize<00:48:35.480><c> the</c><00:48:35.640><c> whole</c><00:48:36.040><c> m</c><00:48:36.240><c> of</c><00:48:36.440><c> x</c><00:48:36.720><c> with</c><00:48:36.880><c> a</c>"
+    },
+    {
+      "start": 2916.91,
+      "duration": 0.0,
+      "text": "parameterize the whole m of x with a"
+    },
+    {
+      "start": 2916.92,
+      "duration": 0.0,
+      "text": "parameterize the whole m of x with a neural<00:48:37.160><c> network</c><00:48:37.960><c> and</c><00:48:38.160><c> then</c><00:48:38.480><c> we</c><00:48:38.600><c> define</c><00:48:39.160><c> delta</c>"
+    },
+    {
+      "start": 2919.59,
+      "duration": 0.0,
+      "text": "neural network and then we define delta"
+    },
+    {
+      "start": 2919.6,
+      "duration": 0.0,
+      "text": "neural network and then we define delta m<00:48:39.920><c> of</c><00:48:40.080><c> x</c><00:48:40.600><c> as</c><00:48:41.040><c> m</c><00:48:41.480><c> theta</c><00:48:41.840><c> minus</c><00:48:42.160><c> phi</c>"
+    },
+    {
+      "start": 2923.15,
+      "duration": 0.0,
+      "text": "m of x as m theta minus phi"
+    },
+    {
+      "start": 2923.16,
+      "duration": 0.0,
+      "text": "m of x as m theta minus phi and<00:48:43.440><c> then</c><00:48:43.760><c> we</c><00:48:43.960><c> can</c><00:48:44.200><c> train</c><00:48:45.120><c> a</c><00:48:45.160><c> neural</c><00:48:45.480><c> network</c>"
+    },
+    {
+      "start": 2925.99,
+      "duration": 0.0,
+      "text": "and then we can train a neural network"
+    },
+    {
+      "start": 2926.0,
+      "duration": 0.0,
+      "text": "and then we can train a neural network delta<00:48:46.359><c> m</c><00:48:47.000><c> of</c><00:48:47.200><c> theta</c><00:48:48.120><c> to</c><00:48:48.240><c> minimize</c><00:48:48.840><c> this</c><00:48:49.040><c> loss</c>"
+    },
+    {
+      "start": 2929.23,
+      "duration": 0.0,
+      "text": "delta m of theta to minimize this loss"
+    },
+    {
+      "start": 2929.24,
+      "duration": 0.0,
+      "text": "delta m of theta to minimize this loss function.<00:48:49.840><c> So</c><00:48:50.000><c> we</c><00:48:50.120><c> have</c><00:48:50.440><c> this</c><00:48:50.760><c> first</c><00:48:51.120><c> term</c>"
+    },
+    {
+      "start": 2932.07,
+      "duration": 0.0,
+      "text": "function. So we have this first term"
+    },
+    {
+      "start": 2932.08,
+      "duration": 0.0,
+      "text": "function. So we have this first term that<00:48:52.560><c> that</c><00:48:52.760><c> essentially</c><00:48:53.400><c> forces</c><00:48:53.920><c> the</c><00:48:54.040><c> neural</c>"
+    },
+    {
+      "start": 2934.27,
+      "duration": 0.0,
+      "text": "that that essentially forces the neural"
+    },
+    {
+      "start": 2934.28,
+      "duration": 0.0,
+      "text": "that that essentially forces the neural network<00:48:54.920><c> to</c><00:48:55.080><c> learn</c><00:48:55.840><c> the</c><00:48:56.720><c> set</c><00:48:57.400><c> of</c><00:48:57.680><c> correlation</c>"
+    },
+    {
+      "start": 2938.23,
+      "duration": 0.0,
+      "text": "network to learn the set of correlation"
+    },
+    {
+      "start": 2938.24,
+      "duration": 0.0,
+      "text": "network to learn the set of correlation functions<00:48:58.840><c> that</c><00:48:59.040><c> we</c><00:48:59.160><c> want</c><00:48:59.760><c> our</c><00:48:59.960><c> system</c><00:49:00.400><c> to</c>"
+    },
+    {
+      "start": 2940.51,
+      "duration": 0.0,
+      "text": "functions that we want our system to"
+    },
+    {
+      "start": 2940.52,
+      "duration": 0.0,
+      "text": "functions that we want our system to reproduce.<00:49:01.640><c> Then</c><00:49:01.880><c> we</c><00:49:02.000><c> have</c><00:49:02.240><c> this</c><00:49:02.440><c> penalty</c>"
+    },
+    {
+      "start": 2942.91,
+      "duration": 0.0,
+      "text": "reproduce. Then we have this penalty"
+    },
+    {
+      "start": 2942.92,
+      "duration": 0.0,
+      "text": "reproduce. Then we have this penalty term<00:49:03.600><c> that</c><00:49:03.760><c> essentially</c><00:49:04.240><c> enforces</c><00:49:05.400><c> that</c><00:49:05.560><c> the</c>"
+    },
+    {
+      "start": 2945.67,
+      "duration": 0.0,
+      "text": "term that essentially enforces that the"
+    },
+    {
+      "start": 2945.68,
+      "duration": 0.0,
+      "text": "term that essentially enforces that the average<00:49:06.080><c> value</c><00:49:06.840><c> of</c><00:49:07.440><c> delta</c><00:49:07.960><c> m</c><00:49:08.480><c> is</c><00:49:08.720><c> equal</c><00:49:08.920><c> to</c><00:49:09.040><c> 0</c>"
+    },
+    {
+      "start": 2949.43,
+      "duration": 0.0,
+      "text": "average value of delta m is equal to 0"
+    },
+    {
+      "start": 2949.44,
+      "duration": 0.0,
+      "text": "average value of delta m is equal to 0 plus<00:49:09.800><c> we</c><00:49:09.960><c> have</c><00:49:10.640><c> a</c><00:49:10.720><c> regularization</c><00:49:11.600><c> term.</c>"
+    },
+    {
+      "start": 2952.95,
+      "duration": 0.0,
+      "text": "plus we have a regularization term."
+    },
+    {
+      "start": 2952.96,
+      "duration": 0.0,
+      "text": "plus we have a regularization term. But<00:49:13.160><c> you</c><00:49:13.240><c> can</c><00:49:13.440><c> see</c><00:49:13.560><c> here</c><00:49:14.160><c> that</c><00:49:14.600><c> we</c><00:49:14.800><c> are</c><00:49:15.359><c> so</c><00:49:15.520><c> we</c>"
+    },
+    {
+      "start": 2955.63,
+      "duration": 0.0,
+      "text": "But you can see here that we are so we"
+    },
+    {
+      "start": 2955.64,
+      "duration": 0.0,
+      "text": "But you can see here that we are so we are<00:49:15.760><c> writing</c>"
+    },
+    {
+      "start": 2957.39,
+      "duration": 0.0,
+      "text": "are writing"
+    },
+    {
+      "start": 2957.4,
+      "duration": 0.0,
+      "text": "are writing a<00:49:17.600><c> loss</c><00:49:17.800><c> function</c><00:49:18.400><c> that</c><00:49:18.560><c> doesn't</c><00:49:18.800><c> depend</c><00:49:19.560><c> on</c><00:49:20.080><c> a</c>"
+    },
+    {
+      "start": 2960.15,
+      "duration": 0.0,
+      "text": "a loss function that doesn't depend on a"
+    },
+    {
+      "start": 2960.16,
+      "duration": 0.0,
+      "text": "a loss function that doesn't depend on a forward<00:49:20.640><c> model</c><00:49:20.920><c> integration.</c><00:49:21.640><c> So</c><00:49:21.840><c> we</c><00:49:21.960><c> never</c>"
+    },
+    {
+      "start": 2962.31,
+      "duration": 0.0,
+      "text": "forward model integration. So we never"
+    },
+    {
+      "start": 2962.32,
+      "duration": 0.0,
+      "text": "forward model integration. So we never have<00:49:22.800><c> to</c><00:49:22.920><c> integrate</c><00:49:23.880><c> our</c><00:49:25.160><c> Langevin</c><00:49:25.680><c> equation</c>"
+    },
+    {
+      "start": 2966.15,
+      "duration": 0.0,
+      "text": "have to integrate our Langevin equation"
+    },
+    {
+      "start": 2966.16,
+      "duration": 0.0,
+      "text": "have to integrate our Langevin equation forward<00:49:26.600><c> in</c><00:49:26.680><c> time.</c>"
+    },
+    {
+      "start": 2967.95,
+      "duration": 0.0,
+      "text": "forward in time."
+    },
+    {
+      "start": 2967.96,
+      "duration": 0.0,
+      "text": "forward in time. We<00:49:28.880><c> just</c><00:49:29.160><c> use</c><00:49:29.560><c> the</c><00:49:29.960><c> the</c><00:49:30.080><c> knowledge</c><00:49:30.680><c> of</c><00:49:31.000><c> the</c>"
+    },
+    {
+      "start": 2971.11,
+      "duration": 0.0,
+      "text": "We just use the the knowledge of the"
+    },
+    {
+      "start": 2971.12,
+      "duration": 0.0,
+      "text": "We just use the the knowledge of the conditional<00:49:31.640><c> score,</c><00:49:32.480><c> the</c><00:49:32.640><c> score</c><00:49:33.120><c> function,</c>"
+    },
+    {
+      "start": 2974.19,
+      "duration": 0.0,
+      "text": "conditional score, the score function,"
+    },
+    {
+      "start": 2974.2,
+      "duration": 0.0,
+      "text": "conditional score, the score function, and<00:49:35.040><c> the</c><00:49:35.480><c> time</c><00:49:35.760><c> derivative</c><00:49:36.320><c> of</c><00:49:36.440><c> the</c>"
+    },
+    {
+      "start": 2976.55,
+      "duration": 0.0,
+      "text": "and the time derivative of the"
+    },
+    {
+      "start": 2976.56,
+      "duration": 0.0,
+      "text": "and the time derivative of the correlation<00:49:37.120><c> functions</c><00:49:37.960><c> to</c><00:49:38.080><c> train</c><00:49:38.760><c> the</c>"
+    },
+    {
+      "start": 2978.87,
+      "duration": 0.0,
+      "text": "correlation functions to train the"
+    },
+    {
+      "start": 2978.88,
+      "duration": 0.0,
+      "text": "correlation functions to train the neural<00:49:39.160><c> network</c><00:49:39.640><c> for</c><00:49:39.840><c> delta</c><00:49:40.080><c> M.</c>"
+    },
+    {
+      "start": 2981.67,
+      "duration": 0.0,
+      "text": "neural network for delta M."
+    },
+    {
+      "start": 2981.68,
+      "duration": 0.0,
+      "text": "neural network for delta M. And<00:49:41.840><c> this</c><00:49:41.960><c> can</c><00:49:42.120><c> be</c><00:49:42.480><c> can</c><00:49:42.600><c> become</c><00:49:43.080><c> extremely</c>"
+    },
+    {
+      "start": 2983.99,
+      "duration": 0.0,
+      "text": "And this can be can become extremely"
+    },
+    {
+      "start": 2984.0,
+      "duration": 0.0,
+      "text": "And this can be can become extremely efficient<00:49:44.960><c> when</c><00:49:45.920><c> the</c><00:49:46.520><c> model</c><00:49:46.960><c> that</c><00:49:47.080><c> we</c><00:49:47.200><c> want</c><00:49:47.440><c> to</c>"
+    },
+    {
+      "start": 2987.51,
+      "duration": 0.0,
+      "text": "efficient when the model that we want to"
+    },
+    {
+      "start": 2987.52,
+      "duration": 0.0,
+      "text": "efficient when the model that we want to integrate<00:49:48.200><c> becomes</c><00:49:48.840><c> very</c><00:49:49.120><c> computationally</c>"
+    },
+    {
+      "start": 2989.71,
+      "duration": 0.0,
+      "text": "integrate becomes very computationally"
+    },
+    {
+      "start": 2989.72,
+      "duration": 0.0,
+      "text": "integrate becomes very computationally expensive."
+    },
+    {
+      "start": 2991.27,
+      "duration": 0.0,
+      "text": "expensive."
+    },
+    {
+      "start": 2991.28,
+      "duration": 0.0,
+      "text": "expensive. Okay,<00:49:51.600><c> so</c><00:49:51.720><c> this</c><00:49:51.920><c> is</c><00:49:52.000><c> the</c><00:49:52.280><c> methodology.</c>"
+    },
+    {
+      "start": 2993.63,
+      "duration": 0.0,
+      "text": "Okay, so this is the methodology."
+    },
+    {
+      "start": 2993.64,
+      "duration": 0.0,
+      "text": "Okay, so this is the methodology. So<00:49:53.720><c> we</c><00:49:53.840><c> have</c><00:49:53.960><c> seen</c><00:49:54.480><c> that's</c>"
+    },
+    {
+      "start": 2995.79,
+      "duration": 0.0,
+      "text": "So we have seen that's"
+    },
+    {
+      "start": 2995.8,
+      "duration": 0.0,
+      "text": "So we have seen that's yeah,<00:49:56.000><c> we</c><00:49:56.280><c> we</c><00:49:56.400><c> are</c><00:49:56.520><c> able</c><00:49:56.840><c> to</c><00:49:56.960><c> train</c><00:49:57.280><c> this</c>"
+    },
+    {
+      "start": 2997.47,
+      "duration": 0.0,
+      "text": "yeah, we we are able to train this"
+    },
+    {
+      "start": 2997.48,
+      "duration": 0.0,
+      "text": "yeah, we we are able to train this neural<00:49:57.720><c> network</c><00:49:58.400><c> without</c><00:49:58.800><c> integrating</c><00:49:59.280><c> the</c>"
+    },
+    {
+      "start": 2999.35,
+      "duration": 0.0,
+      "text": "neural network without integrating the"
+    },
+    {
+      "start": 2999.36,
+      "duration": 0.0,
+      "text": "neural network without integrating the model<00:49:59.640><c> forward</c><00:50:00.400><c> and</c><00:50:01.040><c> when</c><00:50:01.920><c> so</c><00:50:02.120><c> for</c><00:50:02.360><c> some</c>"
+    },
+    {
+      "start": 3002.59,
+      "duration": 0.0,
+      "text": "model forward and when so for some"
+    },
+    {
+      "start": 3002.6,
+      "duration": 0.0,
+      "text": "model forward and when so for some specific<00:50:03.040><c> cases</c><00:50:03.680><c> we</c><00:50:03.800><c> can</c><00:50:04.040><c> simplify</c><00:50:05.440><c> the</c><00:50:05.960><c> shape</c>"
+    },
+    {
+      "start": 3007.15,
+      "duration": 0.0,
+      "text": "specific cases we can simplify the shape"
+    },
+    {
+      "start": 3007.16,
+      "duration": 0.0,
+      "text": "specific cases we can simplify the shape so<00:50:07.360><c> the</c><00:50:07.480><c> functional</c><00:50:08.040><c> form</c><00:50:08.560><c> of</c><00:50:08.840><c> M</c>"
+    },
+    {
+      "start": 3010.03,
+      "duration": 0.0,
+      "text": "so the functional form of M"
+    },
+    {
+      "start": 3010.04,
+      "duration": 0.0,
+      "text": "so the functional form of M by<00:50:10.600><c> replacing</c><00:50:11.200><c> them</c><00:50:11.480><c> with</c><00:50:11.640><c> a</c><00:50:11.680><c> constant</c><00:50:12.400><c> if</c><00:50:12.600><c> we</c>"
+    },
+    {
+      "start": 3012.67,
+      "duration": 0.0,
+      "text": "by replacing them with a constant if we"
+    },
+    {
+      "start": 3012.68,
+      "duration": 0.0,
+      "text": "by replacing them with a constant if we are<00:50:12.800><c> just</c><00:50:13.040><c> interested</c><00:50:13.760><c> in</c><00:50:13.920><c> the</c><00:50:14.040><c> time</c>"
+    },
+    {
+      "start": 3014.27,
+      "duration": 0.0,
+      "text": "are just interested in the time"
+    },
+    {
+      "start": 3014.28,
+      "duration": 0.0,
+      "text": "are just interested in the time correlation<00:50:14.760><c> of</c><00:50:14.840><c> the</c><00:50:14.960><c> systems.</c><00:50:15.960><c> So</c><00:50:16.080><c> now</c><00:50:16.800><c> I</c>"
+    },
+    {
+      "start": 3016.95,
+      "duration": 0.0,
+      "text": "correlation of the systems. So now I"
+    },
+    {
+      "start": 3016.96,
+      "duration": 0.0,
+      "text": "correlation of the systems. So now I will<00:50:17.240><c> conclude</c><00:50:17.880><c> showing</c><00:50:18.360><c> you</c><00:50:18.840><c> some</c>"
+    },
+    {
+      "start": 3018.99,
+      "duration": 0.0,
+      "text": "will conclude showing you some"
+    },
+    {
+      "start": 3019.0,
+      "duration": 0.0,
+      "text": "will conclude showing you some application<00:50:19.640><c> of</c><00:50:19.760><c> these</c><00:50:19.960><c> ideas.</c>"
+    },
+    {
+      "start": 3021.15,
+      "duration": 0.0,
+      "text": "application of these ideas."
+    },
+    {
+      "start": 3021.16,
+      "duration": 0.0,
+      "text": "application of these ideas. So<00:50:21.440><c> I</c><00:50:21.600><c> start</c><00:50:22.280><c> from</c><00:50:22.640><c> an</c><00:50:22.760><c> analytic</c><00:50:23.280><c> warm</c><00:50:23.520><c> up.</c><00:50:24.120><c> So</c>"
+    },
+    {
+      "start": 3024.95,
+      "duration": 0.0,
+      "text": "So I start from an analytic warm up. So"
+    },
+    {
+      "start": 3024.96,
+      "duration": 0.0,
+      "text": "So I start from an analytic warm up. So we<00:50:25.120><c> consider</c><00:50:25.760><c> this</c><00:50:26.440><c> one-dimensional</c><00:50:27.280><c> system</c>"
+    },
+    {
+      "start": 3028.63,
+      "duration": 0.0,
+      "text": "we consider this one-dimensional system"
+    },
+    {
+      "start": 3028.64,
+      "duration": 0.0,
+      "text": "we consider this one-dimensional system for<00:50:28.880><c> which</c><00:50:29.320><c> we</c><00:50:29.480><c> can</c><00:50:29.680><c> determine</c><00:50:30.640><c> analytically</c>"
+    },
+    {
+      "start": 3032.47,
+      "duration": 0.0,
+      "text": "for which we can determine analytically"
+    },
+    {
+      "start": 3032.48,
+      "duration": 0.0,
+      "text": "for which we can determine analytically all<00:50:33.000><c> the</c><00:50:33.120><c> relevant</c><00:50:33.520><c> quantities.</c>"
+    },
+    {
+      "start": 3035.95,
+      "duration": 0.0,
+      "text": "all the relevant quantities."
+    },
+    {
+      "start": 3035.96,
+      "duration": 0.0,
+      "text": "all the relevant quantities. So<00:50:36.080><c> we</c><00:50:36.200><c> can</c><00:50:36.360><c> derive</c><00:50:37.280><c> the</c><00:50:37.400><c> station</c><00:50:37.760><c> the</c>"
+    },
+    {
+      "start": 3038.23,
+      "duration": 0.0,
+      "text": "So we can derive the station the"
+    },
+    {
+      "start": 3038.24,
+      "duration": 0.0,
+      "text": "So we can derive the station the conditional<00:50:38.760><c> score,</c><00:50:39.680><c> the</c><00:50:40.440><c> stationary</c><00:50:41.000><c> score,</c>"
+    },
+    {
+      "start": 3041.99,
+      "duration": 0.0,
+      "text": "conditional score, the stationary score,"
+    },
+    {
+      "start": 3042.0,
+      "duration": 0.0,
+      "text": "conditional score, the stationary score, the<00:50:42.160><c> time</c><00:50:42.440><c> derivative</c><00:50:43.040><c> of</c><00:50:43.160><c> the</c><00:50:43.240><c> correlation</c>"
+    },
+    {
+      "start": 3043.83,
+      "duration": 0.0,
+      "text": "the time derivative of the correlation"
+    },
+    {
+      "start": 3043.84,
+      "duration": 0.0,
+      "text": "the time derivative of the correlation functions<00:50:44.440><c> and</c><00:50:44.600><c> so</c><00:50:44.720><c> on.</c>"
+    },
+    {
+      "start": 3045.79,
+      "duration": 0.0,
+      "text": "functions and so on."
+    },
+    {
+      "start": 3045.8,
+      "duration": 0.0,
+      "text": "functions and so on. These<00:50:46.240><c> are</c><00:50:46.520><c> the</c><00:50:46.680><c> true</c><00:50:47.680><c> values</c><00:50:48.320><c> for</c><00:50:48.720><c> fee</c><00:50:49.160><c> and</c>"
+    },
+    {
+      "start": 3049.35,
+      "duration": 0.0,
+      "text": "These are the true values for fee and"
+    },
+    {
+      "start": 3049.36,
+      "duration": 0.0,
+      "text": "These are the true values for fee and delta<00:50:49.680><c> M.</c>"
+    },
+    {
+      "start": 3051.27,
+      "duration": 0.0,
+      "text": "delta M."
+    },
+    {
+      "start": 3051.28,
+      "duration": 0.0,
+      "text": "delta M. And<00:50:51.560><c> then</c><00:50:51.880><c> by</c><00:50:52.120><c> applying</c><00:50:52.760><c> the</c><00:50:52.920><c> method</c><00:50:53.280><c> I</c>"
+    },
+    {
+      "start": 3054.29,
+      "duration": 0.0,
+      "text": "And then by applying the method I"
+    },
+    {
+      "start": 3054.3,
+      "duration": 0.0,
+      "text": "And then by applying the method I &gt;&gt; [clears throat]"
+    },
+    {
+      "start": 3054.79,
+      "duration": 0.0,
+      "text": "&gt;&gt; [clears throat]"
+    },
+    {
+      "start": 3054.8,
+      "duration": 0.0,
+      "text": "&gt;&gt; [clears throat] &gt;&gt; discussed<00:50:55.400><c> before</c><00:50:56.240><c> we</c><00:50:56.400><c> can</c><00:50:56.600><c> obtain</c><00:50:57.040><c> them</c>"
+    },
+    {
+      "start": 3057.51,
+      "duration": 0.0,
+      "text": "&gt;&gt; discussed before we can obtain them"
+    },
+    {
+      "start": 3057.52,
+      "duration": 0.0,
+      "text": "&gt;&gt; discussed before we can obtain them using<00:50:58.400><c> the</c><00:50:58.560><c> relationship</c><00:50:59.480><c> that</c><00:50:59.640><c> I</c><00:50:59.720><c> showed</c><00:50:59.960><c> you</c>"
+    },
+    {
+      "start": 3060.15,
+      "duration": 0.0,
+      "text": "using the relationship that I showed you"
+    },
+    {
+      "start": 3060.16,
+      "duration": 0.0,
+      "text": "using the relationship that I showed you at<00:51:00.280><c> the</c><00:51:00.360><c> beginning.</c><00:51:01.000><c> So</c><00:51:01.120><c> using</c><00:51:01.440><c> that</c>"
+    },
+    {
+      "start": 3061.63,
+      "duration": 0.0,
+      "text": "at the beginning. So using that"
+    },
+    {
+      "start": 3061.64,
+      "duration": 0.0,
+      "text": "at the beginning. So using that relationship<00:51:02.600><c> we</c><00:51:02.760><c> recover</c><00:51:03.320><c> precisely</c>"
+    },
+    {
+      "start": 3064.59,
+      "duration": 0.0,
+      "text": "relationship we recover precisely"
+    },
+    {
+      "start": 3064.6,
+      "duration": 0.0,
+      "text": "relationship we recover precisely the<00:51:05.040><c> fee</c><00:51:05.360><c> the</c><00:51:05.520><c> correct</c><00:51:05.960><c> fee</c><00:51:06.440><c> and</c><00:51:06.680><c> the</c><00:51:06.760><c> correct</c>"
+    },
+    {
+      "start": 3067.67,
+      "duration": 0.0,
+      "text": "the fee the correct fee and the correct"
+    },
+    {
+      "start": 3067.68,
+      "duration": 0.0,
+      "text": "the fee the correct fee and the correct delta<00:51:07.960><c> M.</c>"
+    },
+    {
+      "start": 3069.23,
+      "duration": 0.0,
+      "text": "delta M."
+    },
+    {
+      "start": 3069.24,
+      "duration": 0.0,
+      "text": "delta M. So<00:51:09.360><c> this</c><00:51:09.520><c> was</c><00:51:09.720><c> just</c><00:51:10.080><c> like</c><00:51:10.280><c> a</c><00:51:10.320><c> test</c><00:51:10.920><c> where</c><00:51:11.240><c> we</c>"
+    },
+    {
+      "start": 3071.39,
+      "duration": 0.0,
+      "text": "So this was just like a test where we"
+    },
+    {
+      "start": 3071.4,
+      "duration": 0.0,
+      "text": "So this was just like a test where we have<00:51:11.960><c> we</c><00:51:12.080><c> know</c><00:51:12.200><c> everything</c><00:51:12.560><c> is</c><00:51:12.640><c> analytically.</c>"
+    },
+    {
+      "start": 3073.75,
+      "duration": 0.0,
+      "text": "have we know everything is analytically."
+    },
+    {
+      "start": 3073.76,
+      "duration": 0.0,
+      "text": "have we know everything is analytically. So<00:51:13.840><c> let's</c><00:51:14.040><c> see</c><00:51:14.840><c> a</c><00:51:14.920><c> different</c><00:51:15.280><c> case.</c><00:51:16.000><c> In</c><00:51:16.160><c> this</c>"
+    },
+    {
+      "start": 3076.31,
+      "duration": 0.0,
+      "text": "So let's see a different case. In this"
+    },
+    {
+      "start": 3076.32,
+      "duration": 0.0,
+      "text": "So let's see a different case. In this case<00:51:16.560><c> we</c><00:51:16.720><c> have</c><00:51:17.240><c> a</c><00:51:17.320><c> two-dimensional</c><00:51:18.000><c> system</c>"
+    },
+    {
+      "start": 3079.23,
+      "duration": 0.0,
+      "text": "case we have a two-dimensional system"
+    },
+    {
+      "start": 3079.24,
+      "duration": 0.0,
+      "text": "case we have a two-dimensional system with<00:51:20.400><c> where</c><00:51:20.560><c> we</c><00:51:20.720><c> have</c><00:51:21.560><c> our</c><00:51:21.840><c> drift</c><00:51:22.160><c> term</c><00:51:22.680><c> which</c>"
+    },
+    {
+      "start": 3082.87,
+      "duration": 0.0,
+      "text": "with where we have our drift term which"
+    },
+    {
+      "start": 3082.88,
+      "duration": 0.0,
+      "text": "with where we have our drift term which contains<00:51:23.400><c> both</c><00:51:24.080><c> a</c><00:51:24.160><c> term</c><00:51:24.680><c> that</c><00:51:24.840><c> can</c><00:51:24.960><c> be</c><00:51:25.080><c> written</c>"
+    },
+    {
+      "start": 3085.43,
+      "duration": 0.0,
+      "text": "contains both a term that can be written"
+    },
+    {
+      "start": 3085.44,
+      "duration": 0.0,
+      "text": "contains both a term that can be written as<00:51:25.560><c> the</c><00:51:25.640><c> gradient</c><00:51:26.080><c> of</c><00:51:26.200><c> a</c><00:51:26.240><c> potential</c><00:51:27.000><c> plus</c><00:51:27.360><c> a</c>"
+    },
+    {
+      "start": 3087.43,
+      "duration": 0.0,
+      "text": "as the gradient of a potential plus a"
+    },
+    {
+      "start": 3087.44,
+      "duration": 0.0,
+      "text": "as the gradient of a potential plus a circulatory<00:51:28.080><c> component.</c><00:51:29.200><c> We</c><00:51:29.360><c> also</c><00:51:29.800><c> have</c><00:51:30.160><c> a</c>"
+    },
+    {
+      "start": 3090.19,
+      "duration": 0.0,
+      "text": "circulatory component. We also have a"
+    },
+    {
+      "start": 3090.2,
+      "duration": 0.0,
+      "text": "circulatory component. We also have a multiplicative<00:51:30.920><c> noise.</c>"
+    },
+    {
+      "start": 3092.43,
+      "duration": 0.0,
+      "text": "multiplicative noise."
+    },
+    {
+      "start": 3092.44,
+      "duration": 0.0,
+      "text": "multiplicative noise. And<00:51:32.640><c> in</c><00:51:32.720><c> this</c><00:51:32.880><c> case</c><00:51:33.360><c> we</c><00:51:33.520><c> cannot</c><00:51:33.960><c> write</c>"
+    },
+    {
+      "start": 3094.35,
+      "duration": 0.0,
+      "text": "And in this case we cannot write"
+    },
+    {
+      "start": 3094.36,
+      "duration": 0.0,
+      "text": "And in this case we cannot write explicitly<00:51:35.200><c> the</c><00:51:35.360><c> score</c><00:51:35.680><c> function</c><00:51:36.840><c> and</c><00:51:37.360><c> the</c>"
+    },
+    {
+      "start": 3098.67,
+      "duration": 0.0,
+      "text": "explicitly the score function and the"
+    },
+    {
+      "start": 3098.68,
+      "duration": 0.0,
+      "text": "explicitly the score function and the conditional<00:51:39.240><c> score.</c><00:51:39.840><c> So</c><00:51:40.040><c> we</c><00:51:40.160><c> need</c><00:51:40.560><c> to</c><00:51:40.680><c> train</c>"
+    },
+    {
+      "start": 3101.35,
+      "duration": 0.0,
+      "text": "conditional score. So we need to train"
+    },
+    {
+      "start": 3101.36,
+      "duration": 0.0,
+      "text": "conditional score. So we need to train two<00:51:41.480><c> neural</c><00:51:41.800><c> networks</c><00:51:42.480><c> for</c><00:51:42.880><c> S</c><00:51:43.400><c> and</c><00:51:43.600><c> for</c><00:51:43.760><c> the</c>"
+    },
+    {
+      "start": 3103.87,
+      "duration": 0.0,
+      "text": "two neural networks for S and for the"
+    },
+    {
+      "start": 3103.88,
+      "duration": 0.0,
+      "text": "two neural networks for S and for the conditional<00:51:44.480><c> score.</c>"
+    },
+    {
+      "start": 3106.43,
+      "duration": 0.0,
+      "text": "conditional score."
+    },
+    {
+      "start": 3106.44,
+      "duration": 0.0,
+      "text": "conditional score. We<00:51:46.640><c> apply</c><00:51:47.120><c> the</c><00:51:47.240><c> methodology</c><00:51:47.840><c> that</c><00:51:48.040><c> I</c>"
+    },
+    {
+      "start": 3108.07,
+      "duration": 0.0,
+      "text": "We apply the methodology that I"
+    },
+    {
+      "start": 3108.08,
+      "duration": 0.0,
+      "text": "We apply the methodology that I described"
+    },
+    {
+      "start": 3109.19,
+      "duration": 0.0,
+      "text": "described"
+    },
+    {
+      "start": 3109.2,
+      "duration": 0.0,
+      "text": "described before<00:51:49.800><c> by</c><00:51:50.000><c> enforcing</c><00:51:51.240><c> the</c><00:51:51.360><c> reproduction</c><00:51:52.280><c> of</c>"
+    },
+    {
+      "start": 3112.67,
+      "duration": 0.0,
+      "text": "before by enforcing the reproduction of"
+    },
+    {
+      "start": 3112.68,
+      "duration": 0.0,
+      "text": "before by enforcing the reproduction of the<00:51:52.800><c> correlation</c><00:51:53.320><c> functions.</c><00:51:54.480><c> We</c><00:51:54.680><c> derive</c>"
+    },
+    {
+      "start": 3116.43,
+      "duration": 0.0,
+      "text": "the correlation functions. We derive"
+    },
+    {
+      "start": 3116.44,
+      "duration": 0.0,
+      "text": "the correlation functions. We derive a<00:51:56.520><c> quite</c><00:51:57.040><c> accurate</c>"
+    },
+    {
+      "start": 3118.71,
+      "duration": 0.0,
+      "text": "a quite accurate"
+    },
+    {
+      "start": 3118.72,
+      "duration": 0.0,
+      "text": "a quite accurate reconstruction<00:51:59.920><c> of</c>"
+    },
+    {
+      "start": 3121.11,
+      "duration": 0.0,
+      "text": "reconstruction of"
+    },
+    {
+      "start": 3121.12,
+      "duration": 0.0,
+      "text": "reconstruction of the<00:52:02.120><c> mobility</c><00:52:02.640><c> fields</c><00:52:03.200><c> so</c><00:52:03.480><c> the</c><00:52:04.160><c> M</c>"
+    },
+    {
+      "start": 3125.51,
+      "duration": 0.0,
+      "text": "the mobility fields so the M"
+    },
+    {
+      "start": 3125.52,
+      "duration": 0.0,
+      "text": "the mobility fields so the M tensor."
+    },
+    {
+      "start": 3127.51,
+      "duration": 0.0,
+      "text": "tensor."
+    },
+    {
+      "start": 3127.52,
+      "duration": 0.0,
+      "text": "tensor. We<00:52:07.680><c> have</c><00:52:08.040><c> some</c><00:52:08.320><c> errors</c><00:52:08.920><c> in</c><00:52:09.040><c> particular</c><00:52:09.600><c> in</c>"
+    },
+    {
+      "start": 3129.71,
+      "duration": 0.0,
+      "text": "We have some errors in particular in"
+    },
+    {
+      "start": 3129.72,
+      "duration": 0.0,
+      "text": "We have some errors in particular in this<00:52:09.960><c> term</c>"
+    },
+    {
+      "start": 3131.19,
+      "duration": 0.0,
+      "text": "this term"
+    },
+    {
+      "start": 3131.2,
+      "duration": 0.0,
+      "text": "this term but<00:52:11.680><c> even</c><00:52:11.960><c> if</c><00:52:12.800><c> so</c><00:52:12.960><c> we</c><00:52:13.080><c> have</c><00:52:14.040><c> like</c><00:52:14.600><c> some</c><00:52:14.880><c> errors</c>"
+    },
+    {
+      "start": 3135.55,
+      "duration": 0.0,
+      "text": "but even if so we have like some errors"
+    },
+    {
+      "start": 3135.56,
+      "duration": 0.0,
+      "text": "but even if so we have like some errors for"
+    },
+    {
+      "start": 3137.27,
+      "duration": 0.0,
+      "text": "for"
+    },
+    {
+      "start": 3137.28,
+      "duration": 0.0,
+      "text": "for the<00:52:17.680><c> M</c><00:52:18.040><c> to</c><00:52:18.200><c> one</c><00:52:18.520><c> terms</c><00:52:19.640><c> when</c><00:52:20.520><c> we</c><00:52:20.720><c> integrate</c><00:52:21.760><c> our</c>"
+    },
+    {
+      "start": 3142.11,
+      "duration": 0.0,
+      "text": "the M to one terms when we integrate our"
+    },
+    {
+      "start": 3142.12,
+      "duration": 0.0,
+      "text": "the M to one terms when we integrate our model<00:52:23.120><c> we</c><00:52:23.240><c> get</c><00:52:23.480><c> a</c><00:52:23.560><c> precise</c><00:52:24.440><c> recovery</c><00:52:25.480><c> of</c><00:52:25.960><c> the</c>"
+    },
+    {
+      "start": 3146.11,
+      "duration": 0.0,
+      "text": "model we get a precise recovery of the"
+    },
+    {
+      "start": 3146.12,
+      "duration": 0.0,
+      "text": "model we get a precise recovery of the univariate<00:52:26.640><c> PDF,</c><00:52:27.400><c> bivariate</c><00:52:27.920><c> PDF</c>"
+    },
+    {
+      "start": 3149.19,
+      "duration": 0.0,
+      "text": "univariate PDF, bivariate PDF"
+    },
+    {
+      "start": 3149.2,
+      "duration": 0.0,
+      "text": "univariate PDF, bivariate PDF all<00:52:29.560><c> the</c><00:52:29.680><c> correlation</c><00:52:30.200><c> functions.</c><00:52:31.080><c> Here</c><00:52:31.600><c> I'm</c>"
+    },
+    {
+      "start": 3151.79,
+      "duration": 0.0,
+      "text": "all the correlation functions. Here I'm"
+    },
+    {
+      "start": 3151.8,
+      "duration": 0.0,
+      "text": "all the correlation functions. Here I'm comparing<00:52:32.760><c> two</c><00:52:32.920><c> different</c><00:52:33.480><c> model</c>"
+    },
+    {
+      "start": 3153.83,
+      "duration": 0.0,
+      "text": "comparing two different model"
+    },
+    {
+      "start": 3153.84,
+      "duration": 0.0,
+      "text": "comparing two different model integrations.<00:52:35.160><c> We</c><00:52:35.360><c> have</c><00:52:35.880><c> the</c><00:52:36.000><c> model</c>"
+    },
+    {
+      "start": 3156.55,
+      "duration": 0.0,
+      "text": "integrations. We have the model"
+    },
+    {
+      "start": 3156.56,
+      "duration": 0.0,
+      "text": "integrations. We have the model integration<00:52:37.400><c> with</c><00:52:37.640><c> the</c><00:52:37.760><c> full</c>"
+    },
+    {
+      "start": 3158.91,
+      "duration": 0.0,
+      "text": "integration with the full"
+    },
+    {
+      "start": 3158.92,
+      "duration": 0.0,
+      "text": "integration with the full mobility<00:52:39.440><c> matrix</c><00:52:40.120><c> M</c><00:52:40.320><c> of</c><00:52:40.480><c> X</c>"
+    },
+    {
+      "start": 3161.59,
+      "duration": 0.0,
+      "text": "mobility matrix M of X"
+    },
+    {
+      "start": 3161.6,
+      "duration": 0.0,
+      "text": "mobility matrix M of X and<00:52:42.400><c> a</c><00:52:42.480><c> model</c><00:52:42.800><c> integration</c><00:52:43.440><c> where</c><00:52:43.680><c> I'm</c>"
+    },
+    {
+      "start": 3164.11,
+      "duration": 0.0,
+      "text": "and a model integration where I'm"
+    },
+    {
+      "start": 3164.12,
+      "duration": 0.0,
+      "text": "and a model integration where I'm replacing<00:52:44.840><c> the</c><00:52:45.000><c> full</c><00:52:45.200><c> mobility</c><00:52:45.680><c> matrix</c><00:52:46.640><c> with</c>"
+    },
+    {
+      "start": 3167.07,
+      "duration": 0.0,
+      "text": "replacing the full mobility matrix with"
+    },
+    {
+      "start": 3167.08,
+      "duration": 0.0,
+      "text": "replacing the full mobility matrix with a<00:52:47.120><c> fee</c><00:52:47.560><c> so</c><00:52:47.720><c> with</c><00:52:47.960><c> this</c><00:52:48.400><c> constant</c><00:52:48.880><c> closure</c><00:52:49.320><c> that</c>"
+    },
+    {
+      "start": 3169.59,
+      "duration": 0.0,
+      "text": "a fee so with this constant closure that"
+    },
+    {
+      "start": 3169.6,
+      "duration": 0.0,
+      "text": "a fee so with this constant closure that I<00:52:50.080><c> introduced</c><00:52:50.520><c> before.</c>"
+    },
+    {
+      "start": 3171.79,
+      "duration": 0.0,
+      "text": "I introduced before."
+    },
+    {
+      "start": 3171.8,
+      "duration": 0.0,
+      "text": "I introduced before. We<00:52:52.000><c> can</c><00:52:52.200><c> see</c><00:52:52.400><c> here</c><00:52:53.000><c> that</c><00:52:53.200><c> using</c><00:52:54.240><c> the</c><00:52:54.360><c> full</c>"
+    },
+    {
+      "start": 3174.59,
+      "duration": 0.0,
+      "text": "We can see here that using the full"
+    },
+    {
+      "start": 3174.6,
+      "duration": 0.0,
+      "text": "We can see here that using the full mobility<00:52:55.080><c> matrix</c><00:52:55.800><c> obtained</c><00:52:56.400><c> by</c><00:52:56.520><c> training</c>"
+    },
+    {
+      "start": 3178.27,
+      "duration": 0.0,
+      "text": "mobility matrix obtained by training"
+    },
+    {
+      "start": 3178.28,
+      "duration": 0.0,
+      "text": "mobility matrix obtained by training a<00:52:58.400><c> neural</c><00:52:58.720><c> network</c><00:52:59.240><c> for</c><00:52:59.440><c> M</c><00:52:59.920><c> we</c><00:53:00.080><c> get</c><00:53:00.440><c> a</c><00:53:00.520><c> more</c>"
+    },
+    {
+      "start": 3180.75,
+      "duration": 0.0,
+      "text": "a neural network for M we get a more"
+    },
+    {
+      "start": 3180.76,
+      "duration": 0.0,
+      "text": "a neural network for M we get a more precise<00:53:01.440><c> recovery</c><00:53:02.600><c> of</c><00:53:03.400><c> the</c><00:53:03.520><c> correlation</c>"
+    },
+    {
+      "start": 3184.07,
+      "duration": 0.0,
+      "text": "precise recovery of the correlation"
+    },
+    {
+      "start": 3184.08,
+      "duration": 0.0,
+      "text": "precise recovery of the correlation functions<00:53:04.680><c> in</c><00:53:04.800><c> particular</c><00:53:05.520><c> for</c><00:53:05.680><c> this</c><00:53:05.880><c> cross</c>"
+    },
+    {
+      "start": 3186.23,
+      "duration": 0.0,
+      "text": "functions in particular for this cross"
+    },
+    {
+      "start": 3186.24,
+      "duration": 0.0,
+      "text": "functions in particular for this cross correlation."
+    },
+    {
+      "start": 3187.83,
+      "duration": 0.0,
+      "text": "correlation."
+    },
+    {
+      "start": 3187.84,
+      "duration": 0.0,
+      "text": "correlation. And<00:53:08.040><c> also</c><00:53:08.560><c> if</c><00:53:08.920><c> I</c><00:53:09.400><c> now</c><00:53:09.760><c> consider</c><00:53:11.080><c> the</c><00:53:11.240><c> target</c>"
+    },
+    {
+      "start": 3192.55,
+      "duration": 0.0,
+      "text": "And also if I now consider the target"
+    },
+    {
+      "start": 3192.56,
+      "duration": 0.0,
+      "text": "And also if I now consider the target dynamical<00:53:13.560><c> observables</c><00:53:14.720><c> so</c><00:53:14.880><c> the</c><00:53:15.040><c> target</c>"
+    },
+    {
+      "start": 3196.35,
+      "duration": 0.0,
+      "text": "dynamical observables so the target"
+    },
+    {
+      "start": 3196.36,
+      "duration": 0.0,
+      "text": "dynamical observables so the target correlation<00:53:16.960><c> functions</c><00:53:17.720><c> that</c><00:53:17.920><c> I</c><00:53:18.000><c> used</c><00:53:18.400><c> to</c>"
+    },
+    {
+      "start": 3198.51,
+      "duration": 0.0,
+      "text": "correlation functions that I used to"
+    },
+    {
+      "start": 3198.52,
+      "duration": 0.0,
+      "text": "correlation functions that I used to train<00:53:18.800><c> the</c><00:53:18.920><c> neural</c><00:53:19.120><c> network</c><00:53:20.160><c> when</c><00:53:20.440><c> I</c><00:53:20.520><c> evaluate</c>"
+    },
+    {
+      "start": 3201.11,
+      "duration": 0.0,
+      "text": "train the neural network when I evaluate"
+    },
+    {
+      "start": 3201.12,
+      "duration": 0.0,
+      "text": "train the neural network when I evaluate them<00:53:21.880><c> from</c><00:53:23.160><c> the</c><00:53:23.360><c> trajectory</c><00:53:24.000><c> that</c><00:53:24.240><c> I</c><00:53:24.960><c> obtained</c>"
+    },
+    {
+      "start": 3205.55,
+      "duration": 0.0,
+      "text": "them from the trajectory that I obtained"
+    },
+    {
+      "start": 3205.56,
+      "duration": 0.0,
+      "text": "them from the trajectory that I obtained by<00:53:25.720><c> integrating</c><00:53:26.840><c> my</c><00:53:28.000><c> model</c>"
+    },
+    {
+      "start": 3209.35,
+      "duration": 0.0,
+      "text": "by integrating my model"
+    },
+    {
+      "start": 3209.36,
+      "duration": 0.0,
+      "text": "by integrating my model I<00:53:29.520><c> get</c><00:53:29.920><c> yeah</c><00:53:30.400><c> a</c><00:53:30.440><c> quite</c><00:53:30.720><c> better</c>"
+    },
+    {
+      "start": 3211.95,
+      "duration": 0.0,
+      "text": "I get yeah a quite better"
+    },
+    {
+      "start": 3211.96,
+      "duration": 0.0,
+      "text": "I get yeah a quite better recovery<00:53:32.840><c> with</c><00:53:33.040><c> respect</c><00:53:33.720><c> to</c><00:53:33.840><c> the</c><00:53:33.960><c> constant</c><00:53:35.040><c> M</c>"
+    },
+    {
+      "start": 3215.63,
+      "duration": 0.0,
+      "text": "recovery with respect to the constant M"
+    },
+    {
+      "start": 3215.64,
+      "duration": 0.0,
+      "text": "recovery with respect to the constant M matrix<00:53:36.200><c> closure.</c>"
+    },
+    {
+      "start": 3217.59,
+      "duration": 0.0,
+      "text": "matrix closure."
+    },
+    {
+      "start": 3217.6,
+      "duration": 0.0,
+      "text": "matrix closure. So<00:53:37.920><c> essentially</c><00:53:38.400><c> this</c><00:53:38.640><c> is</c><00:53:38.800><c> to</c><00:53:38.920><c> show</c><00:53:39.240><c> that</c><00:53:39.560><c> yeah</c>"
+    },
+    {
+      "start": 3219.87,
+      "duration": 0.0,
+      "text": "So essentially this is to show that yeah"
+    },
+    {
+      "start": 3219.88,
+      "duration": 0.0,
+      "text": "So essentially this is to show that yeah by"
+    },
+    {
+      "start": 3220.99,
+      "duration": 0.0,
+      "text": "by"
+    },
+    {
+      "start": 3221.0,
+      "duration": 0.0,
+      "text": "by applying<00:53:41.560><c> this</c><00:53:42.160><c> algorithm</c>"
+    },
+    {
+      "start": 3223.51,
+      "duration": 0.0,
+      "text": "applying this algorithm"
+    },
+    {
+      "start": 3223.52,
+      "duration": 0.0,
+      "text": "applying this algorithm we<00:53:43.680><c> are</c><00:53:43.880><c> able</c><00:53:44.920><c> to</c><00:53:45.240><c> estimate</c><00:53:46.400><c> the</c><00:53:46.520><c> mobility</c>"
+    },
+    {
+      "start": 3226.99,
+      "duration": 0.0,
+      "text": "we are able to estimate the mobility"
+    },
+    {
+      "start": 3227.0,
+      "duration": 0.0,
+      "text": "we are able to estimate the mobility matrix<00:53:47.720><c> M</c><00:53:47.920><c> of</c><00:53:48.120><c> X</c><00:53:48.880><c> together</c><00:53:49.280><c> with</c><00:53:49.440><c> the</c><00:53:49.520><c> score</c>"
+    },
+    {
+      "start": 3230.11,
+      "duration": 0.0,
+      "text": "matrix M of X together with the score"
+    },
+    {
+      "start": 3230.12,
+      "duration": 0.0,
+      "text": "matrix M of X together with the score and<00:53:50.240><c> the</c><00:53:50.360><c> conditional</c><00:53:50.840><c> score</c><00:53:51.480><c> then</c><00:53:51.800><c> combining</c>"
+    },
+    {
+      "start": 3232.39,
+      "duration": 0.0,
+      "text": "and the conditional score then combining"
+    },
+    {
+      "start": 3232.4,
+      "duration": 0.0,
+      "text": "and the conditional score then combining those<00:53:52.600><c> pieces</c><00:53:52.920><c> together</c>"
+    },
+    {
+      "start": 3234.349,
+      "duration": 0.0,
+      "text": "those pieces together"
+    },
+    {
+      "start": 3234.359,
+      "duration": 0.0,
+      "text": "those pieces together we<00:53:54.920><c> obtain</c><00:53:55.880><c> an</c><00:53:56.000><c> expression</c><00:53:56.800><c> for</c><00:53:57.000><c> the</c><00:53:57.120><c> drift</c>"
+    },
+    {
+      "start": 3237.47,
+      "duration": 0.0,
+      "text": "we obtain an expression for the drift"
+    },
+    {
+      "start": 3237.48,
+      "duration": 0.0,
+      "text": "we obtain an expression for the drift term<00:53:58.080><c> that</c><00:53:58.280><c> is</c><00:53:58.520><c> able</c>"
+    },
+    {
+      "start": 3239.67,
+      "duration": 0.0,
+      "text": "term that is able"
+    },
+    {
+      "start": 3239.68,
+      "duration": 0.0,
+      "text": "term that is able to<00:53:59.800><c> reproduce</c><00:54:01.000><c> the</c><00:54:01.400><c> steady</c><00:54:01.640><c> state</c><00:54:02.120><c> density</c>"
+    },
+    {
+      "start": 3243.43,
+      "duration": 0.0,
+      "text": "to reproduce the steady state density"
+    },
+    {
+      "start": 3243.44,
+      "duration": 0.0,
+      "text": "to reproduce the steady state density the<00:54:03.760><c> time</c><00:54:04.080><c> correlations</c><00:54:05.080><c> together</c><00:54:05.560><c> with</c><00:54:06.400><c> all</c>"
+    },
+    {
+      "start": 3247.07,
+      "duration": 0.0,
+      "text": "the time correlations together with all"
+    },
+    {
+      "start": 3247.08,
+      "duration": 0.0,
+      "text": "the time correlations together with all the<00:54:07.160><c> correlations</c><00:54:07.960><c> that</c><00:54:08.240><c> we</c><00:54:08.920><c> enforced</c>"
+    },
+    {
+      "start": 3250.23,
+      "duration": 0.0,
+      "text": "the correlations that we enforced"
+    },
+    {
+      "start": 3250.24,
+      "duration": 0.0,
+      "text": "the correlations that we enforced in<00:54:10.359><c> the</c><00:54:10.440><c> training.</c>"
+    },
+    {
+      "start": 3251.79,
+      "duration": 0.0,
+      "text": "in the training."
+    },
+    {
+      "start": 3251.8,
+      "duration": 0.0,
+      "text": "in the training. Okay,<00:54:12.080><c> so</c><00:54:12.200><c> now</c><00:54:12.480><c> let's</c><00:54:12.720><c> consider</c><00:54:13.240><c> more</c>"
+    },
+    {
+      "start": 3253.47,
+      "duration": 0.0,
+      "text": "Okay, so now let's consider more"
+    },
+    {
+      "start": 3253.48,
+      "duration": 0.0,
+      "text": "Okay, so now let's consider more high-dimensional<00:54:14.160><c> systems.</c>"
+    },
+    {
+      "start": 3255.349,
+      "duration": 0.0,
+      "text": "high-dimensional systems."
+    },
+    {
+      "start": 3255.359,
+      "duration": 0.0,
+      "text": "high-dimensional systems. So<00:54:15.680><c> for</c><00:54:16.080><c> the</c><00:54:16.200><c> next</c><00:54:16.560><c> two</c><00:54:16.720><c> systems</c><00:54:17.720><c> I</c><00:54:17.840><c> will</c><00:54:18.080><c> only</c>"
+    },
+    {
+      "start": 3258.39,
+      "duration": 0.0,
+      "text": "So for the next two systems I will only"
+    },
+    {
+      "start": 3258.4,
+      "duration": 0.0,
+      "text": "So for the next two systems I will only consider<00:54:19.200><c> the</c><00:54:19.600><c> constant</c><00:54:20.080><c> closure</c><00:54:20.560><c> for</c><00:54:20.800><c> M.</c><00:54:21.240><c> So</c>"
+    },
+    {
+      "start": 3261.39,
+      "duration": 0.0,
+      "text": "consider the constant closure for M. So"
+    },
+    {
+      "start": 3261.4,
+      "duration": 0.0,
+      "text": "consider the constant closure for M. So essentially<00:54:21.880><c> I</c><00:54:21.960><c> approximate</c><00:54:22.880><c> M</c><00:54:23.040><c> of</c><00:54:23.200><c> X</c><00:54:23.600><c> with</c>"
+    },
+    {
+      "start": 3264.27,
+      "duration": 0.0,
+      "text": "essentially I approximate M of X with"
+    },
+    {
+      "start": 3264.28,
+      "duration": 0.0,
+      "text": "essentially I approximate M of X with its<00:54:24.680><c> average</c><00:54:25.120><c> value</c><00:54:25.560><c> so</c><00:54:25.720><c> with</c><00:54:25.920><c> fee.</c>"
+    },
+    {
+      "start": 3266.91,
+      "duration": 0.0,
+      "text": "its average value so with fee."
+    },
+    {
+      "start": 3266.92,
+      "duration": 0.0,
+      "text": "its average value so with fee. In<00:54:27.040><c> this</c><00:54:27.200><c> case</c><00:54:27.520><c> I'm</c><00:54:27.640><c> integrating</c><00:54:28.520><c> this</c>"
+    },
+    {
+      "start": 3268.75,
+      "duration": 0.0,
+      "text": "In this case I'm integrating this"
+    },
+    {
+      "start": 3268.76,
+      "duration": 0.0,
+      "text": "In this case I'm integrating this Kuramoto-Sivashinsky<00:54:30.200><c> PDE.</c>"
+    },
+    {
+      "start": 3271.39,
+      "duration": 0.0,
+      "text": "Kuramoto-Sivashinsky PDE."
+    },
+    {
+      "start": 3271.4,
+      "duration": 0.0,
+      "text": "Kuramoto-Sivashinsky PDE. I'm<00:54:31.880><c> integrating</c><00:54:32.680><c> this</c><00:54:32.920><c> partial</c>"
+    },
+    {
+      "start": 3273.27,
+      "duration": 0.0,
+      "text": "I'm integrating this partial"
+    },
+    {
+      "start": 3273.28,
+      "duration": 0.0,
+      "text": "I'm integrating this partial differential<00:54:33.760><c> equation</c><00:54:34.359><c> with</c><00:54:34.680><c> 512</c><00:54:35.720><c> Fourier</c>"
+    },
+    {
+      "start": 3275.99,
+      "duration": 0.0,
+      "text": "differential equation with 512 Fourier"
+    },
+    {
+      "start": 3276.0,
+      "duration": 0.0,
+      "text": "differential equation with 512 Fourier modes.<00:54:36.680><c> I</c><00:54:36.800><c> obtain</c><00:54:37.480><c> a</c><00:54:37.520><c> 1024-dimensional</c>"
+    },
+    {
+      "start": 3280.39,
+      "duration": 0.0,
+      "text": "modes. I obtain a 1024-dimensional"
+    },
+    {
+      "start": 3280.4,
+      "duration": 0.0,
+      "text": "modes. I obtain a 1024-dimensional um"
+    },
+    {
+      "start": 3282.03,
+      "duration": 0.0,
+      "text": "um"
+    },
+    {
+      "start": 3282.04,
+      "duration": 0.0,
+      "text": "um um<00:54:42.400><c> time</c><00:54:42.680><c> series.</c><00:54:43.600><c> I'm</c>"
+    },
+    {
+      "start": 3284.91,
+      "duration": 0.0,
+      "text": "um time series. I'm"
+    },
+    {
+      "start": 3284.92,
+      "duration": 0.0,
+      "text": "um time series. I'm considering<00:54:45.880><c> just</c><00:54:46.480><c> one</c>"
+    },
+    {
+      "start": 3287.79,
+      "duration": 0.0,
+      "text": "considering just one"
+    },
+    {
+      "start": 3287.8,
+      "duration": 0.0,
+      "text": "considering just one mode<00:54:48.560><c> every</c><00:54:48.880><c> 32.</c><00:54:49.600><c> So</c><00:54:49.720><c> essentially</c><00:54:50.280><c> I'm</c>"
+    },
+    {
+      "start": 3291.23,
+      "duration": 0.0,
+      "text": "mode every 32. So essentially I'm"
+    },
+    {
+      "start": 3291.24,
+      "duration": 0.0,
+      "text": "mode every 32. So essentially I'm subsampling<00:54:52.160><c> this</c><00:54:52.480><c> 1024-dimensional</c>"
+    },
+    {
+      "start": 3294.27,
+      "duration": 0.0,
+      "text": "subsampling this 1024-dimensional"
+    },
+    {
+      "start": 3294.28,
+      "duration": 0.0,
+      "text": "subsampling this 1024-dimensional state<00:54:55.320><c> to</c><00:54:55.920><c> a</c><00:54:55.960><c> 32-dimensional</c><00:54:56.840><c> state.</c>"
+    },
+    {
+      "start": 3297.95,
+      "duration": 0.0,
+      "text": "state to a 32-dimensional state."
+    },
+    {
+      "start": 3297.96,
+      "duration": 0.0,
+      "text": "state to a 32-dimensional state. And<00:54:58.160><c> then</c><00:54:58.320><c> using</c><00:54:58.760><c> those</c>"
+    },
+    {
+      "start": 3299.55,
+      "duration": 0.0,
+      "text": "And then using those"
+    },
+    {
+      "start": 3299.56,
+      "duration": 0.0,
+      "text": "And then using those those<00:55:00.240><c> 32-dimensional</c><00:55:01.280><c> modes</c><00:55:01.960><c> to</c><00:55:02.080><c> build</c><00:55:02.480><c> my</c>"
+    },
+    {
+      "start": 3302.63,
+      "duration": 0.0,
+      "text": "those 32-dimensional modes to build my"
+    },
+    {
+      "start": 3302.64,
+      "duration": 0.0,
+      "text": "those 32-dimensional modes to build my Langevin<00:55:03.000><c> equation.</c><00:55:03.920><c> So</c><00:55:04.000><c> essentially</c><00:55:04.400><c> here</c>"
+    },
+    {
+      "start": 3305.03,
+      "duration": 0.0,
+      "text": "Langevin equation. So essentially here"
+    },
+    {
+      "start": 3305.04,
+      "duration": 0.0,
+      "text": "Langevin equation. So essentially here I'm"
+    },
+    {
+      "start": 3306.349,
+      "duration": 0.0,
+      "text": "I'm"
+    },
+    {
+      "start": 3306.359,
+      "duration": 0.0,
+      "text": "I'm building<00:55:07.240><c> so</c><00:55:07.480><c> I'm</c><00:55:07.640><c> starting</c><00:55:08.200><c> from</c><00:55:08.640><c> a</c><00:55:08.760><c> fully</c>"
+    },
+    {
+      "start": 3309.43,
+      "duration": 0.0,
+      "text": "building so I'm starting from a fully"
+    },
+    {
+      "start": 3309.44,
+      "duration": 0.0,
+      "text": "building so I'm starting from a fully fully<00:55:09.800><c> deterministic</c><00:55:11.240><c> partial</c><00:55:11.680><c> differential</c>"
+    },
+    {
+      "start": 3312.27,
+      "duration": 0.0,
+      "text": "fully deterministic partial differential"
+    },
+    {
+      "start": 3312.28,
+      "duration": 0.0,
+      "text": "fully deterministic partial differential equation<00:55:13.000><c> partial</c>"
+    },
+    {
+      "start": 3313.91,
+      "duration": 0.0,
+      "text": "equation partial"
+    },
+    {
+      "start": 3313.92,
+      "duration": 0.0,
+      "text": "equation partial which<00:55:14.200><c> is</c><00:55:14.440><c> partially</c><00:55:14.840><c> observed</c>"
+    },
+    {
+      "start": 3316.15,
+      "duration": 0.0,
+      "text": "which is partially observed"
+    },
+    {
+      "start": 3316.16,
+      "duration": 0.0,
+      "text": "which is partially observed and<00:55:16.359><c> then</c><00:55:16.520><c> using</c><00:55:16.920><c> a</c><00:55:17.000><c> completely</c><00:55:17.480><c> different</c>"
+    },
+    {
+      "start": 3318.83,
+      "duration": 0.0,
+      "text": "and then using a completely different"
+    },
+    {
+      "start": 3318.84,
+      "duration": 0.0,
+      "text": "and then using a completely different model<00:55:19.600><c> to</c>"
+    },
+    {
+      "start": 3320.87,
+      "duration": 0.0,
+      "text": "model to"
+    },
+    {
+      "start": 3320.88,
+      "duration": 0.0,
+      "text": "model to um"
+    },
+    {
+      "start": 3321.67,
+      "duration": 0.0,
+      "text": "um"
+    },
+    {
+      "start": 3321.68,
+      "duration": 0.0,
+      "text": "um be<00:55:22.080><c> so</c><00:55:22.320><c> to</c><00:55:23.120><c> predict</c><00:55:23.800><c> its</c><00:55:24.000><c> dynamics</c><00:55:24.720><c> using</c><00:55:25.840><c> my</c>"
+    },
+    {
+      "start": 3327.11,
+      "duration": 0.0,
+      "text": "be so to predict its dynamics using my"
+    },
+    {
+      "start": 3327.12,
+      "duration": 0.0,
+      "text": "be so to predict its dynamics using my stochastic<00:55:27.680><c> closure.</c>"
+    },
+    {
+      "start": 3329.15,
+      "duration": 0.0,
+      "text": "stochastic closure."
+    },
+    {
+      "start": 3329.16,
+      "duration": 0.0,
+      "text": "stochastic closure. And<00:55:29.359><c> here</c><00:55:29.880><c> are</c><00:55:30.000><c> the</c><00:55:30.120><c> results.</c><00:55:30.840><c> So</c><00:55:31.160><c> this</c><00:55:32.280><c> is</c><00:55:32.880><c> the</c>"
+    },
+    {
+      "start": 3333.03,
+      "duration": 0.0,
+      "text": "And here are the results. So this is the"
+    },
+    {
+      "start": 3333.04,
+      "duration": 0.0,
+      "text": "And here are the results. So this is the time<00:55:33.320><c> series</c><00:55:34.040><c> obtained</c><00:55:34.680><c> by</c><00:55:34.840><c> integrating</c><00:55:35.480><c> my</c>"
+    },
+    {
+      "start": 3335.63,
+      "duration": 0.0,
+      "text": "time series obtained by integrating my"
+    },
+    {
+      "start": 3335.64,
+      "duration": 0.0,
+      "text": "time series obtained by integrating my Langevin<00:55:36.000><c> equation.</c><00:55:36.840><c> This</c><00:55:37.400><c> is</c><00:55:37.800><c> the</c><00:55:37.960><c> real</c>"
+    },
+    {
+      "start": 3339.51,
+      "duration": 0.0,
+      "text": "Langevin equation. This is the real"
+    },
+    {
+      "start": 3339.52,
+      "duration": 0.0,
+      "text": "Langevin equation. This is the real observed<00:55:40.320><c> time</c><00:55:40.520><c> series</c><00:55:41.680><c> and</c><00:55:41.880><c> here</c><00:55:42.160><c> I'm</c>"
+    },
+    {
+      "start": 3342.31,
+      "duration": 0.0,
+      "text": "observed time series and here I'm"
+    },
+    {
+      "start": 3342.32,
+      "duration": 0.0,
+      "text": "observed time series and here I'm plotting<00:55:43.120><c> the</c><00:55:43.240><c> comparison</c><00:55:44.080><c> between</c><00:55:44.720><c> the</c>"
+    },
+    {
+      "start": 3345.79,
+      "duration": 0.0,
+      "text": "plotting the comparison between the"
+    },
+    {
+      "start": 3345.8,
+      "duration": 0.0,
+      "text": "plotting the comparison between the um"
+    },
+    {
+      "start": 3346.79,
+      "duration": 0.0,
+      "text": "um"
+    },
+    {
+      "start": 3346.8,
+      "duration": 0.0,
+      "text": "um the<00:55:46.920><c> bivariate</c><00:55:47.720><c> and</c><00:55:47.920><c> the</c><00:55:48.000><c> univariate</c><00:55:48.680><c> PDFs</c>"
+    },
+    {
+      "start": 3349.47,
+      "duration": 0.0,
+      "text": "the bivariate and the univariate PDFs"
+    },
+    {
+      "start": 3349.48,
+      "duration": 0.0,
+      "text": "the bivariate and the univariate PDFs obtained<00:55:50.359><c> from</c><00:55:51.359><c> the</c><00:55:51.840><c> observations</c><00:55:53.040><c> and</c><00:55:53.520><c> the</c>"
+    },
+    {
+      "start": 3353.63,
+      "duration": 0.0,
+      "text": "obtained from the observations and the"
+    },
+    {
+      "start": 3353.64,
+      "duration": 0.0,
+      "text": "obtained from the observations and the one<00:55:53.920><c> obtained</c><00:55:54.520><c> from</c>"
+    },
+    {
+      "start": 3355.71,
+      "duration": 0.0,
+      "text": "one obtained from"
+    },
+    {
+      "start": 3355.72,
+      "duration": 0.0,
+      "text": "one obtained from a<00:55:55.800><c> model</c><00:55:56.120><c> integration</c><00:55:56.760><c> of</c><00:55:56.880><c> my</c><00:55:57.040><c> Langevin</c>"
+    },
+    {
+      "start": 3357.39,
+      "duration": 0.0,
+      "text": "a model integration of my Langevin"
+    },
+    {
+      "start": 3357.4,
+      "duration": 0.0,
+      "text": "a model integration of my Langevin equation.<00:55:58.240><c> And</c><00:55:58.440><c> here</c><00:55:58.680><c> instead</c><00:55:59.120><c> is</c><00:55:59.320><c> the</c>"
+    },
+    {
+      "start": 3359.39,
+      "duration": 0.0,
+      "text": "equation. And here instead is the"
+    },
+    {
+      "start": 3359.4,
+      "duration": 0.0,
+      "text": "equation. And here instead is the autocorrelation<00:56:00.080><c> function</c><00:56:00.920><c> for</c><00:56:01.960><c> both</c><00:56:02.840><c> the</c>"
+    },
+    {
+      "start": 3363.07,
+      "duration": 0.0,
+      "text": "autocorrelation function for both the"
+    },
+    {
+      "start": 3363.08,
+      "duration": 0.0,
+      "text": "autocorrelation function for both the observations<00:56:04.120><c> and</c><00:56:04.440><c> my</c><00:56:04.640><c> Langevin</c>"
+    },
+    {
+      "start": 3365.03,
+      "duration": 0.0,
+      "text": "observations and my Langevin"
+    },
+    {
+      "start": 3365.04,
+      "duration": 0.0,
+      "text": "observations and my Langevin integration."
+    },
+    {
+      "start": 3366.63,
+      "duration": 0.0,
+      "text": "integration."
+    },
+    {
+      "start": 3366.64,
+      "duration": 0.0,
+      "text": "integration. Then<00:56:06.920><c> finally</c><00:56:07.840><c> I</c><00:56:08.120><c> considered</c>"
+    },
+    {
+      "start": 3369.71,
+      "duration": 0.0,
+      "text": "Then finally I considered"
+    },
+    {
+      "start": 3369.72,
+      "duration": 0.0,
+      "text": "Then finally I considered the<00:56:09.920><c> sea</c><00:56:10.080><c> surface</c><00:56:10.440><c> temperature</c><00:56:11.040><c> data</c><00:56:11.640><c> from</c>"
+    },
+    {
+      "start": 3372.19,
+      "duration": 0.0,
+      "text": "the sea surface temperature data from"
+    },
+    {
+      "start": 3372.2,
+      "duration": 0.0,
+      "text": "the sea surface temperature data from Plasim<00:56:12.840><c> so</c><00:56:13.040><c> which</c><00:56:13.240><c> is</c><00:56:13.480><c> a</c><00:56:14.280><c> um</c>"
+    },
+    {
+      "start": 3375.67,
+      "duration": 0.0,
+      "text": "Plasim so which is a um"
+    },
+    {
+      "start": 3375.68,
+      "duration": 0.0,
+      "text": "Plasim so which is a um a<00:56:15.760><c> global</c><00:56:16.240><c> circulation</c><00:56:17.080><c> model</c><00:56:17.520><c> of</c>"
+    },
+    {
+      "start": 3377.67,
+      "duration": 0.0,
+      "text": "a global circulation model of"
+    },
+    {
+      "start": 3377.68,
+      "duration": 0.0,
+      "text": "a global circulation model of intermediate"
+    },
+    {
+      "start": 3379.27,
+      "duration": 0.0,
+      "text": "intermediate"
+    },
+    {
+      "start": 3379.28,
+      "duration": 0.0,
+      "text": "intermediate intermediate<00:56:20.359><c> complexity.</c>"
+    },
+    {
+      "start": 3381.71,
+      "duration": 0.0,
+      "text": "intermediate complexity."
+    },
+    {
+      "start": 3381.72,
+      "duration": 0.0,
+      "text": "intermediate complexity. I<00:56:21.960><c> have</c>"
+    },
+    {
+      "start": 3383.75,
+      "duration": 0.0,
+      "text": "I have"
+    },
+    {
+      "start": 3383.76,
+      "duration": 0.0,
+      "text": "I have the<00:56:23.840><c> data</c><00:56:24.400><c> for</c><00:56:24.600><c> the</c><00:56:24.760><c> sea</c><00:56:24.960><c> surface</c>"
+    },
+    {
+      "start": 3385.83,
+      "duration": 0.0,
+      "text": "the data for the sea surface"
+    },
+    {
+      "start": 3385.84,
+      "duration": 0.0,
+      "text": "the data for the sea surface for<00:56:26.000><c> the</c><00:56:26.120><c> global</c><00:56:26.920><c> sea</c><00:56:27.080><c> surface</c><00:56:27.520><c> temperature</c>"
+    },
+    {
+      "start": 3388.87,
+      "duration": 0.0,
+      "text": "for the global sea surface temperature"
+    },
+    {
+      "start": 3388.88,
+      "duration": 0.0,
+      "text": "for the global sea surface temperature evolution<00:56:30.160><c> and</c><00:56:30.400><c> I</c><00:56:30.480><c> want</c><00:56:31.680><c> a</c><00:56:31.760><c> model</c><00:56:32.400><c> that</c><00:56:32.560><c> is</c>"
+    },
+    {
+      "start": 3392.71,
+      "duration": 0.0,
+      "text": "evolution and I want a model that is"
+    },
+    {
+      "start": 3392.72,
+      "duration": 0.0,
+      "text": "evolution and I want a model that is able<00:56:33.000><c> essentially</c><00:56:33.440><c> to</c><00:56:33.840><c> predict</c><00:56:34.560><c> and</c><00:56:34.720><c> to</c><00:56:34.840><c> model</c>"
+    },
+    {
+      "start": 3395.83,
+      "duration": 0.0,
+      "text": "able essentially to predict and to model"
+    },
+    {
+      "start": 3395.84,
+      "duration": 0.0,
+      "text": "able essentially to predict and to model this<00:56:36.200><c> sea</c><00:56:36.359><c> surface</c><00:56:36.680><c> temperature</c><00:56:37.120><c> data.</c>"
+    },
+    {
+      "start": 3398.27,
+      "duration": 0.0,
+      "text": "this sea surface temperature data."
+    },
+    {
+      "start": 3398.28,
+      "duration": 0.0,
+      "text": "this sea surface temperature data. So<00:56:38.560><c> the</c><00:56:38.680><c> data</c><00:56:39.160><c> set</c><00:56:39.800><c> is</c><00:56:40.080><c> around</c>"
+    },
+    {
+      "start": 3401.03,
+      "duration": 0.0,
+      "text": "So the data set is around"
+    },
+    {
+      "start": 3401.04,
+      "duration": 0.0,
+      "text": "So the data set is around 2000-dimensional."
+    },
+    {
+      "start": 3403.07,
+      "duration": 0.0,
+      "text": "2000-dimensional."
+    },
+    {
+      "start": 3403.08,
+      "duration": 0.0,
+      "text": "2000-dimensional. I<00:56:43.720><c> did</c><00:56:44.040><c> a</c><00:56:44.080><c> dimensionality</c><00:56:44.960><c> reduction</c><00:56:45.800><c> taking</c>"
+    },
+    {
+      "start": 3406.19,
+      "duration": 0.0,
+      "text": "I did a dimensionality reduction taking"
+    },
+    {
+      "start": 3406.2,
+      "duration": 0.0,
+      "text": "I did a dimensionality reduction taking the<00:56:46.320><c> first</c><00:56:46.800><c> 20</c><00:56:47.720><c> principal</c><00:56:48.200><c> components.</c>"
+    },
+    {
+      "start": 3409.83,
+      "duration": 0.0,
+      "text": "the first 20 principal components."
+    },
+    {
+      "start": 3409.84,
+      "duration": 0.0,
+      "text": "the first 20 principal components. And<00:56:50.120><c> here</c><00:56:50.480><c> since</c><00:56:50.760><c> I</c><00:56:50.840><c> have</c><00:56:51.040><c> a</c><00:56:51.080><c> strong</c>"
+    },
+    {
+      "start": 3411.55,
+      "duration": 0.0,
+      "text": "And here since I have a strong"
+    },
+    {
+      "start": 3411.56,
+      "duration": 0.0,
+      "text": "And here since I have a strong periodicity<00:56:52.640><c> I</c><00:56:53.000><c> augmented</c><00:56:53.800><c> the</c><00:56:53.920><c> state</c><00:56:54.240><c> space</c>"
+    },
+    {
+      "start": 3414.79,
+      "duration": 0.0,
+      "text": "periodicity I augmented the state space"
+    },
+    {
+      "start": 3414.8,
+      "duration": 0.0,
+      "text": "periodicity I augmented the state space by<00:56:55.000><c> including</c><00:56:55.680><c> some</c><00:56:56.080><c> harmonic</c><00:56:56.600><c> functions.</c>"
+    },
+    {
+      "start": 3418.51,
+      "duration": 0.0,
+      "text": "by including some harmonic functions."
+    },
+    {
+      "start": 3418.52,
+      "duration": 0.0,
+      "text": "by including some harmonic functions. And<00:56:59.720><c> these</c><00:56:59.880><c> are</c><00:57:00.000><c> the</c><00:57:00.120><c> results.</c>"
+    },
+    {
+      "start": 3421.349,
+      "duration": 0.0,
+      "text": "And these are the results."
+    },
+    {
+      "start": 3421.359,
+      "duration": 0.0,
+      "text": "And these are the results. I<00:57:01.920><c> yeah</c><00:57:02.440><c> was</c><00:57:02.880><c> predicting</c><00:57:04.000><c> the</c><00:57:04.400><c> probability</c>"
+    },
+    {
+      "start": 3425.349,
+      "duration": 0.0,
+      "text": "I yeah was predicting the probability"
+    },
+    {
+      "start": 3425.359,
+      "duration": 0.0,
+      "text": "I yeah was predicting the probability the<00:57:05.480><c> conditional</c><00:57:06.040><c> probability</c><00:57:06.600><c> density</c>"
+    },
+    {
+      "start": 3427.99,
+      "duration": 0.0,
+      "text": "the conditional probability density"
+    },
+    {
+      "start": 3428.0,
+      "duration": 0.0,
+      "text": "the conditional probability density of<00:57:08.880><c> the</c>"
+    },
+    {
+      "start": 3431.56,
+      "duration": 0.0,
+      "text": "20<00:57:12.640><c> principal</c><00:57:13.120><c> components</c><00:57:13.840><c> together</c><00:57:14.359><c> with</c>"
+    },
+    {
+      "start": 3434.75,
+      "duration": 0.0,
+      "text": "20 principal components together with"
+    },
+    {
+      "start": 3434.76,
+      "duration": 0.0,
+      "text": "20 principal components together with their<00:57:15.359><c> autocovariance.</c><00:57:16.560><c> So</c>"
+    },
+    {
+      "start": 3437.31,
+      "duration": 0.0,
+      "text": "their autocovariance. So"
+    },
+    {
+      "start": 3437.32,
+      "duration": 0.0,
+      "text": "their autocovariance. So as<00:57:17.480><c> you</c><00:57:17.560><c> can</c><00:57:17.760><c> see</c><00:57:18.320><c> we</c><00:57:18.480><c> can</c><00:57:18.680><c> have</c><00:57:18.920><c> a</c><00:57:19.000><c> quite</c>"
+    },
+    {
+      "start": 3439.27,
+      "duration": 0.0,
+      "text": "as you can see we can have a quite"
+    },
+    {
+      "start": 3439.28,
+      "duration": 0.0,
+      "text": "as you can see we can have a quite decent"
+    },
+    {
+      "start": 3440.43,
+      "duration": 0.0,
+      "text": "decent"
+    },
+    {
+      "start": 3440.44,
+      "duration": 0.0,
+      "text": "decent reconstruction<00:57:21.840><c> of</c><00:57:22.400><c> the</c><00:57:23.240><c> PDFs</c><00:57:24.200><c> and</c><00:57:24.480><c> the</c><00:57:24.720><c> ACFs</c>"
+    },
+    {
+      "start": 3445.75,
+      "duration": 0.0,
+      "text": "reconstruction of the PDFs and the ACFs"
+    },
+    {
+      "start": 3445.76,
+      "duration": 0.0,
+      "text": "reconstruction of the PDFs and the ACFs of<00:57:26.359><c> all</c><00:57:26.880><c> the</c><00:57:27.000><c> 20</c><00:57:28.080><c> principal</c><00:57:28.560><c> components.</c><00:57:29.320><c> Here</c>"
+    },
+    {
+      "start": 3449.51,
+      "duration": 0.0,
+      "text": "of all the 20 principal components. Here"
+    },
+    {
+      "start": 3449.52,
+      "duration": 0.0,
+      "text": "of all the 20 principal components. Here I'm<00:57:29.680><c> plotting</c><00:57:30.120><c> just</c><00:57:30.400><c> the</c><00:57:30.480><c> first</c><00:57:30.840><c> 10.</c>"
+    },
+    {
+      "start": 3452.11,
+      "duration": 0.0,
+      "text": "I'm plotting just the first 10."
+    },
+    {
+      "start": 3452.12,
+      "duration": 0.0,
+      "text": "I'm plotting just the first 10. And<00:57:32.320><c> also</c><00:57:33.120><c> we</c><00:57:33.280><c> were</c><00:57:33.600><c> able</c><00:57:34.359><c> to</c><00:57:34.480><c> capture</c><00:57:35.760><c> the</c>"
+    },
+    {
+      "start": 3456.19,
+      "duration": 0.0,
+      "text": "And also we were able to capture the"
+    },
+    {
+      "start": 3456.2,
+      "duration": 0.0,
+      "text": "And also we were able to capture the nonlinear<00:57:37.120><c> so</c><00:57:37.320><c> and</c><00:57:37.600><c> the</c><00:57:37.840><c> non-Gaussian</c>"
+    },
+    {
+      "start": 3458.63,
+      "duration": 0.0,
+      "text": "nonlinear so and the non-Gaussian"
+    },
+    {
+      "start": 3458.64,
+      "duration": 0.0,
+      "text": "nonlinear so and the non-Gaussian probability<00:57:39.400><c> density</c><00:57:39.800><c> function</c><00:57:40.840><c> evaluated</c>"
+    },
+    {
+      "start": 3461.83,
+      "duration": 0.0,
+      "text": "probability density function evaluated"
+    },
+    {
+      "start": 3461.84,
+      "duration": 0.0,
+      "text": "probability density function evaluated at<00:57:42.080><c> every</c><00:57:42.520><c> grid</c><00:57:42.840><c> point</c>"
+    },
+    {
+      "start": 3464.07,
+      "duration": 0.0,
+      "text": "at every grid point"
+    },
+    {
+      "start": 3464.08,
+      "duration": 0.0,
+      "text": "at every grid point um<00:57:44.400><c> from</c><00:57:44.640><c> our</c><00:57:44.800><c> simulation.</c><00:57:45.560><c> So</c><00:57:45.720><c> here</c>"
+    },
+    {
+      "start": 3465.91,
+      "duration": 0.0,
+      "text": "um from our simulation. So here"
+    },
+    {
+      "start": 3465.92,
+      "duration": 0.0,
+      "text": "um from our simulation. So here essentially<00:57:46.440><c> I'm</c><00:57:46.600><c> plotting</c><00:57:47.440><c> the</c><00:57:47.560><c> probability</c>"
+    },
+    {
+      "start": 3468.19,
+      "duration": 0.0,
+      "text": "essentially I'm plotting the probability"
+    },
+    {
+      "start": 3468.2,
+      "duration": 0.0,
+      "text": "essentially I'm plotting the probability density<00:57:48.840><c> at</c><00:57:49.000><c> different</c><00:57:49.400><c> season</c>"
+    },
+    {
+      "start": 3470.59,
+      "duration": 0.0,
+      "text": "density at different season"
+    },
+    {
+      "start": 3470.6,
+      "duration": 0.0,
+      "text": "density at different season of<00:57:51.280><c> the</c><00:57:51.960><c> temperature</c><00:57:52.920><c> at</c><00:57:53.120><c> a</c><00:57:53.160><c> given</c><00:57:53.880><c> grid</c><00:57:54.160><c> point</c>"
+    },
+    {
+      "start": 3474.91,
+      "duration": 0.0,
+      "text": "of the temperature at a given grid point"
+    },
+    {
+      "start": 3474.92,
+      "duration": 0.0,
+      "text": "of the temperature at a given grid point and<00:57:55.120><c> I'm</c><00:57:55.320><c> doing</c><00:57:55.720><c> that</c><00:57:56.520><c> using</c><00:57:57.680><c> the</c><00:57:57.840><c> full</c>"
+    },
+    {
+      "start": 3477.99,
+      "duration": 0.0,
+      "text": "and I'm doing that using the full"
+    },
+    {
+      "start": 3478.0,
+      "duration": 0.0,
+      "text": "and I'm doing that using the full observation<00:57:58.920><c> so</c><00:57:59.000><c> essentially</c><00:57:59.480><c> all</c><00:57:59.760><c> the</c>"
+    },
+    {
+      "start": 3479.87,
+      "duration": 0.0,
+      "text": "observation so essentially all the"
+    },
+    {
+      "start": 3479.88,
+      "duration": 0.0,
+      "text": "observation so essentially all the principal<00:58:00.320><c> components</c>"
+    },
+    {
+      "start": 3481.51,
+      "duration": 0.0,
+      "text": "principal components"
+    },
+    {
+      "start": 3481.52,
+      "duration": 0.0,
+      "text": "principal components just<00:58:02.320><c> the</c><00:58:02.480><c> first</c><00:58:02.800><c> 20</c><00:58:03.280><c> principal</c><00:58:03.720><c> components</c>"
+    },
+    {
+      "start": 3484.99,
+      "duration": 0.0,
+      "text": "just the first 20 principal components"
+    },
+    {
+      "start": 3485.0,
+      "duration": 0.0,
+      "text": "just the first 20 principal components and"
+    },
+    {
+      "start": 3486.59,
+      "duration": 0.0,
+      "text": "and"
+    },
+    {
+      "start": 3486.6,
+      "duration": 0.0,
+      "text": "and the<00:58:07.160><c> 20-dimensional</c>"
+    },
+    {
+      "start": 3488.99,
+      "duration": 0.0,
+      "text": "the 20-dimensional"
+    },
+    {
+      "start": 3489.0,
+      "duration": 0.0,
+      "text": "the 20-dimensional stochastic<00:58:09.720><c> model</c><00:58:10.120><c> that</c><00:58:10.320><c> I</c><00:58:10.359><c> trained</c><00:58:11.400><c> on</c><00:58:11.880><c> these</c>"
+    },
+    {
+      "start": 3492.19,
+      "duration": 0.0,
+      "text": "stochastic model that I trained on these"
+    },
+    {
+      "start": 3492.2,
+      "duration": 0.0,
+      "text": "stochastic model that I trained on these first<00:58:12.520><c> 20</c><00:58:12.840><c> components</c><00:58:13.960><c> and</c><00:58:14.200><c> I</c><00:58:14.280><c> integrated</c>"
+    },
+    {
+      "start": 3494.99,
+      "duration": 0.0,
+      "text": "first 20 components and I integrated"
+    },
+    {
+      "start": 3495.0,
+      "duration": 0.0,
+      "text": "first 20 components and I integrated forward."
+    },
+    {
+      "start": 3496.27,
+      "duration": 0.0,
+      "text": "forward."
+    },
+    {
+      "start": 3496.28,
+      "duration": 0.0,
+      "text": "forward. And<00:58:16.440><c> as</c><00:58:16.560><c> you</c><00:58:16.640><c> can</c><00:58:16.800><c> see</c><00:58:17.200><c> so</c><00:58:17.320><c> even</c><00:58:17.520><c> if</c><00:58:18.080><c> I</c><00:58:18.200><c> did</c><00:58:18.480><c> a</c>"
+    },
+    {
+      "start": 3498.55,
+      "duration": 0.0,
+      "text": "And as you can see so even if I did a"
+    },
+    {
+      "start": 3498.56,
+      "duration": 0.0,
+      "text": "And as you can see so even if I did a dimensionality<00:58:19.200><c> reduction</c><00:58:19.800><c> of</c><00:58:19.960><c> the</c>"
+    },
+    {
+      "start": 3501.31,
+      "duration": 0.0,
+      "text": "dimensionality reduction of the"
+    },
+    {
+      "start": 3501.32,
+      "duration": 0.0,
+      "text": "dimensionality reduction of the data<00:58:21.640><c> set</c><00:58:22.640><c> I</c><00:58:22.760><c> was</c><00:58:22.960><c> still</c><00:58:23.240><c> able</c><00:58:23.560><c> to</c><00:58:23.680><c> get</c><00:58:24.040><c> this</c>"
+    },
+    {
+      "start": 3504.27,
+      "duration": 0.0,
+      "text": "data set I was still able to get this"
+    },
+    {
+      "start": 3504.28,
+      "duration": 0.0,
+      "text": "data set I was still able to get this nonlinear<00:58:25.720><c> probability</c><00:58:26.320><c> density</c><00:58:26.720><c> functions</c>"
+    },
+    {
+      "start": 3508.19,
+      "duration": 0.0,
+      "text": "nonlinear probability density functions"
+    },
+    {
+      "start": 3508.2,
+      "duration": 0.0,
+      "text": "nonlinear probability density functions using<00:58:28.960><c> this</c>"
+    },
+    {
+      "start": 3510.03,
+      "duration": 0.0,
+      "text": "using this"
+    },
+    {
+      "start": 3510.04,
+      "duration": 0.0,
+      "text": "using this quite<00:58:30.240><c> simple</c>"
+    },
+    {
+      "start": 3511.67,
+      "duration": 0.0,
+      "text": "quite simple"
+    },
+    {
+      "start": 3511.68,
+      "duration": 0.0,
+      "text": "quite simple stochastic<00:58:32.200><c> model</c><00:58:33.120><c> that</c><00:58:33.280><c> I</c><00:58:33.359><c> built</c><00:58:34.120><c> using</c><00:58:34.880><c> this</c>"
+    },
+    {
+      "start": 3516.31,
+      "duration": 0.0,
+      "text": "stochastic model that I built using this"
+    },
+    {
+      "start": 3516.32,
+      "duration": 0.0,
+      "text": "stochastic model that I built using this constant<00:58:36.800><c> closure</c><00:58:37.800><c> for</c><00:58:38.080><c> my</c>"
+    },
+    {
+      "start": 3519.55,
+      "duration": 0.0,
+      "text": "constant closure for my"
+    },
+    {
+      "start": 3519.56,
+      "duration": 0.0,
+      "text": "constant closure for my mobility<00:58:40.280><c> matrix.</c>"
+    },
+    {
+      "start": 3521.95,
+      "duration": 0.0,
+      "text": "mobility matrix."
+    },
+    {
+      "start": 3521.96,
+      "duration": 0.0,
+      "text": "mobility matrix. Okay,<00:58:42.280><c> so</c><00:58:42.440><c> these</c><00:58:42.800><c> are</c>"
+    },
+    {
+      "start": 3524.11,
+      "duration": 0.0,
+      "text": "Okay, so these are"
+    },
+    {
+      "start": 3524.12,
+      "duration": 0.0,
+      "text": "Okay, so these are some<00:58:44.400><c> of</c><00:58:44.480><c> the</c><00:58:44.600><c> papers</c><00:58:45.480><c> on</c>"
+    },
+    {
+      "start": 3526.91,
+      "duration": 0.0,
+      "text": "some of the papers on"
+    },
+    {
+      "start": 3526.92,
+      "duration": 0.0,
+      "text": "some of the papers on that<00:58:47.200><c> I</c><00:58:47.640><c> so</c><00:58:47.760><c> either</c><00:58:47.960><c> published</c><00:58:48.600><c> or</c><00:58:48.880><c> put</c><00:58:49.080><c> on</c>"
+    },
+    {
+      "start": 3529.19,
+      "duration": 0.0,
+      "text": "that I so either published or put on"
+    },
+    {
+      "start": 3529.2,
+      "duration": 0.0,
+      "text": "that I so either published or put on archive<00:58:50.080><c> on</c><00:58:50.200><c> this</c><00:58:50.359><c> topic.</c><00:58:51.040><c> We</c><00:58:51.160><c> tried</c><00:58:51.560><c> the</c>"
+    },
+    {
+      "start": 3531.63,
+      "duration": 0.0,
+      "text": "archive on this topic. We tried the"
+    },
+    {
+      "start": 3531.64,
+      "duration": 0.0,
+      "text": "archive on this topic. We tried the different<00:58:52.760><c> directions</c><00:58:53.440><c> that</c><00:58:53.640><c> I</c><00:58:53.720><c> haven't</c>"
+    },
+    {
+      "start": 3533.99,
+      "duration": 0.0,
+      "text": "different directions that I haven't"
+    },
+    {
+      "start": 3534.0,
+      "duration": 0.0,
+      "text": "different directions that I haven't presented<00:58:54.520><c> here.</c>"
+    },
+    {
+      "start": 3535.79,
+      "duration": 0.0,
+      "text": "presented here."
+    },
+    {
+      "start": 3535.8,
+      "duration": 0.0,
+      "text": "presented here. But"
+    },
+    {
+      "start": 3537.75,
+      "duration": 0.0,
+      "text": "But"
+    },
+    {
+      "start": 3537.76,
+      "duration": 0.0,
+      "text": "But so<00:58:58.160><c> these</c><00:58:58.720><c> were</c><00:58:59.600><c> so</c><00:58:59.800><c> the</c><00:59:00.520><c> main</c><00:59:00.960><c> references</c>"
+    },
+    {
+      "start": 3542.43,
+      "duration": 0.0,
+      "text": "so these were so the main references"
+    },
+    {
+      "start": 3542.44,
+      "duration": 0.0,
+      "text": "so these were so the main references and<00:59:02.600><c> to</c><00:59:02.680><c> conclude</c><00:59:03.440><c> so</c><00:59:03.600><c> we</c><00:59:03.720><c> have</c><00:59:03.880><c> seen</c>"
+    },
+    {
+      "start": 3544.99,
+      "duration": 0.0,
+      "text": "and to conclude so we have seen"
+    },
+    {
+      "start": 3545.0,
+      "duration": 0.0,
+      "text": "and to conclude so we have seen how<00:59:06.280><c> to</c><00:59:06.520><c> model</c><00:59:07.080><c> high-dimensional</c>"
+    },
+    {
+      "start": 3548.63,
+      "duration": 0.0,
+      "text": "how to model high-dimensional"
+    },
+    {
+      "start": 3548.64,
+      "duration": 0.0,
+      "text": "how to model high-dimensional partially<00:59:09.040><c> observed</c><00:59:09.880><c> chaotic</c><00:59:10.320><c> systems</c>"
+    },
+    {
+      "start": 3551.99,
+      "duration": 0.0,
+      "text": "partially observed chaotic systems"
+    },
+    {
+      "start": 3552.0,
+      "duration": 0.0,
+      "text": "partially observed chaotic systems um"
+    },
+    {
+      "start": 3553.349,
+      "duration": 0.0,
+      "text": "um"
+    },
+    {
+      "start": 3553.359,
+      "duration": 0.0,
+      "text": "um how<00:59:14.160><c> the</c><00:59:14.880><c> knowledge</c><00:59:15.320><c> of</c><00:59:15.400><c> the</c><00:59:15.520><c> score</c><00:59:15.800><c> function</c>"
+    },
+    {
+      "start": 3556.91,
+      "duration": 0.0,
+      "text": "how the knowledge of the score function"
+    },
+    {
+      "start": 3556.92,
+      "duration": 0.0,
+      "text": "how the knowledge of the score function plays<00:59:17.520><c> a</c><00:59:17.640><c> key</c><00:59:17.920><c> role</c><00:59:18.840><c> in</c><00:59:19.640><c> allowing</c><00:59:20.200><c> this</c>"
+    },
+    {
+      "start": 3560.99,
+      "duration": 0.0,
+      "text": "plays a key role in allowing this"
+    },
+    {
+      "start": 3561.0,
+      "duration": 0.0,
+      "text": "plays a key role in allowing this modeling<00:59:21.480><c> this</c><00:59:21.960><c> modeling</c><00:59:22.359><c> strategies.</c><00:59:23.600><c> We</c>"
+    },
+    {
+      "start": 3563.67,
+      "duration": 0.0,
+      "text": "modeling this modeling strategies. We"
+    },
+    {
+      "start": 3563.68,
+      "duration": 0.0,
+      "text": "modeling this modeling strategies. We have<00:59:23.840><c> seen</c><00:59:24.080><c> two</c><00:59:24.240><c> different</c><00:59:24.560><c> directions.</c><00:59:25.400><c> In</c>"
+    },
+    {
+      "start": 3565.51,
+      "duration": 0.0,
+      "text": "have seen two different directions. In"
+    },
+    {
+      "start": 3565.52,
+      "duration": 0.0,
+      "text": "have seen two different directions. In the<00:59:25.600><c> first</c><00:59:25.920><c> one</c><00:59:26.240><c> we</c><00:59:26.400><c> have</c><00:59:26.640><c> a</c><00:59:26.680><c> model</c><00:59:27.040><c> answers</c>"
+    },
+    {
+      "start": 3568.47,
+      "duration": 0.0,
+      "text": "the first one we have a model answers"
+    },
+    {
+      "start": 3568.48,
+      "duration": 0.0,
+      "text": "the first one we have a model answers and<00:59:28.920><c> we</c><00:59:29.000><c> are</c><00:59:29.120><c> just</c>"
+    },
+    {
+      "start": 3570.47,
+      "duration": 0.0,
+      "text": "and we are just"
+    },
+    {
+      "start": 3570.48,
+      "duration": 0.0,
+      "text": "and we are just calibrating<00:59:31.240><c> the</c><00:59:31.359><c> model</c><00:59:31.640><c> parameters</c><00:59:32.600><c> using</c><00:59:33.000><c> a</c>"
+    },
+    {
+      "start": 3573.07,
+      "duration": 0.0,
+      "text": "calibrating the model parameters using a"
+    },
+    {
+      "start": 3573.08,
+      "duration": 0.0,
+      "text": "calibrating the model parameters using a combination<00:59:33.800><c> between</c><00:59:34.840><c> the</c><00:59:34.960><c> generalized</c>"
+    },
+    {
+      "start": 3575.63,
+      "duration": 0.0,
+      "text": "combination between the generalized"
+    },
+    {
+      "start": 3575.64,
+      "duration": 0.0,
+      "text": "combination between the generalized fluctuation<00:59:36.080><c> distribution</c><00:59:36.560><c> theorem</c><00:59:37.440><c> and</c><00:59:38.200><c> the</c>"
+    },
+    {
+      "start": 3579.11,
+      "duration": 0.0,
+      "text": "fluctuation distribution theorem and the"
+    },
+    {
+      "start": 3579.12,
+      "duration": 0.0,
+      "text": "fluctuation distribution theorem and the score<00:59:39.480><c> modeling.</c>"
+    },
+    {
+      "start": 3580.99,
+      "duration": 0.0,
+      "text": "score modeling."
+    },
+    {
+      "start": 3581.0,
+      "duration": 0.0,
+      "text": "score modeling. The<00:59:41.120><c> other</c><00:59:41.320><c> direction</c><00:59:41.840><c> instead</c>"
+    },
+    {
+      "start": 3583.23,
+      "duration": 0.0,
+      "text": "The other direction instead"
+    },
+    {
+      "start": 3583.24,
+      "duration": 0.0,
+      "text": "The other direction instead doesn't<00:59:44.120><c> have</c><00:59:44.600><c> any</c><00:59:45.120><c> model</c><00:59:45.520><c> answer.</c>"
+    },
+    {
+      "start": 3587.43,
+      "duration": 0.0,
+      "text": "doesn't have any model answer."
+    },
+    {
+      "start": 3587.44,
+      "duration": 0.0,
+      "text": "doesn't have any model answer. We<00:59:47.560><c> just</c><00:59:47.800><c> try</c><00:59:48.440><c> to</c>"
+    },
+    {
+      "start": 3589.99,
+      "duration": 0.0,
+      "text": "We just try to"
+    },
+    {
+      "start": 3590.0,
+      "duration": 0.0,
+      "text": "We just try to starting<00:59:50.640><c> from</c><00:59:51.120><c> a</c><00:59:51.200><c> set</c><00:59:51.480><c> of</c><00:59:51.600><c> statistical</c><00:59:52.160><c> and</c>"
+    },
+    {
+      "start": 3592.27,
+      "duration": 0.0,
+      "text": "starting from a set of statistical and"
+    },
+    {
+      "start": 3592.28,
+      "duration": 0.0,
+      "text": "starting from a set of statistical and dynamical<00:59:52.920><c> observables</c><00:59:53.840><c> to</c><00:59:53.880><c> build</c><00:59:54.200><c> a</c><00:59:54.280><c> model</c>"
+    },
+    {
+      "start": 3594.75,
+      "duration": 0.0,
+      "text": "dynamical observables to build a model"
+    },
+    {
+      "start": 3594.76,
+      "duration": 0.0,
+      "text": "dynamical observables to build a model that<00:59:54.920><c> by</c><00:59:55.080><c> construction</c><00:59:56.160><c> reproduces</c><00:59:56.840><c> all</c><00:59:56.920><c> of</c>"
+    },
+    {
+      "start": 3597.03,
+      "duration": 0.0,
+      "text": "that by construction reproduces all of"
+    },
+    {
+      "start": 3597.04,
+      "duration": 0.0,
+      "text": "that by construction reproduces all of them<00:59:57.600><c> without</c><00:59:58.280><c> integrating</c><00:59:58.960><c> the</c><00:59:59.040><c> model</c>"
+    },
+    {
+      "start": 3599.31,
+      "duration": 0.0,
+      "text": "them without integrating the model"
+    },
+    {
+      "start": 3599.32,
+      "duration": 0.0,
+      "text": "them without integrating the model forward."
+    },
+    {
+      "start": 3600.83,
+      "duration": 0.0,
+      "text": "forward."
+    },
+    {
+      "start": 3600.84,
+      "duration": 0.0,
+      "text": "forward. And<01:00:00.960><c> then</c><01:00:01.200><c> we've</c><01:00:01.400><c> seen</c><01:00:01.640><c> how</c><01:00:02.040><c> this</c><01:00:02.840><c> approach</c>"
+    },
+    {
+      "start": 3603.51,
+      "duration": 0.0,
+      "text": "And then we've seen how this approach"
+    },
+    {
+      "start": 3603.52,
+      "duration": 0.0,
+      "text": "And then we've seen how this approach can<01:00:03.920><c> scale</c><01:00:04.520><c> on</c><01:00:05.120><c> different</c><01:00:05.480><c> systems</c><01:00:06.360><c> from</c><01:00:06.760><c> toy</c>"
+    },
+    {
+      "start": 3606.91,
+      "duration": 0.0,
+      "text": "can scale on different systems from toy"
+    },
+    {
+      "start": 3606.92,
+      "duration": 0.0,
+      "text": "can scale on different systems from toy models<01:00:07.760><c> to</c><01:00:08.400><c> very</c><01:00:08.680><c> high</c><01:00:08.840><c> dimensional</c><01:00:09.920><c> systems.</c>"
+    },
+    {
+      "start": 3611.47,
+      "duration": 0.0,
+      "text": "models to very high dimensional systems."
+    },
+    {
+      "start": 3611.48,
+      "duration": 0.0,
+      "text": "models to very high dimensional systems. Okay,<01:00:11.880><c> thanks</c><01:00:12.560><c> for</c><01:00:12.680><c> listening</c><01:00:13.640><c> and</c><01:00:13.760><c> let</c><01:00:13.920><c> me</c>"
+    },
+    {
+      "start": 3613.99,
+      "duration": 0.0,
+      "text": "Okay, thanks for listening and let me"
+    },
+    {
+      "start": 3614.0,
+      "duration": 0.0,
+      "text": "Okay, thanks for listening and let me know<01:00:14.120><c> if</c><01:00:14.280><c> you</c><01:00:14.360><c> have</c>"
+    },
+    {
+      "start": 3615.23,
+      "duration": 0.0,
+      "text": "know if you have"
+    },
+    {
+      "start": 3615.24,
+      "duration": 0.0,
+      "text": "know if you have any<01:00:15.480><c> question.</c>"
+    },
+    {
+      "start": 3618.48,
+      "duration": 0.0,
+      "text": "Thank<01:00:18.680><c> you,</c><01:00:18.800><c> Ludovico.</c>"
+    },
+    {
+      "start": 3620.19,
+      "duration": 0.0,
+      "text": "Thank you, Ludovico."
+    },
+    {
+      "start": 3620.2,
+      "duration": 0.0,
+      "text": "Thank you, Ludovico. Any<01:00:20.440><c> questions?</c>"
+    },
+    {
+      "start": 3622.07,
+      "duration": 0.0,
+      "text": "Any questions?"
+    },
+    {
+      "start": 3622.08,
+      "duration": 0.0,
+      "text": "Any questions? I<01:00:22.760><c> have</c><01:00:22.920><c> a</c><01:00:22.960><c> question.</c><01:00:23.840><c> So,</c>"
+    },
+    {
+      "start": 3624.95,
+      "duration": 0.0,
+      "text": "I have a question. So,"
+    },
+    {
+      "start": 3624.96,
+      "duration": 0.0,
+      "text": "I have a question. So, uh"
+    },
+    {
+      "start": 3625.51,
+      "duration": 0.0,
+      "text": "uh"
+    },
+    {
+      "start": 3625.52,
+      "duration": 0.0,
+      "text": "uh So,<01:00:25.600><c> I'm</c><01:00:25.680><c> wondering</c><01:00:26.360><c> according</c><01:00:26.800><c> to</c><01:00:26.960><c> your</c>"
+    },
+    {
+      "start": 3627.31,
+      "duration": 0.0,
+      "text": "So, I'm wondering according to your"
+    },
+    {
+      "start": 3627.32,
+      "duration": 0.0,
+      "text": "So, I'm wondering according to your formulation,<01:00:28.400><c> is</c><01:00:28.720><c> does</c><01:00:28.960><c> your</c><01:00:29.160><c> method</c><01:00:29.600><c> allow</c>"
+    },
+    {
+      "start": 3630.07,
+      "duration": 0.0,
+      "text": "formulation, is does your method allow"
+    },
+    {
+      "start": 3630.08,
+      "duration": 0.0,
+      "text": "formulation, is does your method allow you<01:00:30.600><c> that</c>"
+    },
+    {
+      "start": 3631.43,
+      "duration": 0.0,
+      "text": "you that"
+    },
+    {
+      "start": 3631.44,
+      "duration": 0.0,
+      "text": "you that have<01:00:32.480><c> allow</c><01:00:32.800><c> you</c>"
+    },
+    {
+      "start": 3634.07,
+      "duration": 0.0,
+      "text": "have allow you"
+    },
+    {
+      "start": 3634.08,
+      "duration": 0.0,
+      "text": "have allow you to<01:00:34.400><c> to</c><01:00:34.520><c> work</c><01:00:34.760><c> on</c><01:00:34.960><c> data</c><01:00:35.200><c> set</c><01:00:35.440><c> that</c><01:00:35.680><c> has</c>"
+    },
+    {
+      "start": 3636.11,
+      "duration": 0.0,
+      "text": "to to work on data set that has"
+    },
+    {
+      "start": 3636.12,
+      "duration": 0.0,
+      "text": "to to work on data set that has absolutely<01:00:36.760><c> no</c><01:00:37.120><c> time</c><01:00:37.440><c> information?</c>"
+    },
+    {
+      "start": 3639.39,
+      "duration": 0.0,
+      "text": "absolutely no time information?"
+    },
+    {
+      "start": 3639.4,
+      "duration": 0.0,
+      "text": "absolutely no time information? What<01:00:39.600><c> do</c><01:00:39.680><c> you</c><01:00:39.760><c> mean</c><01:00:39.960><c> with</c><01:00:40.200><c> absolutely</c><01:00:41.280><c> no</c><01:00:41.480><c> time</c>"
+    },
+    {
+      "start": 3641.71,
+      "duration": 0.0,
+      "text": "What do you mean with absolutely no time"
+    },
+    {
+      "start": 3641.72,
+      "duration": 0.0,
+      "text": "What do you mean with absolutely no time information?<01:00:42.560><c> So,</c><01:00:42.760><c> like</c>"
+    },
+    {
+      "start": 3643.95,
+      "duration": 0.0,
+      "text": "information? So, like"
+    },
+    {
+      "start": 3643.96,
+      "duration": 0.0,
+      "text": "information? So, like time<01:00:44.240><c> series</c><01:00:45.440><c> uh</c>"
+    },
+    {
+      "start": 3646.15,
+      "duration": 0.0,
+      "text": "time series uh"
+    },
+    {
+      "start": 3646.16,
+      "duration": 0.0,
+      "text": "time series uh where<01:00:46.680><c> every</c><01:00:47.200><c> snapshot</c><01:00:47.880><c> is</c><01:00:48.000><c> completely</c>"
+    },
+    {
+      "start": 3648.83,
+      "duration": 0.0,
+      "text": "where every snapshot is completely"
+    },
+    {
+      "start": 3648.84,
+      "duration": 0.0,
+      "text": "where every snapshot is completely uncorrelated?<01:00:50.200><c> Yeah,</c><01:00:50.480><c> yeah.</c><01:00:51.200><c> In</c><01:00:51.320><c> that</c><01:00:51.520><c> case,</c>"
+    },
+    {
+      "start": 3652.31,
+      "duration": 0.0,
+      "text": "uncorrelated? Yeah, yeah. In that case,"
+    },
+    {
+      "start": 3652.32,
+      "duration": 0.0,
+      "text": "uncorrelated? Yeah, yeah. In that case, yes.<01:00:53.000><c> So,</c><01:00:53.400><c> so</c><01:00:53.520><c> you</c><01:00:53.640><c> can</c><01:00:53.800><c> do</c><01:00:54.000><c> it,</c><01:00:54.480><c> but</c><01:00:55.480><c> you</c><01:00:56.160><c> will</c>"
+    },
+    {
+      "start": 3656.31,
+      "duration": 0.0,
+      "text": "yes. So, so you can do it, but you will"
+    },
+    {
+      "start": 3656.32,
+      "duration": 0.0,
+      "text": "yes. So, so you can do it, but you will be<01:00:56.440><c> able</c><01:00:57.120><c> to</c><01:00:57.240><c> build</c><01:00:57.520><c> a</c><01:00:57.560><c> mathematical</c><01:00:58.120><c> model</c>"
+    },
+    {
+      "start": 3658.59,
+      "duration": 0.0,
+      "text": "be able to build a mathematical model"
+    },
+    {
+      "start": 3658.6,
+      "duration": 0.0,
+      "text": "be able to build a mathematical model that<01:00:58.800><c> reproduces</c><01:00:59.440><c> the</c><01:00:59.560><c> steady</c><01:00:59.880><c> state</c>"
+    },
+    {
+      "start": 3660.03,
+      "duration": 0.0,
+      "text": "that reproduces the steady state"
+    },
+    {
+      "start": 3660.04,
+      "duration": 0.0,
+      "text": "that reproduces the steady state distribution,"
+    },
+    {
+      "start": 3661.59,
+      "duration": 0.0,
+      "text": "distribution,"
+    },
+    {
+      "start": 3661.6,
+      "duration": 0.0,
+      "text": "distribution, but<01:01:01.880><c> not</c><01:01:02.200><c> the</c><01:01:02.320><c> dynamics</c><01:01:02.920><c> because</c><01:01:03.280><c> you</c><01:01:03.480><c> don't</c>"
+    },
+    {
+      "start": 3663.67,
+      "duration": 0.0,
+      "text": "but not the dynamics because you don't"
+    },
+    {
+      "start": 3663.68,
+      "duration": 0.0,
+      "text": "but not the dynamics because you don't have<01:01:03.920><c> any</c><01:01:04.400><c> information</c><01:01:05.000><c> about</c><01:01:05.360><c> the</c><01:01:05.800><c> dynamics.</c>"
+    },
+    {
+      "start": 3666.31,
+      "duration": 0.0,
+      "text": "have any information about the dynamics."
+    },
+    {
+      "start": 3666.32,
+      "duration": 0.0,
+      "text": "have any information about the dynamics. So,<01:01:06.480><c> what</c><01:01:06.680><c> you</c><01:01:06.760><c> can</c><01:01:06.920><c> do</c><01:01:07.040><c> in</c><01:01:07.280><c> in</c><01:01:07.400><c> that</c><01:01:07.600><c> case</c>"
+    },
+    {
+      "start": 3668.99,
+      "duration": 0.0,
+      "text": "So, what you can do in in that case"
+    },
+    {
+      "start": 3669.0,
+      "duration": 0.0,
+      "text": "So, what you can do in in that case and<01:01:09.120><c> that</c><01:01:09.320><c> will</c><01:01:09.520><c> be</c><01:01:10.200><c> yeah,</c><01:01:10.960><c> much</c><01:01:11.360><c> more</c><01:01:12.320><c> simple,</c>"
+    },
+    {
+      "start": 3674.03,
+      "duration": 0.0,
+      "text": "and that will be yeah, much more simple,"
+    },
+    {
+      "start": 3674.04,
+      "duration": 0.0,
+      "text": "and that will be yeah, much more simple, is<01:01:14.320><c> to</c><01:01:14.440><c> replace</c>"
+    },
+    {
+      "start": 3675.83,
+      "duration": 0.0,
+      "text": "is to replace"
+    },
+    {
+      "start": 3675.84,
+      "duration": 0.0,
+      "text": "is to replace here<01:01:16.400><c> uh</c>"
+    },
+    {
+      "start": 3677.67,
+      "duration": 0.0,
+      "text": "here uh"
+    },
+    {
+      "start": 3677.68,
+      "duration": 0.0,
+      "text": "here uh M<01:01:18.080><c> of</c><01:01:18.360><c> X</c><01:01:19.080><c> just</c><01:01:19.840><c> with</c><01:01:20.040><c> the</c><01:01:20.160><c> identity,</c><01:01:21.160><c> right?</c>"
+    },
+    {
+      "start": 3681.51,
+      "duration": 0.0,
+      "text": "M of X just with the identity, right?"
+    },
+    {
+      "start": 3681.52,
+      "duration": 0.0,
+      "text": "M of X just with the identity, right? So,<01:01:21.640><c> if</c><01:01:21.800><c> you're</c><01:01:22.400><c> only</c><01:01:22.720><c> caring</c><01:01:23.160><c> about</c><01:01:23.720><c> the</c>"
+    },
+    {
+      "start": 3683.83,
+      "duration": 0.0,
+      "text": "So, if you're only caring about the"
+    },
+    {
+      "start": 3683.84,
+      "duration": 0.0,
+      "text": "So, if you're only caring about the steady<01:01:24.120><c> state</c><01:01:24.720><c> distribution</c><01:01:26.040><c> and</c><01:01:26.200><c> also</c><01:01:26.480><c> you</c>"
+    },
+    {
+      "start": 3686.59,
+      "duration": 0.0,
+      "text": "steady state distribution and also you"
+    },
+    {
+      "start": 3686.6,
+      "duration": 0.0,
+      "text": "steady state distribution and also you don't<01:01:26.800><c> have</c><01:01:27.000><c> any</c><01:01:27.160><c> information</c>"
+    },
+    {
+      "start": 3688.71,
+      "duration": 0.0,
+      "text": "don't have any information"
+    },
+    {
+      "start": 3688.72,
+      "duration": 0.0,
+      "text": "don't have any information to<01:01:28.880><c> build</c><01:01:29.960><c> the</c>"
+    },
+    {
+      "start": 3690.83,
+      "duration": 0.0,
+      "text": "to build the"
+    },
+    {
+      "start": 3690.84,
+      "duration": 0.0,
+      "text": "to build the so<01:01:30.920><c> so</c>"
+    },
+    {
+      "start": 3691.95,
+      "duration": 0.0,
+      "text": "so so"
+    },
+    {
+      "start": 3691.96,
+      "duration": 0.0,
+      "text": "so so to<01:01:32.040><c> estimate</c><01:01:32.560><c> the</c><01:01:32.680><c> correlation</c><01:01:33.240><c> functions,</c>"
+    },
+    {
+      "start": 3694.79,
+      "duration": 0.0,
+      "text": "to estimate the correlation functions,"
+    },
+    {
+      "start": 3694.8,
+      "duration": 0.0,
+      "text": "to estimate the correlation functions, then<01:01:34.920><c> it</c><01:01:35.000><c> essentially</c><01:01:35.360><c> means</c><01:01:35.680><c> that</c><01:01:35.960><c> any</c><01:01:36.520><c> So,</c>"
+    },
+    {
+      "start": 3696.63,
+      "duration": 0.0,
+      "text": "then it essentially means that any So,"
+    },
+    {
+      "start": 3696.64,
+      "duration": 0.0,
+      "text": "then it essentially means that any So, you<01:01:36.800><c> cannot</c><01:01:37.160><c> infer</c><01:01:37.720><c> M</c><01:01:37.880><c> of</c><01:01:38.040><c> X</c><01:01:38.280><c> because</c><01:01:38.560><c> M</c><01:01:38.680><c> of</c><01:01:38.840><c> X</c>"
+    },
+    {
+      "start": 3699.15,
+      "duration": 0.0,
+      "text": "you cannot infer M of X because M of X"
+    },
+    {
+      "start": 3699.16,
+      "duration": 0.0,
+      "text": "you cannot infer M of X because M of X is<01:01:39.480><c> carrying</c><01:01:39.800><c> information</c><01:01:40.400><c> about</c><01:01:40.680><c> the</c>"
+    },
+    {
+      "start": 3700.75,
+      "duration": 0.0,
+      "text": "is carrying information about the"
+    },
+    {
+      "start": 3700.76,
+      "duration": 0.0,
+      "text": "is carrying information about the dynamics.<01:01:41.880><c> So,</c><01:01:42.000><c> you</c><01:01:42.160><c> can</c><01:01:42.440><c> replace</c><01:01:43.160><c> M</c><01:01:43.360><c> of</c><01:01:43.520><c> X</c>"
+    },
+    {
+      "start": 3703.83,
+      "duration": 0.0,
+      "text": "dynamics. So, you can replace M of X"
+    },
+    {
+      "start": 3703.84,
+      "duration": 0.0,
+      "text": "dynamics. So, you can replace M of X with<01:01:44.000><c> the</c><01:01:44.080><c> identity.</c>"
+    },
+    {
+      "start": 3705.55,
+      "duration": 0.0,
+      "text": "with the identity."
+    },
+    {
+      "start": 3705.56,
+      "duration": 0.0,
+      "text": "with the identity. Uh<01:01:45.800><c> so</c><01:01:46.080><c> so</c><01:01:46.240><c> you</c><01:01:46.720><c> So,</c><01:01:47.040><c> if</c><01:01:47.280><c> I</c><01:01:47.400><c> train</c><01:01:47.680><c> a</c><01:01:47.760><c> model,</c><01:01:48.280><c> I</c>"
+    },
+    {
+      "start": 3708.39,
+      "duration": 0.0,
+      "text": "Uh so so you So, if I train a model, I"
+    },
+    {
+      "start": 3708.4,
+      "duration": 0.0,
+      "text": "Uh so so you So, if I train a model, I only<01:01:48.640><c> need</c><01:01:48.840><c> to</c><01:01:49.000><c> train</c><01:01:49.400><c> the</c><01:01:49.560><c> M,</c><01:01:49.960><c> right?</c>"
+    },
+    {
+      "start": 3710.91,
+      "duration": 0.0,
+      "text": "only need to train the M, right?"
+    },
+    {
+      "start": 3710.92,
+      "duration": 0.0,
+      "text": "only need to train the M, right? Uh<01:01:51.120><c> if</c><01:01:51.320><c> you</c><01:01:51.480><c> train</c><01:01:52.040><c> So,</c><01:01:52.200><c> if</c><01:01:52.320><c> you</c><01:01:52.720><c> only</c><01:01:53.000><c> want</c><01:01:53.520><c> to</c>"
+    },
+    {
+      "start": 3713.59,
+      "duration": 0.0,
+      "text": "Uh if you train So, if you only want to"
+    },
+    {
+      "start": 3713.6,
+      "duration": 0.0,
+      "text": "Uh if you train So, if you only want to reproduce<01:01:54.160><c> the</c><01:01:54.320><c> steady</c><01:01:54.600><c> state</c><01:01:55.000><c> distribution</c>"
+    },
+    {
+      "start": 3715.79,
+      "duration": 0.0,
+      "text": "reproduce the steady state distribution"
+    },
+    {
+      "start": 3715.8,
+      "duration": 0.0,
+      "text": "reproduce the steady state distribution because<01:01:56.240><c> you</c><01:01:56.400><c> don't</c><01:01:56.640><c> have</c><01:01:56.840><c> information</c><01:01:57.440><c> about</c>"
+    },
+    {
+      "start": 3718.71,
+      "duration": 0.0,
+      "text": "because you don't have information about"
+    },
+    {
+      "start": 3718.72,
+      "duration": 0.0,
+      "text": "because you don't have information about the<01:01:59.000><c> dynamics,</c>"
+    },
+    {
+      "start": 3720.23,
+      "duration": 0.0,
+      "text": "the dynamics,"
+    },
+    {
+      "start": 3720.24,
+      "duration": 0.0,
+      "text": "the dynamics, you<01:02:00.440><c> just</c><01:02:00.880><c> need</c><01:02:01.600><c> to</c><01:02:01.720><c> train</c><01:02:02.640><c> a</c><01:02:02.720><c> neural</c><01:02:03.000><c> network</c>"
+    },
+    {
+      "start": 3723.51,
+      "duration": 0.0,
+      "text": "you just need to train a neural network"
+    },
+    {
+      "start": 3723.52,
+      "duration": 0.0,
+      "text": "you just need to train a neural network to<01:02:03.720><c> learn</c><01:02:04.280><c> the</c><01:02:04.400><c> score</c><01:02:04.720><c> function.</c>"
+    },
+    {
+      "start": 3726.47,
+      "duration": 0.0,
+      "text": "to learn the score function."
+    },
+    {
+      "start": 3726.48,
+      "duration": 0.0,
+      "text": "to learn the score function. So,<01:02:06.640><c> M</c><01:02:07.120><c> can</c><01:02:07.360><c> be</c><01:02:07.520><c> just</c><01:02:07.840><c> replaced</c><01:02:08.520><c> with</c><01:02:08.680><c> the</c>"
+    },
+    {
+      "start": 3728.75,
+      "duration": 0.0,
+      "text": "So, M can be just replaced with the"
+    },
+    {
+      "start": 3728.76,
+      "duration": 0.0,
+      "text": "So, M can be just replaced with the identity."
+    },
+    {
+      "start": 3730.59,
+      "duration": 0.0,
+      "text": "identity."
+    },
+    {
+      "start": 3730.6,
+      "duration": 0.0,
+      "text": "identity. Hm.<01:02:10.960><c> Okay,</c><01:02:11.200><c> because</c><01:02:11.560><c> then</c><01:02:11.880><c> so</c><01:02:12.080><c> any</c><01:02:12.360><c> value</c><01:02:13.000><c> So,</c>"
+    },
+    {
+      "start": 3733.19,
+      "duration": 0.0,
+      "text": "Hm. Okay, because then so any value So,"
+    },
+    {
+      "start": 3733.2,
+      "duration": 0.0,
+      "text": "Hm. Okay, because then so any value So, any<01:02:13.440><c> shape</c><01:02:13.840><c> of</c><01:02:14.000><c> M</c><01:02:14.160><c> of</c><01:02:14.360><c> X</c><01:02:15.000><c> will</c><01:02:15.680><c> uh</c>"
+    },
+    {
+      "start": 3736.55,
+      "duration": 0.0,
+      "text": "any shape of M of X will uh"
+    },
+    {
+      "start": 3736.56,
+      "duration": 0.0,
+      "text": "any shape of M of X will uh uh<01:02:17.120><c> give</c><01:02:17.560><c> you</c>"
+    },
+    {
+      "start": 3739.63,
+      "duration": 0.0,
+      "text": "uh give you"
+    },
+    {
+      "start": 3739.64,
+      "duration": 0.0,
+      "text": "uh give you the<01:02:19.760><c> correct</c><01:02:20.240><c> steady</c><01:02:20.520><c> state</c><01:02:21.320><c> distribution.</c>"
+    },
+    {
+      "start": 3742.35,
+      "duration": 0.0,
+      "text": "the correct steady state distribution."
+    },
+    {
+      "start": 3742.36,
+      "duration": 0.0,
+      "text": "the correct steady state distribution. So,<01:02:22.560><c> you</c><01:02:22.720><c> can</c><01:02:23.160><c> just</c><01:02:23.440><c> choose</c><01:02:23.800><c> M</c><01:02:24.000><c> of</c><01:02:24.200><c> X</c><01:02:24.600><c> equal</c><01:02:25.000><c> to</c>"
+    },
+    {
+      "start": 3745.11,
+      "duration": 0.0,
+      "text": "So, you can just choose M of X equal to"
+    },
+    {
+      "start": 3745.12,
+      "duration": 0.0,
+      "text": "So, you can just choose M of X equal to the<01:02:25.200><c> identity.</c>"
+    },
+    {
+      "start": 3747.19,
+      "duration": 0.0,
+      "text": "the identity."
+    },
+    {
+      "start": 3747.2,
+      "duration": 0.0,
+      "text": "the identity. Also,<01:02:27.760><c> if</c><01:02:27.920><c> you</c><01:02:28.040><c> choose</c><01:02:28.360><c> M</c><01:02:28.560><c> of</c><01:02:28.720><c> X</c><01:02:29.040><c> equal</c><01:02:29.320><c> to</c><01:02:29.400><c> the</c>"
+    },
+    {
+      "start": 3749.51,
+      "duration": 0.0,
+      "text": "Also, if you choose M of X equal to the"
+    },
+    {
+      "start": 3749.52,
+      "duration": 0.0,
+      "text": "Also, if you choose M of X equal to the identity,<01:02:30.000><c> it</c><01:02:30.160><c> probably</c><01:02:30.600><c> is</c><01:02:30.800><c> an</c><01:02:30.920><c> optimal</c>"
+    },
+    {
+      "start": 3751.31,
+      "duration": 0.0,
+      "text": "identity, it probably is an optimal"
+    },
+    {
+      "start": 3751.32,
+      "duration": 0.0,
+      "text": "identity, it probably is an optimal choice<01:02:31.840><c> because</c><01:02:32.720><c> uh</c>"
+    },
+    {
+      "start": 3753.55,
+      "duration": 0.0,
+      "text": "choice because uh"
+    },
+    {
+      "start": 3753.56,
+      "duration": 0.0,
+      "text": "choice because uh um<01:02:34.880><c> you</c><01:02:35.760><c> have</c><01:02:36.640><c> the</c><01:02:36.760><c> fastest</c><01:02:37.520><c> convergence</c>"
+    },
+    {
+      "start": 3758.87,
+      "duration": 0.0,
+      "text": "um you have the fastest convergence"
+    },
+    {
+      "start": 3758.88,
+      "duration": 0.0,
+      "text": "um you have the fastest convergence towards<01:02:39.440><c> the</c><01:02:39.560><c> steady</c><01:02:39.800><c> state</c><01:02:40.200><c> density.</c><01:02:41.120><c> So,</c>"
+    },
+    {
+      "start": 3761.31,
+      "duration": 0.0,
+      "text": "towards the steady state density. So,"
+    },
+    {
+      "start": 3761.32,
+      "duration": 0.0,
+      "text": "towards the steady state density. So, like<01:02:41.560><c> if</c><01:02:41.720><c> you</c><01:02:41.800><c> integrate</c><01:02:42.440><c> your</c><01:02:42.680><c> model,</c>"
+    },
+    {
+      "start": 3764.15,
+      "duration": 0.0,
+      "text": "like if you integrate your model,"
+    },
+    {
+      "start": 3764.16,
+      "duration": 0.0,
+      "text": "like if you integrate your model, will<01:02:44.440><c> converge</c>"
+    },
+    {
+      "start": 3766.59,
+      "duration": 0.0,
+      "text": "will converge"
+    },
+    {
+      "start": 3766.6,
+      "duration": 0.0,
+      "text": "will converge quite<01:02:46.800><c> fast</c><01:02:47.560><c> towards</c><01:02:47.960><c> the</c><01:02:48.080><c> steady</c><01:02:48.320><c> state</c>"
+    },
+    {
+      "start": 3768.67,
+      "duration": 0.0,
+      "text": "quite fast towards the steady state"
+    },
+    {
+      "start": 3768.68,
+      "duration": 0.0,
+      "text": "quite fast towards the steady state density.<01:02:49.160><c> So,</c><01:02:49.600><c> if</c><01:02:49.800><c> instead</c><01:02:50.720><c> M</c><01:02:50.920><c> of</c><01:02:51.120><c> X</c><01:02:51.560><c> is</c><01:02:51.840><c> a</c>"
+    },
+    {
+      "start": 3771.91,
+      "duration": 0.0,
+      "text": "density. So, if instead M of X is a"
+    },
+    {
+      "start": 3771.92,
+      "duration": 0.0,
+      "text": "density. So, if instead M of X is a constant<01:02:52.560><c> matrix</c><01:02:53.320><c> uh</c>"
+    },
+    {
+      "start": 3773.91,
+      "duration": 0.0,
+      "text": "constant matrix uh"
+    },
+    {
+      "start": 3773.92,
+      "duration": 0.0,
+      "text": "constant matrix uh with<01:02:54.400><c> a</c><01:02:54.440><c> wide</c><01:02:55.520><c> um</c>"
+    },
+    {
+      "start": 3776.27,
+      "duration": 0.0,
+      "text": "with a wide um"
+    },
+    {
+      "start": 3776.28,
+      "duration": 0.0,
+      "text": "with a wide um variety<01:02:56.773><c> [clears throat]</c>"
+    },
+    {
+      "start": 3778.35,
+      "duration": 0.0,
+      "text": "variety [clears throat]"
+    },
+    {
+      "start": 3778.36,
+      "duration": 0.0,
+      "text": "variety [clears throat] wide<01:02:58.800><c> amplitude</c><01:02:59.760><c> in</c><01:03:00.040><c> the</c><01:03:00.560><c> eigenvalues,</c>"
+    },
+    {
+      "start": 3782.23,
+      "duration": 0.0,
+      "text": "wide amplitude in the eigenvalues,"
+    },
+    {
+      "start": 3782.24,
+      "duration": 0.0,
+      "text": "wide amplitude in the eigenvalues, you<01:03:02.400><c> have</c><01:03:02.560><c> essentially</c><01:03:03.320><c> that</c><01:03:03.560><c> some</c><01:03:03.800><c> modes</c>"
+    },
+    {
+      "start": 3784.39,
+      "duration": 0.0,
+      "text": "you have essentially that some modes"
+    },
+    {
+      "start": 3784.4,
+      "duration": 0.0,
+      "text": "you have essentially that some modes will<01:03:05.280><c> decay</c><01:03:05.640><c> faster</c><01:03:06.160><c> than</c><01:03:06.360><c> others</c><01:03:07.040><c> and</c><01:03:07.200><c> so</c><01:03:07.320><c> you</c>"
+    },
+    {
+      "start": 3787.39,
+      "duration": 0.0,
+      "text": "will decay faster than others and so you"
+    },
+    {
+      "start": 3787.4,
+      "duration": 0.0,
+      "text": "will decay faster than others and so you have<01:03:07.560><c> to</c><01:03:07.680><c> wait</c><01:03:08.280><c> like</c><01:03:08.440><c> a</c><01:03:08.480><c> longer</c><01:03:08.800><c> time</c><01:03:09.680><c> to</c><01:03:09.880><c> see</c>"
+    },
+    {
+      "start": 3790.99,
+      "duration": 0.0,
+      "text": "have to wait like a longer time to see"
+    },
+    {
+      "start": 3791.0,
+      "duration": 0.0,
+      "text": "have to wait like a longer time to see thermalization<01:03:11.760><c> of</c><01:03:11.840><c> the</c><01:03:11.960><c> system</c><01:03:12.400><c> towards</c><01:03:12.840><c> the</c>"
+    },
+    {
+      "start": 3792.95,
+      "duration": 0.0,
+      "text": "thermalization of the system towards the"
+    },
+    {
+      "start": 3792.96,
+      "duration": 0.0,
+      "text": "thermalization of the system towards the steady<01:03:13.160><c> state</c><01:03:13.800><c> distribution.</c>"
+    },
+    {
+      "start": 3795.99,
+      "duration": 0.0,
+      "text": "steady state distribution."
+    },
+    {
+      "start": 3796.0,
+      "duration": 0.0,
+      "text": "steady state distribution. Yeah,<01:03:16.120><c> this</c><01:03:16.320><c> is</c><01:03:16.480><c> very</c><01:03:16.720><c> interesting</c><01:03:17.200><c> because</c>"
+    },
+    {
+      "start": 3797.63,
+      "duration": 0.0,
+      "text": "Yeah, this is very interesting because"
+    },
+    {
+      "start": 3797.64,
+      "duration": 0.0,
+      "text": "Yeah, this is very interesting because we<01:03:17.920><c> we</c><01:03:18.080><c> previously</c><01:03:18.520><c> have</c><01:03:18.840><c> a</c>"
+    },
+    {
+      "start": 3799.91,
+      "duration": 0.0,
+      "text": "we we previously have a"
+    },
+    {
+      "start": 3799.92,
+      "duration": 0.0,
+      "text": "we we previously have a have<01:03:20.080><c> a</c><01:03:20.160><c> paper</c><01:03:20.600><c> that</c><01:03:21.520><c> targeting</c><01:03:22.120><c> exactly</c><01:03:22.680><c> on</c>"
+    },
+    {
+      "start": 3803.07,
+      "duration": 0.0,
+      "text": "have a paper that targeting exactly on"
+    },
+    {
+      "start": 3803.08,
+      "duration": 0.0,
+      "text": "have a paper that targeting exactly on no<01:03:23.320><c> time</c><01:03:23.600><c> information</c><01:03:24.400><c> and</c><01:03:25.120><c> we</c><01:03:25.360><c> we</c><01:03:25.520><c> got</c><01:03:25.720><c> some</c>"
+    },
+    {
+      "start": 3805.91,
+      "duration": 0.0,
+      "text": "no time information and we we got some"
+    },
+    {
+      "start": 3805.92,
+      "duration": 0.0,
+      "text": "no time information and we we got some difficulty<01:03:26.880><c> when</c><01:03:27.040><c> we</c><01:03:27.120><c> move</c><01:03:27.440><c> from</c><01:03:27.760><c> low</c>"
+    },
+    {
+      "start": 3807.95,
+      "duration": 0.0,
+      "text": "difficulty when we move from low"
+    },
+    {
+      "start": 3807.96,
+      "duration": 0.0,
+      "text": "difficulty when we move from low dimension<01:03:28.400><c> like</c><01:03:28.880><c> two</c><01:03:29.040><c> or</c><01:03:29.160><c> three</c><01:03:29.560><c> to</c><01:03:30.040><c> to</c>"
+    },
+    {
+      "start": 3810.71,
+      "duration": 0.0,
+      "text": "dimension like two or three to to"
+    },
+    {
+      "start": 3810.72,
+      "duration": 0.0,
+      "text": "dimension like two or three to to thousands<01:03:31.160><c> of</c><01:03:31.280><c> dimension.</c><01:03:32.000><c> In</c><01:03:32.200><c> thousands</c><01:03:32.640><c> of</c>"
+    },
+    {
+      "start": 3812.71,
+      "duration": 0.0,
+      "text": "thousands of dimension. In thousands of"
+    },
+    {
+      "start": 3812.72,
+      "duration": 0.0,
+      "text": "thousands of dimension. In thousands of dimension,<01:03:33.200><c> our</c><01:03:33.480><c> method</c><01:03:33.840><c> basically</c><01:03:34.920><c> uh</c>"
+    },
+    {
+      "start": 3815.23,
+      "duration": 0.0,
+      "text": "dimension, our method basically uh"
+    },
+    {
+      "start": 3815.24,
+      "duration": 0.0,
+      "text": "dimension, our method basically uh almost<01:03:35.640><c> failed</c><01:03:36.280><c> and</c><01:03:36.960><c> uh</c>"
+    },
+    {
+      "start": 3817.51,
+      "duration": 0.0,
+      "text": "almost failed and uh"
+    },
+    {
+      "start": 3817.52,
+      "duration": 0.0,
+      "text": "almost failed and uh so<01:03:38.080><c> yeah,</c><01:03:38.360><c> so</c><01:03:38.640><c> I'm</c><01:03:38.760><c> wondering</c>"
+    },
+    {
+      "start": 3819.79,
+      "duration": 0.0,
+      "text": "so yeah, so I'm wondering"
+    },
+    {
+      "start": 3819.8,
+      "duration": 0.0,
+      "text": "so yeah, so I'm wondering if<01:03:40.400><c> your</c><01:03:40.600><c> method</c><01:03:41.040><c> can</c><01:03:41.800><c> can</c><01:03:41.960><c> be</c><01:03:42.080><c> helpful.</c><01:03:42.880><c> Yeah,</c>"
+    },
+    {
+      "start": 3823.23,
+      "duration": 0.0,
+      "text": "if your method can can be helpful. Yeah,"
+    },
+    {
+      "start": 3823.24,
+      "duration": 0.0,
+      "text": "if your method can can be helpful. Yeah, so<01:03:43.520><c> we</c><01:03:43.720><c> estimated</c><01:03:44.240><c> the</c><01:03:44.320><c> score</c><01:03:44.560><c> function</c><01:03:45.080><c> also</c>"
+    },
+    {
+      "start": 3825.39,
+      "duration": 0.0,
+      "text": "so we estimated the score function also"
+    },
+    {
+      "start": 3825.4,
+      "duration": 0.0,
+      "text": "so we estimated the score function also for<01:03:45.880><c> thousand</c><01:03:46.280><c> dimensional</c><01:03:46.760><c> systems</c><01:03:47.440><c> and</c>"
+    },
+    {
+      "start": 3828.15,
+      "duration": 0.0,
+      "text": "for thousand dimensional systems and"
+    },
+    {
+      "start": 3828.16,
+      "duration": 0.0,
+      "text": "for thousand dimensional systems and yeah,<01:03:48.560><c> like</c><01:03:48.760><c> it's</c><01:03:49.000><c> not</c><01:03:49.280><c> a</c><01:03:49.320><c> problem.</c>"
+    },
+    {
+      "start": 3830.55,
+      "duration": 0.0,
+      "text": "yeah, like it's not a problem."
+    },
+    {
+      "start": 3830.56,
+      "duration": 0.0,
+      "text": "yeah, like it's not a problem. But<01:03:50.760><c> which</c>"
+    },
+    {
+      "start": 3831.91,
+      "duration": 0.0,
+      "text": "But which"
+    },
+    {
+      "start": 3831.92,
+      "duration": 0.0,
+      "text": "But which So,<01:03:52.080><c> how</c><01:03:52.280><c> have</c><01:03:52.400><c> you</c><01:03:52.520><c> done</c><01:03:52.760><c> this?</c><01:03:53.000><c> So,</c><01:03:53.600><c> did</c><01:03:53.760><c> you</c>"
+    },
+    {
+      "start": 3833.83,
+      "duration": 0.0,
+      "text": "So, how have you done this? So, did you"
+    },
+    {
+      "start": 3833.84,
+      "duration": 0.0,
+      "text": "So, how have you done this? So, did you use<01:03:54.080><c> the</c><01:03:54.160><c> neural</c><01:03:54.400><c> network</c><01:03:55.160><c> to</c>"
+    },
+    {
+      "start": 3835.91,
+      "duration": 0.0,
+      "text": "use the neural network to"
+    },
+    {
+      "start": 3835.92,
+      "duration": 0.0,
+      "text": "use the neural network to to<01:03:56.040><c> estimate</c><01:03:56.379><c> [clears throat]</c><01:03:56.480><c> this</c><01:03:56.720><c> the</c>"
+    },
+    {
+      "start": 3836.83,
+      "duration": 0.0,
+      "text": "to estimate [clears throat] this the"
+    },
+    {
+      "start": 3836.84,
+      "duration": 0.0,
+      "text": "to estimate [clears throat] this the score<01:03:57.120><c> function?</c><01:03:58.440><c> Yeah,</c><01:03:58.720><c> yeah.</c><01:03:59.000><c> We</c><01:03:59.359><c> we</c>"
+    },
+    {
+      "start": 3839.47,
+      "duration": 0.0,
+      "text": "score function? Yeah, yeah. We we"
+    },
+    {
+      "start": 3839.48,
+      "duration": 0.0,
+      "text": "score function? Yeah, yeah. We we basically<01:04:00.000><c> first</c><01:04:00.359><c> train</c><01:04:00.880><c> to</c><01:04:01.080><c> get</c><01:04:01.560><c> a</c><01:04:01.720><c> score</c>"
+    },
+    {
+      "start": 3841.99,
+      "duration": 0.0,
+      "text": "basically first train to get a score"
+    },
+    {
+      "start": 3842.0,
+      "duration": 0.0,
+      "text": "basically first train to get a score function,<01:04:02.520><c> then</c><01:04:02.920><c> we</c><01:04:03.160><c> train</c><01:04:03.440><c> a</c><01:04:03.480><c> dynamic</c>"
+    },
+    {
+      "start": 3843.91,
+      "duration": 0.0,
+      "text": "function, then we train a dynamic"
+    },
+    {
+      "start": 3843.92,
+      "duration": 0.0,
+      "text": "function, then we train a dynamic function.<01:04:04.520><c> But</c><01:04:04.720><c> that</c><01:04:04.960><c> dynamic</c><01:04:05.320><c> function</c><01:04:05.680><c> is</c>"
+    },
+    {
+      "start": 3845.83,
+      "duration": 0.0,
+      "text": "function. But that dynamic function is"
+    },
+    {
+      "start": 3845.84,
+      "duration": 0.0,
+      "text": "function. But that dynamic function is also<01:04:06.080><c> a</c><01:04:06.120><c> neural</c><01:04:06.359><c> network.</c><01:04:06.840><c> So,</c><01:04:07.040><c> we</c><01:04:07.320><c> we</c><01:04:07.520><c> do</c><01:04:07.800><c> we</c>"
+    },
+    {
+      "start": 3847.91,
+      "duration": 0.0,
+      "text": "also a neural network. So, we we do we"
+    },
+    {
+      "start": 3847.92,
+      "duration": 0.0,
+      "text": "also a neural network. So, we we do we do<01:04:08.040><c> not</c>"
+    },
+    {
+      "start": 3848.23,
+      "duration": 0.0,
+      "text": "do not"
+    },
+    {
+      "start": 3848.24,
+      "duration": 0.0,
+      "text": "do not &gt;&gt; But<01:04:08.280><c> you</c><01:04:08.440><c> don't</c><01:04:08.640><c> need</c><01:04:08.880><c> that</c><01:04:09.240><c> because</c><01:04:09.640><c> if</c><01:04:09.880><c> you</c>"
+    },
+    {
+      "start": 3849.95,
+      "duration": 0.0,
+      "text": "&gt;&gt; But you don't need that because if you"
+    },
+    {
+      "start": 3849.96,
+      "duration": 0.0,
+      "text": "&gt;&gt; But you don't need that because if you just<01:04:10.280><c> care</c><01:04:10.560><c> about</c><01:04:11.320><c> the</c><01:04:11.840><c> the</c><01:04:12.280><c> steady</c><01:04:12.840><c> So,</c><01:04:13.000><c> a</c>"
+    },
+    {
+      "start": 3853.03,
+      "duration": 0.0,
+      "text": "just care about the the steady So, a"
+    },
+    {
+      "start": 3853.04,
+      "duration": 0.0,
+      "text": "just care about the the steady So, a system<01:04:13.480><c> that</c><01:04:13.640><c> reproduces</c><01:04:14.400><c> the</c><01:04:14.520><c> steady</c><01:04:14.800><c> state</c>"
+    },
+    {
+      "start": 3855.19,
+      "duration": 0.0,
+      "text": "system that reproduces the steady state"
+    },
+    {
+      "start": 3855.2,
+      "duration": 0.0,
+      "text": "system that reproduces the steady state density,"
+    },
+    {
+      "start": 3856.63,
+      "duration": 0.0,
+      "text": "density,"
+    },
+    {
+      "start": 3856.64,
+      "duration": 0.0,
+      "text": "density, you<01:04:16.800><c> can</c><01:04:17.040><c> just</c><01:04:17.400><c> integrate</c><01:04:18.080><c> this</c><01:04:18.359><c> Langevin</c>"
+    },
+    {
+      "start": 3858.79,
+      "duration": 0.0,
+      "text": "you can just integrate this Langevin"
+    },
+    {
+      "start": 3858.8,
+      "duration": 0.0,
+      "text": "you can just integrate this Langevin equation<01:04:19.400><c> without</c>"
+    },
+    {
+      "start": 3860.79,
+      "duration": 0.0,
+      "text": "equation without"
+    },
+    {
+      "start": 3860.8,
+      "duration": 0.0,
+      "text": "equation without any"
+    },
+    {
+      "start": 3862.15,
+      "duration": 0.0,
+      "text": "any"
+    },
+    {
+      "start": 3862.16,
+      "duration": 0.0,
+      "text": "any other<01:04:22.480><c> network.</c><01:04:23.680><c> So,</c><01:04:24.240><c> is</c><01:04:24.400><c> this</c><01:04:24.720><c> a</c><01:04:24.840><c> cover</c><01:04:25.160><c> full</c>"
+    },
+    {
+      "start": 3865.51,
+      "duration": 0.0,
+      "text": "other network. So, is this a cover full"
+    },
+    {
+      "start": 3865.52,
+      "duration": 0.0,
+      "text": "other network. So, is this a cover full solution<01:04:26.080><c> or</c><01:04:26.520><c> just</c><01:04:26.760><c> a</c><01:04:26.840><c> subset</c><01:04:27.359><c> of</c><01:04:27.520><c> solution?</c>"
+    },
+    {
+      "start": 3868.55,
+      "duration": 0.0,
+      "text": "solution or just a subset of solution?"
+    },
+    {
+      "start": 3868.56,
+      "duration": 0.0,
+      "text": "solution or just a subset of solution? No,<01:04:28.720><c> this</c><01:04:28.920><c> is</c><01:04:29.080><c> a</c><01:04:29.160><c> general</c><01:04:29.480><c> solution.</c><01:04:30.160><c> So,</c><01:04:30.320><c> this</c>"
+    },
+    {
+      "start": 3870.83,
+      "duration": 0.0,
+      "text": "No, this is a general solution. So, this"
+    },
+    {
+      "start": 3870.84,
+      "duration": 0.0,
+      "text": "No, this is a general solution. So, this expression<01:04:31.440><c> for</c><01:04:31.600><c> f</c><01:04:31.760><c> of</c><01:04:31.920><c> x</c><01:04:32.480><c> is</c><01:04:32.600><c> a</c><01:04:32.680><c> general</c>"
+    },
+    {
+      "start": 3872.99,
+      "duration": 0.0,
+      "text": "expression for f of x is a general"
+    },
+    {
+      "start": 3873.0,
+      "duration": 0.0,
+      "text": "expression for f of x is a general solution."
+    },
+    {
+      "start": 3874.55,
+      "duration": 0.0,
+      "text": "solution."
+    },
+    {
+      "start": 3874.56,
+      "duration": 0.0,
+      "text": "solution. So,<01:04:34.680><c> it's</c><01:04:34.800><c> essentially</c><01:04:35.520><c> is</c><01:04:36.160><c> So,</c><01:04:36.320><c> given</c><01:04:36.800><c> this</c>"
+    },
+    {
+      "start": 3877.03,
+      "duration": 0.0,
+      "text": "So, it's essentially is So, given this"
+    },
+    {
+      "start": 3877.04,
+      "duration": 0.0,
+      "text": "So, it's essentially is So, given this Langevin<01:04:37.359><c> equation,</c><01:04:38.320><c> if</c><01:04:38.520><c> you</c><01:04:38.640><c> ask</c><01:04:39.359><c> what</c><01:04:39.600><c> is</c>"
+    },
+    {
+      "start": 3879.71,
+      "duration": 0.0,
+      "text": "Langevin equation, if you ask what is"
+    },
+    {
+      "start": 3879.72,
+      "duration": 0.0,
+      "text": "Langevin equation, if you ask what is the<01:04:39.880><c> most</c><01:04:40.200><c> general</c><01:04:40.640><c> expression</c><01:04:41.960><c> for</c><01:04:42.359><c> the</c>"
+    },
+    {
+      "start": 3882.51,
+      "duration": 0.0,
+      "text": "the most general expression for the"
+    },
+    {
+      "start": 3882.52,
+      "duration": 0.0,
+      "text": "the most general expression for the drift<01:04:42.920><c> term</c><01:04:43.400><c> in</c><01:04:43.520><c> such</c><01:04:43.760><c> a</c><01:04:43.840><c> way</c><01:04:44.880><c> that</c><01:04:45.800><c> uh</c>"
+    },
+    {
+      "start": 3887.55,
+      "duration": 0.0,
+      "text": "drift term in such a way that uh"
+    },
+    {
+      "start": 3887.56,
+      "duration": 0.0,
+      "text": "drift term in such a way that uh So,<01:04:47.800><c> it</c><01:04:48.080><c> reproduces</c><01:04:48.880><c> the</c><01:04:49.000><c> observed</c><01:04:49.640><c> steady</c>"
+    },
+    {
+      "start": 3889.95,
+      "duration": 0.0,
+      "text": "So, it reproduces the observed steady"
+    },
+    {
+      "start": 3889.96,
+      "duration": 0.0,
+      "text": "So, it reproduces the observed steady state<01:04:50.440><c> distribution.</c><01:04:51.320><c> So,</c><01:04:51.480><c> essentially</c><01:04:52.200><c> that</c>"
+    },
+    {
+      "start": 3892.39,
+      "duration": 0.0,
+      "text": "state distribution. So, essentially that"
+    },
+    {
+      "start": 3892.4,
+      "duration": 0.0,
+      "text": "state distribution. So, essentially that solve"
+    },
+    {
+      "start": 3893.63,
+      "duration": 0.0,
+      "text": "solve"
+    },
+    {
+      "start": 3893.64,
+      "duration": 0.0,
+      "text": "solve the<01:04:54.000><c> stationary</c><01:04:54.720><c> Fokker-Planck</c><01:04:55.280><c> equation,</c>"
+    },
+    {
+      "start": 3896.59,
+      "duration": 0.0,
+      "text": "the stationary Fokker-Planck equation,"
+    },
+    {
+      "start": 3896.6,
+      "duration": 0.0,
+      "text": "the stationary Fokker-Planck equation, then<01:04:56.840><c> this</c><01:04:57.040><c> is</c><01:04:57.160><c> the</c><01:04:57.280><c> most</c><01:04:57.520><c> general</c>"
+    },
+    {
+      "start": 3898.19,
+      "duration": 0.0,
+      "text": "then this is the most general"
+    },
+    {
+      "start": 3898.2,
+      "duration": 0.0,
+      "text": "then this is the most general expression."
+    },
+    {
+      "start": 3899.67,
+      "duration": 0.0,
+      "text": "expression."
+    },
+    {
+      "start": 3899.68,
+      "duration": 0.0,
+      "text": "expression. Interesting.<01:05:00.280><c> Yeah.</c><01:05:00.720><c> But</c><01:05:01.200><c> since</c><01:05:02.160><c> So,</c><01:05:02.400><c> here</c>"
+    },
+    {
+      "start": 3902.63,
+      "duration": 0.0,
+      "text": "Interesting. Yeah. But since So, here"
+    },
+    {
+      "start": 3902.64,
+      "duration": 0.0,
+      "text": "Interesting. Yeah. But since So, here the<01:05:02.760><c> main</c><01:05:03.120><c> point</c><01:05:03.880><c> is</c><01:05:04.080><c> to</c><01:05:04.200><c> estimate</c><01:05:05.600><c> So,</c><01:05:05.720><c> it's</c>"
+    },
+    {
+      "start": 3905.83,
+      "duration": 0.0,
+      "text": "the main point is to estimate So, it's"
+    },
+    {
+      "start": 3905.84,
+      "duration": 0.0,
+      "text": "the main point is to estimate So, it's to<01:05:05.920><c> reproduce</c><01:05:06.480><c> the</c><01:05:06.600><c> dynamics.</c><01:05:07.280><c> So,</c><01:05:07.400><c> this</c><01:05:07.560><c> is</c>"
+    },
+    {
+      "start": 3907.67,
+      "duration": 0.0,
+      "text": "to reproduce the dynamics. So, this is"
+    },
+    {
+      "start": 3907.68,
+      "duration": 0.0,
+      "text": "to reproduce the dynamics. So, this is the<01:05:07.800><c> non-trivial</c><01:05:08.440><c> part.</c>"
+    },
+    {
+      "start": 3909.63,
+      "duration": 0.0,
+      "text": "the non-trivial part."
+    },
+    {
+      "start": 3909.64,
+      "duration": 0.0,
+      "text": "the non-trivial part. If<01:05:09.760><c> you're</c><01:05:09.880><c> just</c><01:05:10.200><c> interested</c><01:05:10.880><c> in</c><01:05:10.960><c> the</c>"
+    },
+    {
+      "start": 3911.07,
+      "duration": 0.0,
+      "text": "If you're just interested in the"
+    },
+    {
+      "start": 3911.08,
+      "duration": 0.0,
+      "text": "If you're just interested in the statistics,<01:05:11.920><c> then</c><01:05:12.680><c> yeah,</c><01:05:12.960><c> take</c><01:05:13.359><c> M</c><01:05:13.520><c> of</c><01:05:13.720><c> X</c><01:05:13.960><c> equal</c>"
+    },
+    {
+      "start": 3914.349,
+      "duration": 0.0,
+      "text": "statistics, then yeah, take M of X equal"
+    },
+    {
+      "start": 3914.359,
+      "duration": 0.0,
+      "text": "statistics, then yeah, take M of X equal to<01:05:14.480><c> the</c><01:05:14.560><c> identity</c>"
+    },
+    {
+      "start": 3916.31,
+      "duration": 0.0,
+      "text": "to the identity"
+    },
+    {
+      "start": 3916.32,
+      "duration": 0.0,
+      "text": "to the identity and<01:05:16.600><c> that's</c><01:05:16.760><c> it.</c>"
+    },
+    {
+      "start": 3918.03,
+      "duration": 0.0,
+      "text": "and that's it."
+    },
+    {
+      "start": 3918.04,
+      "duration": 0.0,
+      "text": "and that's it. Okay.<01:05:18.640><c> Thank</c><01:05:18.840><c> you</c><01:05:18.920><c> very</c><01:05:19.080><c> much.</c><01:05:19.400><c> I</c><01:05:19.560><c> I</c><01:05:19.640><c> will</c>"
+    },
+    {
+      "start": 3919.87,
+      "duration": 0.0,
+      "text": "Okay. Thank you very much. I I will"
+    },
+    {
+      "start": 3919.88,
+      "duration": 0.0,
+      "text": "Okay. Thank you very much. I I will write<01:05:20.120><c> an</c><01:05:20.160><c> email</c><01:05:20.440><c> to</c><01:05:20.560><c> you.</c><01:05:21.200><c> Uh</c><01:05:21.240><c> Yeah.</c>"
+    },
+    {
+      "start": 3928.24,
+      "duration": 0.0,
+      "text": "By<01:05:28.359><c> the</c><01:05:28.480><c> way,</c>"
+    },
+    {
+      "start": 3929.71,
+      "duration": 0.0,
+      "text": "By the way,"
+    },
+    {
+      "start": 3929.72,
+      "duration": 0.0,
+      "text": "By the way, when<01:05:30.359><c> some</c><01:05:30.600><c> system</c><01:05:31.000><c> have</c><01:05:31.400><c> a</c><01:05:31.440><c> source</c><01:05:31.960><c> and</c><01:05:32.200><c> sink</c>"
+    },
+    {
+      "start": 3933.27,
+      "duration": 0.0,
+      "text": "when some system have a source and sink"
+    },
+    {
+      "start": 3933.28,
+      "duration": 0.0,
+      "text": "when some system have a source and sink and<01:05:34.320><c> does</c><01:05:34.560><c> your</c><01:05:34.720><c> method</c><01:05:35.160><c> can</c>"
+    },
+    {
+      "start": 3936.15,
+      "duration": 0.0,
+      "text": "and does your method can"
+    },
+    {
+      "start": 3936.16,
+      "duration": 0.0,
+      "text": "and does your method can can<01:05:36.800><c> cover</c><01:05:37.280><c> those</c><01:05:37.520><c> situations?</c>"
+    },
+    {
+      "start": 3939.87,
+      "duration": 0.0,
+      "text": "can cover those situations?"
+    },
+    {
+      "start": 3939.88,
+      "duration": 0.0,
+      "text": "can cover those situations? Yeah,<01:05:40.160><c> so</c><01:05:40.480><c> if</c>"
+    },
+    {
+      "start": 3942.55,
+      "duration": 0.0,
+      "text": "Yeah, so if"
+    },
+    {
+      "start": 3942.56,
+      "duration": 0.0,
+      "text": "Yeah, so if So,<01:05:42.720><c> there</c><01:05:42.960><c> is</c><01:05:43.200><c> no</c><01:05:43.400><c> time</c><01:05:43.840><c> modulation</c>"
+    },
+    {
+      "start": 3946.19,
+      "duration": 0.0,
+      "text": "So, there is no time modulation"
+    },
+    {
+      "start": 3946.2,
+      "duration": 0.0,
+      "text": "So, there is no time modulation of<01:05:46.400><c> them.</c><01:05:46.960><c> So,</c><01:05:47.359><c> so</c><01:05:47.440><c> essentially</c><01:05:48.040><c> if</c><01:05:48.240><c> you</c><01:05:48.359><c> can</c>"
+    },
+    {
+      "start": 3948.83,
+      "duration": 0.0,
+      "text": "of them. So, so essentially if you can"
+    },
+    {
+      "start": 3948.84,
+      "duration": 0.0,
+      "text": "of them. So, so essentially if you can define"
+    },
+    {
+      "start": 3951.59,
+      "duration": 0.0,
+      "text": "define"
+    },
+    {
+      "start": 3951.6,
+      "duration": 0.0,
+      "text": "define a<01:05:51.680><c> steady</c><01:05:52.040><c> state</c><01:05:52.560><c> distribution,</c>"
+    },
+    {
+      "start": 3955.11,
+      "duration": 0.0,
+      "text": "a steady state distribution,"
+    },
+    {
+      "start": 3955.12,
+      "duration": 0.0,
+      "text": "a steady state distribution, then<01:05:55.880><c> yes.</c>"
+    },
+    {
+      "start": 3958.64,
+      "duration": 0.0,
+      "text": "But<01:05:59.040><c> yeah,</c><01:05:59.240><c> I</c><01:05:59.280><c> haven't</c><01:05:59.520><c> tested</c><01:05:59.880><c> them</c><01:06:00.280><c> that</c>"
+    },
+    {
+      "start": 3960.47,
+      "duration": 0.0,
+      "text": "But yeah, I haven't tested them that"
+    },
+    {
+      "start": 3960.48,
+      "duration": 0.0,
+      "text": "But yeah, I haven't tested them that much.<01:06:00.600><c> So,</c><01:06:01.080><c> Thank</c><01:06:01.240><c> you.</c><01:06:01.600><c> I</c><01:06:01.800><c> showed</c><01:06:02.040><c> you</c><01:06:02.120><c> the</c>"
+    },
+    {
+      "start": 3962.63,
+      "duration": 0.0,
+      "text": "much. So, Thank you. I showed you the"
+    },
+    {
+      "start": 3962.64,
+      "duration": 0.0,
+      "text": "much. So, Thank you. I showed you the system<01:06:03.120><c> that</c><01:06:03.720><c> Yeah,</c><01:06:04.200><c> I</c><01:06:04.280><c> tested</c><01:06:04.680><c> the</c>"
+    },
+    {
+      "start": 3964.75,
+      "duration": 0.0,
+      "text": "system that Yeah, I tested the"
+    },
+    {
+      "start": 3964.76,
+      "duration": 0.0,
+      "text": "system that Yeah, I tested the algorithm."
+    }
+  ],
+  "plain": "for inviting me at your\ngroup meeting. Um\nSo, in today's presentation, uh I will\ntalk about building mathematical models\nfrom high high-dimensional partially\nobserved\ndynamical systems.\nSo, I think this is a central topic in\nmany scientific fields ranging from\ngeophysical fluid dynamics, which is the\nfield that is the closest uh\nto my research, but also other physical\nsystems like molecular systems or\nneuroscience. So, so every times we have\naccess to data, so we have observations\nof the underlying system, and we want to\nbuild a mathematical model that is able\nto reproduce and capture the main\ncausality relationships between the\nvariable of the system, and also\nhopefully to perform predictions and\nthat uncertainty quantification.\nSo, of course for this specific class of\nsystem\nthat are extremely high-dimensional and\nalso and also multiscale and partially\nobserved, uh\nit is quite meaningless to try to\nperform trajectory shadowing. So, it's\nquite meaningless to try to build a\nmodel that is able to precisely\npredict the trajectory of the system.\nInstead, what we are usually So, what\nour goal becomes usually is to\nto reproduce some key statistical and\ndynamical observable of the\nthe data set that we are observing.\nThose observables can be, for example,\nsome moments of the steady state\ndistribution, the whole steady state\ndistribution,\nand also some\nuh state or temporal correlations\nbetween the\nobserved variables of the system.\nSo, this is the goal of this talk. So,\ndevelop a mathematical model that is\nable uh\nuh to reproduce this target statistical\nand dynamical observables.\nSo, because of this timescale separation\nof the data, so the fact that the data\ncan be multiscale, I will use\na stochastic model where we have a first\ncomponent, the drift term f of x, which\nuh will model the deterministic and slow\nvariable component of my data set.\nAnd I\nAnd I also have a stochastic component\nalso have a stochastic component\nthat takes into account the faster\ndynamics,\nthe unresolved fast fluctuations of the\ndata set.\nSo, this will be the model\nthat I will that I want to construct\nfrom the the observations\nwith the\nfinal goal\nof being able to reproduce\nin\nan efficient way this target statistical\nOkay, so right now I have been extremely\ngeneral. So, I have\ntalked about a general model to solve\nthis very general problem.\nIn the next two slides, I want to give a\nbit more details about the first what\nare the requirements that we want from\nour model,\nand then also what are the assumptions\nthat we're doing on the observed data\nset.\nSo, first\nlet's see what are the requirement that\nwe want\nfrom our modeling strategy.\nSo, as I said, we would like to model\nreal data that can be extremely\nhigh-dimensional\nand also maybe also unevenly sampled or\nhave\na very low sampling frequency. So, these\nare all\nfeatures of our real data set that we\nhave to take into account when we are\nbuilding a mathematical model for it.\nSo, this essentially implies first\nthat since\nwe want our modeling strategy to scale\nvery well with the dimension,\nand since also the model that we want to\ndevelop can be extremely\nhigh-dimensional and\ncomputationally expensive to integrate,\nwe would like to be able to reconstruct\nthe mathematical model from data using\nas few model integrations\nas possible.\nSo, a naive approach to build a\nmathematical model can be to start with\na model ansatz, integrate it forward for\na given time,\nthen calculate\nall the statistical and dynamical\nobservables, compare them with the\ntargets,\nevaluate a loss function, and use that\nto update the model. But, this will\nrequire many model integrations that can\nbecome extremely computationally\nexpensive. So, our modeling strategy\nSo, one of the main goal of our modeling\nstrategy would be to use as few model\nintegrations as possible.\nAnother requirement that we would that\nour modeling strategy to have is to\navoid the state space clustering.\nSo, we would like to avoid clustering\nthe state space to estimate the average\nvelocity field\nbecause essentially\nSo, clustering the state space, even if\nwe are using an extremely efficient\nclustering algorithm like, I don't know,\nbisecting k-means clustering algorithm,\nsuffer from the curse of dimensionality.\nSo, if we are considering a data set or\na system with dimension larger than a\nfew dozens, clustering becomes extremely\ncomputationally expensive, so we would\nlike to avoid clustering.\nAnd also we would like to use a finite\ndifference estimation to estimate the\nvelocity from trajectories. So, we are\nassuming that our realistic data set can\nbe can have a low frequen- a low\nsampling frequency, which essentially\nmeans that we cannot trust the data set\nto recover\nthe velocity\nof our dynamics using finite difference.\nSo, these are the\nthree main\num\nconstraints that we want our modeling\nstrategy to have in order to be scalable\nto high-dimensional system and also\nbeing used to realistic settings.\nThen, let's see what are instead the\nconstraints\nthat we want to impose on our system.\nSo, on which kind of systems this So,\nthe model the modeling strategy that I\nwill propose in this talk can be\napplied.\nSo, first of all, I will assume that the\nsystem I'm studying has statistical\nstationarity. So, essentially that we\ncan define a steady state distribution.\nSo, of course this can be generalized\nalso to cyclostationary data by\naugmenting the state space or, for\nexample, to\ndata set that show a low trends that can\nbe detrended and then we can get a\nstationary\nI will also assume that the system has\num so, is ergodic and mixing, which\nessentially means that time averages\nalong\nlong trajectory\nwill converge to ensemble averages, and\nalso that\nthe correlation function decays\nsufficiently fast.\nAnd then I'm also assuming that we have\nan effective timescale separation. So,\nthis essentially allow us to model the\ndata set using\nthe Langevin equation that I showed you\nbefore. So, essentially treat the fast\ntimescale as a noise process, and then\nbuild a drift term for the slow\ntimescales.\nI will also assume to have enough\nobservables of the slow timescales,\nand so in such a way I can get\na full recovery of the slow variables.\nEven if we don't have a complete\nobservables for the slow variables, we\ncan still augment the state space using\nsome delay embedded version of the data\nAnd then finally, even if\nuh we will not have probably some very\nfine resolution\nof the data sample,\nwe still assume to have enough\nresolution to be able to define\ncorrelation functions. So, we don't have\nenough resolution to\nevaluate the velocity at every\ntime point by using finite difference,\nbut still we have enough data\nessentially to define a correlation\nfunction.\nSo, these are the goals that I would\nlike\nmy modeling strategy to achieve,\nand also those are the main assumptions\nthat I'm doing on the\nsystem that I'm studying.\nSo, in this talk, I will present two\ndifferent directions. So, two different\nmodeling strategies that I'm currently\npursuing.\nThe first one So, in the first one, I'm\nassuming\nto have a model ansatz. So, essentially\nI have\na knowledge of the functional form of my\nmodel, so a knowledge that be derived\ndirectly from physics.\nAnd this model answers depends on a set\nof parameters that I would like to\ndetermine.\nI have an initial guess for those\nparameters, and I would like to\ncalibrate those model parameters in\norder to reproduce a set of statistical\nobservables\nthat I derive from my observations.\nThe second direction instead is a bit\nmore ambitious.\nI assume not to know\nany model answers. So, I don't start\nfrom\nmodel answers assumption, and I want to\nderive from my data\nthe whole functional form of the model.\nIn in this case, what I'm interested to\ndo precisely is to start from the set of\nstatistical and dynamical target\nobservables,\nand I would like to infer from them\nthe most general class of dynamical\nsystems that by construction reproduces\nthose dynamical observables.\nAnd I would like to do it. So, this is\nthe most ambitious part of the\nof the method because I would like to do\nit without ever integrating my model\nforward. So, just from the knowledge of\nthe target statistical and dynamical\nobservable, I would like to be able to\ninfer the model without ever integrating\nmy model forward in time.\nSo, these are the two directions that I\nwill present today.\nLet's um for both directions, the key\nelement that allow\nessentially\nthose two directions\nis the score function. So, essentially,\nI will show how from the knowledge of\nthe score function,\nwe are able to\nbuild those two modeling strategies. So,\nthe score function is defined as the\ngradient of the logarithm of the steady\nstate distribution.\nUm\nUm so, the score function is one of the\ncentral So, it's a central quantity in\nmachine learning\nspecifically in score-based generative\nmodeling\nbecause essentially it's a quantity that\nallow\nuh\nthe generation of new data sample\naccording to a specific probability\ndensity without the knowledge of the\nnormalization constant that\nthat otherwise someone needed to to\nevaluate in order to to define a\nnormalized probability density function.\nSo, essentially, it's a way to sample\nnew data sample according to a specific\nprobability density without the\nknowledge of the steady state\ndistribution, which can be extremely\nchallenging to be constructed from high\ndimensional systems.\nIn the In this talk, I will So, there\nare of course many different methodology\nto estimate the score function from\ndata. In this talk, I will use the\ndenoising score matching method,\nwhich essentially consists in taking the\ndata set that I would like to use to\nestimate the score function, I perturb\nit by adding a tiny Gaussian white noise\nwith amplitude sigma,\nand then I will use the the denoising\nscore matching identity, according to\nwhich I can write the score function of\nthis perturbed probability density\nin fun So, in\nin terms of the expected value of zeta\nconditioned on the perturbed value x\nsigma. So, given So, I am at the end\nwhat I'm doing is to train a neural\nnetwork to predict\nthe value of the noise zeta\ngiven an input the perturbed value of\nthe data set point x sigma.\nUm so, if sigma So, by considering sigma\nSo, the perturbation amplitude very\nsmall, I'm essentially estimating the\nscore function of a perturbed density,\nwhich is extremely close\nto the observed one. And so, the score\nfunction that I derive using this method\nit will be\nvery close\nto the correct score function.\nThe advantage of this algo of this\nalgorithm is that it scales extremely\nwell with the dimension because\nessentially we are recasting\nthe score estimation problem, which will\notherwise imply the differentiation of\nthe logarithm of the steady state\ndensity,\nwith a regression problem.\nAnd this regression problem scales very\nwell with\nwith the dimension. This essentially\nmeans that we can estimate the score\nfunction\nquite efficiently also for very high\ndimensional systems. And then we'll use\nthe knowledge of the score function\nto infer the full mathematical model\nthat explain that is able to reproduce\nthis set of target observables that I\nmentioned\nat the beginning.\nSo, let's start from the first modeling\nstrategy.\nSo, as I said,\nI'm assuming to have a model answers, so\nto have an answers for the functional\nform of my Langevin equation with\nmultiplicative noise.\nof parameters alpha and beta.\nWe call alpha all the parameters inside\nthe drift term, and beta all the\nparameters inside the diffusion term.\nSo, the problem consists now in finding\nthe values for alpha and beta\nthat reproduces the target statistical\nobservables.\nSo, learning\nSo,\nin order to solve this calibration\nproblem, what we need is the parameter\nsensitivity. So, if we call phi m the\nset of observables that we want our\nmodel to reproduce,\nwhat we need to know is how\nthose observable phi of m will change by\nchanging by a tiny amount each of the\nmodel parameters.\nSo, if we know these these quantities,\nwe can essentially update\nthe parameters accordingly in order to\nminimize the distance between the target\nobservables and the observables\npredicted by the model.\nWe can estimate the statistical\nJacobians\nusing a naive approach by just\nintegrating the model forward many\ntimes. Every time we perturb just one\nparameter by a tiny amount,\nand then using finite difference, we\nestimate this statistical Jacobians.\nThis method of course works, but it has\nthe problem that it requires an\nextremely large number of model\nintegrations. So, we need at least\none model integration for every\nparameter. And if the model is very\nlarge,\nit becomes extremely computationally\nexpensive to integrate, in particular,\nif we have many parameters that we want\nto calibrate.\nSo, the\nthe key idea here is to recast this\ncalibration problem\nto a perturbation problem. So, if we\nimagine to Taylor expand at first order\nthe drift and and the diffusion\ncoefficients after we perturb by a tiny\namount each of those parameters,\nso essentially, what we obtain is a\nperturbed version of this Langevin\nequation, where we have an additional\nterm in the drift and in the diffusion\ncoefficients, which is given by the\nperturbation amplitude of the parameter\nmultiplied by the partial derivatives of\nthe drift and the diffusion term with\nrespect to that specific parameter. So,\nessentially, it's like if we are\nstudying a perturbed version\nof our model, and we would like to\npredict how all those statistical\nobservables will change after we add\nthis perturbation.\nOkay. So, now we have this\nperturbation problem,\nwhich can be addressed using a very\npowerful\ntool from\nstatistical physics,\nwhich is the generalized\nfluctuation-dissipation theorem or GFT.\nSo, GFT\nis\na mathematical machinery that allows to\npredict how a dynamical system respond\nto a perturbation without actually\nperturbing it, just from the knowledge\nof its statistics.\nSo, without entering into the\nmathematical details of this problem, we\ncan essentially write the response of an\nobservable A\nto a perturbation given by\nuh U of so a drift perturbation given by\nU of X and a diffusion perturbation\ngiven by V of X.\nSo, those are nothing but\nthese perturbations\nthat we defined before.\nAnd we can write the response of this\nobservable A in terms of an integral of\nthe correlation between the observable\nitself and this conjugate variable B,\nwhich depends on the\nperturbation that we are applying. So, U\nof X\nand V of X.\nAnd importantly on the score function S.\nSo, essentially if we have the knowledge\nof the score function since\nwe know the analytic expression for our\nmodel. And also we know the analytic\nexpression for the perturbation that we\nare applying U and V.\nWe can predict how our system, so in\nthis case how the observable A will\nrespond to an external perturbation.\nSo, if we identify the observable A with\nthe set of observables that we want our\nmodel to reproduce.\nWe can essentially construct using the\ngeneralized fluctuation-dissipation\ntheorem the parameter sensitivities that\nwe are interested in.\nJust from the knowledge of the score\nfunction itself.\nSo, this is the main connection. So,\nusing GFDT we are able to estimate the\nparameter sensitivities\nwithout the running the model forward\nfor every parameter.\nWe just needed to run the model forward\none time to estimate the score function.\nAnd from this single model integration\nwe are able to estimate all the\nparameter sensitivities that we can use\nfor calibration.\nOkay, so as we have seen this entirely\nso this so the applicability of the\ntheorem to this specific problem relies\nonly on the estimation of the score\nSo, at this point you may ask okay, but\nwhy nobody like followed this direction?\nAnd the reason is that estimating the\nscore function for high-dimensional\nsystem has always been the main\nbottleneck to use this generalized\nfluctuation-dissipation theorem inside\nrealistic and practical problems.\nSo, usually people in the literature\nhave always struggled in estimating the\nscore function. And so, it was possible\nfor low-dimensional systems. Instead for\nhigh-dimensional systems,\nwhat has been usually done in the\nliterature was to use the so-called\nquasi-Gaussian approximations,\nwhich consists in approximating the\nsteady state distribution with a\nmultivariate Gaussian, which allows to\nwrite the score function in terms of a\nlinear function that depends on the\ncovariance matrix of the data set, which\nof course is extremely easy to be\nestimated.\nBut the main problem is that\nwhen the dynamics is highly nonlinear,\nso when the steady state distribution\nalso is highly\nnon-Gaussian,\nthen this approach introduces some\nstrong biases,\nwhich\ndoesn't allow to get quite precise\nprediction of the system responses using\nGFDT.\nBut we we have seen before how yeah,\nrecent advances in score estimation\nmethods using a neural network\nallow us to get a very precise and\nefficient estimation for the score\nfunction also for very high-dimensional\nsystems.\nAnd so, this knowledge allow us to\nconstruct and to estimate the system\nresponses using the generalized\nfluctuation-dissipation theorem.\nSo, we applied those ideas already\nto evaluate\nand to predict system responses for\nquite high-dimensional systems. We\nstarted PDEs discretized on around 10 at\nthe third grid points.\nAnd we considered more specifically\ntwo-dimensional turbulent data and\nAlan-Khan reaction-diffusion data. So,\nthese are the two papers where we\npublished this\nconnection between the score-based\ngenerative modeling and the generalized\nfluctuation-dissipation theorem. And so,\nnow the idea is to use this mathematical\nmachinery to evaluate the parameter\nsensitivities.\nAnd\nso, to do that I\nso, I described how we can do it. So,\nfrom the knowledge of the response\nfunction we can estimate the parameter\nsensitivities. Now, let's see how to do\nthat in practice.\nMore specifically, let's consider two\nexamples now. So, the first one is a\nvery low-dimensional model. We have a\nthree-dimensional SDE with a\nmultiplicative noise. So, this model is\nused in geophysical fluid dynamics to\ndescribe\nEl Niño-Southern Oscillation, which is a\ninterannual so, it's a annual\nvariability phenomenon of the sea\nsurface temperature in the tropical\nPacific. We have two slow variables, one\nfast variable, which are coupled.\nThis model depends on six coefficients.\nAnd so, what we are going to do is to\nstart so, is to first run this model\nwith the correct values of these\ncoefficients to have\nan obser so, to build our observations.\nThen\nwe will\ntry to recover the correct values of\nthose coefficients and starting from an\ninitial guess\nobtained by perturbing by around 20%\neach of those coefficients and running\nour calibration algorithm. So, using\nGFDT to estimate the\nand use\nthis knowledge inside a Newton algorithm\nto estimate the correct values of the\ncoefficient.\nThose are the six observables that we\nwould like to recover. So, we start with\na parameter guess.\nUm we use this parameter guess to\nintegrate the model forward.\nWe use this data set to estimate the\nparameter sensitivities. And then we are\nupdating the parameters using the\nknowledge of the parameter\nAnd we are iterating this procedure\nuntil\nthe statistics of our predicted system\nwill converge\nto the target statistical observables.\nWe are doing this procedure using three\ndifferent methods to evaluate the\nparameter sensitivities. We are first\nusing finite difference. So, we are just\nintegrating many time the trajectory\nforward one for each parameter. And then\nwe are using finite difference to\nestimate the parameter Jacobian.\nAnd we are using that information to\nupdate the parameter value.\nThen we are using the generalized\nfluctuation-dissipation theorem using\ntwo different ways to estimate the score\nWe first used the quasi-Gaussian\napproximation. So, we just\nwrote the score function in terms of a\ncovariance matrix of the data. And then\nwe are using the\ndenoising score matching approach to\nestimate the score function.\nAnd we will be comparing those three\napproaches.\nThese are the results.\nSo, here on the left you can see the L2\nnorm between predicted versus\ntarget statistical observables as a\nfunction of the\nalgorithm iteration. So, in this case\nwe introduced the breaking point when\nthe\nL2 norm\nwas falling below 10 at the minus three.\nAnd as you can see the blue curve and\nthe gray and the gray curve, which\nrepresent the calibration algorithm\nusing the denoising score matching score\nfunction plus GFDT and the naive\nand and the naive\nfinite difference estimation for the\nparameter Jacobians in just five\niteration we are falling below the\nthreshold.\nWhich essentially means in five\niteration we were able, as you can see\nhere in this panel showing the parameter\ndeviation, to precisely recover the\ncorrect parameters of the model.\nAnd instead using the Gaussian\napproximation for the score, so like a\nmore so, less precise estimation of the\nscore function was very difficult to\nhave the algorithm\ncon\nconverged\nto the correct value.\nBut so, here what we can see that using\nthe generalized fluctuation-dissipation\ntheorem plus the denoising score\nmatching to estimate this score\nfunction, we were able to have very\nsimilar performances\nwith respect to the naive\nfinite difference method at a fraction\nof the computational cost because\nwe\nSo, for every iteration, we needed to\nintegrate the system forward only one\ntime\ninstead of six time\nSo, the number of the parameters that we\nwant to calibrate or like in in this\ncase 12 because we use the center\ndifference for\nparameter for the\nuh um\nparameter Jacobian estimation. So,\nessentially here we have an algorithm\nthat's doesn't scale So, doesn't So, the\nfor which the computational time doesn't\nscale\nlinearly with the number of parameters,\nbut is constant since we only need to\nrun the model forward one\none single time for every iteration.\nOkay. So, in this case we considered a\nquite a low dimensional system.\nNext, I will consider this coupled\nLorenz '96\nsystem, which is around a 400\ndimensional system. We have 36 slow mode\nand we have a for each slow mode we have\n10 fast modes. Plus, we also have some\nwhite noise in each of um\nthose\nvariables.\nAnd what we want to do now is to do\nsomething different. So, we would like\nto build a stochastic closure for the X\nSo, the slow variables. So, essentially\nwe would like to build a 36 dimensional\nmodel\ninstead of this model here that is\naround a 400 dimensional\nwhich is able to precisely recover\nthe target statistical observables\nevaluated from the high dimensional\nmodel. So, we have observations for X\nwhich have been generated integrating\nthis very high dimensional system and we\nwould like to build this reduced order\nmodel which is only 36 dimensional\nwith the correct values of alphas of the\nalpha coefficients and the sigma\ncoefficients such that they reproduce\nthis set of target statistical\nobservables which are the mean uh the\nvariance, skewness, excess kurtosis, and\nuh covariance C1.\nWe have in total five parameters that we\nwant to calibrate\n&gt;&gt; [gasps]\n&gt;&gt; uh on the\nuh on the\non these five different statistical\nAnd so, we used also in this case these\nthree different methods.\nWe have in orange GFDT plus Gaussian\nuh estimation of the score function. And\nthen in gray and in blue, we have\nthe finite difference method and then\nthe GFDT plus the noise score matching\nfor the score\nestimation. So, essentially in in this\ncase we can see like yeah, clear\nadvantage in using the\nthe noise score matching to build the\nscore function. And from the knowledge\nof the score function, also in this case\nwhich is quite high dimensional, we can\nobserve\nhow we get quite similar performance\nthan of using finite difference at a\nfraction of the computational cost. And\nso, essentially at at a fraction of the\nnumber of time that we have to integrate\nour model forward.\nOkay. So, this\nis the first direction. So, start from a\nmodel answers and use this combination\nbetween the generalized fluctuation\ndissipation theorem from non-equilibrium\nstatistical physics with the noise score\nmatching from generative modeling\nto to to estimate\nuh the parameter sensitivities\nwith a very limited number of model\nintegrations.\nNow, let's see the second direction.\nSo, in this case we don't have any model\nanswers for\nthe functional form of our mathematical\nmodel.\nWe have a set of statistical and\ndynamical observables that we want our\nmodel to reproduce, which in this case\nare the full steady state distribution\nand a set of correlation functions where\nphi m and phi n are\nSo, it's a set of observables of the the\nstate variable of the system.\nSo, given these constraints, we would\nlike to build a mathematical model that\nby construction reproduces those\nconstraints without integrating our\nmodel forward.\nAnd again, we will use the score\nfunction to do that.\nSpecifically, we will use two different\nscore functions in this case. We have\nthe plain score function that we've seen\nbefore\nand also we will use the conditional\nscore function. This essentially is the\ngradient\nof with respect to X0 of the logarithm\nof the conditional probability density\nfunction. So, the probability density\nfunction of X at time T conditioned on\nof X at time T conditioned on\nX0.\nThe conditional score function can be\nconstructed using the noise score\nmatching precisely as we did it for the\nplain score function. In fact, we can\nwrite the conditional score function in\nterms of the joint score function and\nthe plain score function.\nFor the joint score function, we just\ntake our data set. We use a delay\nembedding\nin order to build\na time series of X0 and XT.\nWe do it for different value of the time\ndelay and in this way we estimate the\nscore function the joint score function.\nAnd then we can combine it with us and\nusing the same the noise score matching\nmachinery we have seen before, we can\nestimate both\nthe conditional score and the score\nfunction from data.\nAnd as we have seen before, both these\nalgorithms scale quite well with the\ndimension of the system.\nSo, the idea here is then to use those\ntwo quantities\nwhere the first quantity essentially\ntakes into account the geometry of the\nsteady state distribution.\nInstead, the second quantity essentially\ntakes into account how the system\nrelaxes towards the steady state\ndistribution. So, it's carrying\ninformation also about the\nthe dynamics of the system and not only\nabout the statistics.\nSo, this is the intuition. So, try to\nuse those two quantities that can be\nevaluated quite well also for very high\ndimensional systems\nto build our\nstochastic modeling approach.\nSo, let's start from our\nLangevin equation. So, this is the same\nLangevin equation I wrote at the\nI wrote at the\nbeginning. Yeah, just have here a factor\nsquare root of two.\nAnd then let's first impose\nstationarity. So, we want for a given\nsigma X to find our drift term F\nsuch that by construction\nreproduces the steady state\ndistribution.\nAnd to do that, we can write the\nFokker-Planck equation relative to the\nLangevin [clears throat] equation,\nimpose the stationarity\nand we can show that without losing any\ngenerality\nwe can write\nthe drift term\nin this way. So, in terms of the score\nfunction that we defined before\nand the diffusion matrix. So, this\nsymmetric matrix D of X\nand another anti-symmetric matrix R of\nX.\nSo, this is a very general\nexpression. We're not doing any\napproximation here. We are We are just\nfinding the most general way to express\nthe drift for a given the diffusion in\nsuch a way that it reproduces the steady\nstate distribution by construction,\nwhich essentially means in such a way\nthat F So, this specific shape of F\nsolve the stationary Fokker-Planck\nequation.\nNow, we can So, we can see that we have\ntwo different tensors D of X and R of X.\nD of X is symmetric and represent the\ndiffusion tensor.\nInstead, R which is the anti-symmetric\npart can be interpreted as the term that\nbreaks the tail balance and that\nintroduces some rotational component to\nour system without changing the steady\nstate distribution. So, this can be\nrelated to an Helmholtz decomposition of\nthe drift term. So, we have a\na term a symmetric term which\nsatisfies the detail balance and give us\na system which is just a Brownian motion\ninside a potential. And then we have\nthis other circulatory term which\nintroduces some rotational component\nthat breaks detailed balance.\nOkay, so now by using this expression\nhere for the drift term, we are\nguaranteed\nto recover the steady state\ndistribution. So we\nachieved the first goal of our modeling\nstrategy which is to build a stochastic\nmodel that by construction reproduces\nthe observed the steady state\ndistribution of the data set. Now let's\ntry to impose also the second constraint\nwhich is\nwe want to reproduce also the the time\ncorrelations.\nSo we would like to\nreproduce this time correlations for a\nset of observables phi n.\nSo without going into the mathematical\ndetails of this derivation, we can show\nthat\nthe time derivative of this correlation\nfunction for this specific model, so for\nthis specific Langevin equation with\ndrift term given by this expression over\nhere,\ncan be written in this way.\nSo essentially we can relate the time\nderivative of this correlation function\nwith this\nexpression here which contains the two\nphi, so the two observable phi m and phi\nn, the conditional score function, and\nthe matrix, so the tensor m.\nAnd the tensor m is the only term here\nthat we don't know because we can\nestimate this quantity here from data.\nWe just evaluate\nthe correlation function and then we\nestimate the derivative.\nWe can estimate the conditional score.\nWe know the analytical expression for\nboth phi m and phi n because this is the\nlibraries of observable that we are\nconsidering.\nThe only term that we don't know is this\nmatrix\nm x of 0. So now let's see how we can\nderive this matrix m x of 0.\nSo first\nlet's do this decomposition. So let's\ndecompose m x in terms of a constant\nterm plus a fluctuation.\nThis fluctuation is so the average value\nover the stationary density of this\nfluctuation must be equal to 0. So\nessentially here we have\na yeah, a constant term plus\nzero mean fluctuation term delta m. So\nlet's use this expression here for m\ninside the this equation over there\nand we can then rewrite c dot in terms\nof two terms. This first term which\ndepends on phi\ndoes not depend on the conditional\nscore, depends only on the stationary\nscore.\n.\nOkay, so we have the first term here\nwhich is much easier to evaluate because\nyeah, we don't need to estimate the\nconditional score\nand it only depends on this constant\nmatrix phi and this\nplain score itself\nminus this additional so this additional\nterm which is nothing but this one over\nhere written in terms of delta m instead\nof m.\nSo the key idea here is that if we have\na library of observable which is rich\nenough such\nsuch that m of x is uniquely determined,\nthen we can estimate m m of x, so this\ntensor m of x which is the missing\nelement for so in our stochastic model\nby essentially using this relationship\nover here.\nSo using this relationship here\nand a library of observable which is\nrich enough, we can estimate both phi\nand delta m.\nSo let's see now how we can estimate phi\nfirst.\nSo to do that, let's consider\njust the coordinate observable. Okay, so\nwe have here in theory like a very large\nlibraries of observable. Now let's focus\non a few of them and few of them of x\nequal to x. So we're just considering\nthe observable coordinate.\nBy doing this replacement, the first\nterm becomes this expected value\nmultiplied by phi.\nNow if we consider t equal to 0,\nthen this\nexpected value becomes minus the\nidentity because of the Stein identity.\nThis term here becomes equal to 0\nbecause we have so you can just\nintegrate the conditional score term at\nt equal to 0 and you will get 0.\nSo this essentially means that if if you\nconsider the coordinate observable and t\nequal to 0, we are able to derive a\nrelationship for phi.\nSo we can essentially fix the average\nvalue of the matrix m\nand we can write it in terms of the\ncoordinate the time derivative of the\ncoordinate correlation at t equal to 0.\ncorrelation at t equal to 0.\nNow let's see what this implies. So we\nhave then\nfixed the phi. We have this additional\nterm, this correction term e of x.\nWhen we consider\nphi phi phi phi 1 of x equal to x, we\nthen have this coordinate term at the\nbeginning.\nWe can rewrite this expected value in\nterms of the gradient with respect to x\n0 of the expected value of x of t\nconditioned on x 0 multiplied by delta\nm.\nSo by just considering the coordinate\nobservable case, we can derive phi\nand then we can write this relationship\nfor the correction term. But at this\npoint we can notice that if\nm of x, so if the expected value of x of\nt conditioned on x 0 is approximately\naffine which essentially means if we can\nwrite the expected value of x t\nconditioned on x 0\nin terms of a linear function of x 0,\nthen when we take the gradient, we will\nget a constant term with respect to x\nand then by construction the average\nvalue of x of delta m is equal to 0\nwhich essentially means that if the\nconditional mean is approximately affine\nwhich essentially\nis is is the case if\nthe joint probability density function\nof x x t is a Gaussian,\nwe can then use so we can then replace\nour matrix m of x which is state\ndependent with just the matrix phi\nand we have a model that by construction\nreproduces both the temporal\ncorrelations and the steady state\nOkay, so if\nthis term so if m of t is linear in x 0\nwhich is often the case because if the\nconditional probability density so if\nthe joint probability density of x 0 and\nand x t can be approximated with a\nGaussian distribution, then m of t\ndepends linearly on x 0.\nSo if this term is negligible,\nthen we can\nreproduce the time correlations of the\nobserved data just using phi, so this\nconstant matrix phi that we can easily\ndetermine from the correlation function\ninstead of the state dependent matrix m.\nAnd then we have built a Langevin\nequation that by construction reproduces\nboth the steady state distribution and\nthe time correlations.\nIf we want instead to add more\nconstraint on the correlations, so we\nwant to add more constraints on the\ndynamics adding more correlations, then\nwe have to obtain\nalso the matrix delta m that we can\nparameterize with a neural network.\nSo in this specific case, we\nparameterize the whole m of x with a\nneural network and then we define delta\nm of x as m theta minus phi\nand then we can train a neural network\ndelta m of theta to minimize this loss\nfunction. So we have this first term\nthat that essentially forces the neural\nnetwork to learn the set of correlation\nfunctions that we want our system to\nreproduce. Then we have this penalty\nterm that essentially enforces that the\naverage value of delta m is equal to 0\nplus we have a regularization term.\nBut you can see here that we are so we\nare writing\na loss function that doesn't depend on a\nforward model integration. So we never\nhave to integrate our Langevin equation\nforward in time.\nWe just use the the knowledge of the\nconditional score, the score function,\nand the time derivative of the\ncorrelation functions to train the\nneural network for delta M.\nAnd this can be can become extremely\nefficient when the model that we want to\nintegrate becomes very computationally\nexpensive.\nOkay, so this is the methodology.\nSo we have seen that's\nyeah, we we are able to train this\nneural network without integrating the\nmodel forward and when so for some\nspecific cases we can simplify the shape\nso the functional form of M\nby replacing them with a constant if we\nare just interested in the time\ncorrelation of the systems. So now I\nwill conclude showing you some\napplication of these ideas.\nSo I start from an analytic warm up. So\nwe consider this one-dimensional system\nfor which we can determine analytically\nall the relevant quantities.\nSo we can derive the station the\nconditional score, the stationary score,\nthe time derivative of the correlation\nfunctions and so on.\nThese are the true values for fee and\ndelta M.\nAnd then by applying the method I\n&gt;&gt; [clears throat]\n&gt;&gt; discussed before we can obtain them\ndiscussed before we can obtain them\nusing the relationship that I showed you\nat the beginning. So using that\nrelationship we recover precisely\nthe fee the correct fee and the correct\nSo this was just like a test where we\nhave we know everything is analytically.\nSo let's see a different case. In this\ncase we have a two-dimensional system\nwith where we have our drift term which\ncontains both a term that can be written\nas the gradient of a potential plus a\ncirculatory component. We also have a\nAnd in this case we cannot write\nexplicitly the score function and the\nconditional score. So we need to train\ntwo neural networks for S and for the\nconditional score.\nWe apply the methodology that I\ndescribed\nbefore by enforcing the reproduction of\nthe correlation functions. We derive\na quite accurate\nreconstruction of\nthe mobility fields so the M\ntensor.\nWe have some errors in particular in\nthis term\nbut even if so we have like some errors\nfor\nthe M to one terms when we integrate our\nmodel we get a precise recovery of the\nunivariate PDF, bivariate PDF\nall the correlation functions. Here I'm\ncomparing two different model\nintegrations. We have the model\nintegration with the full\nwith the full\nmobility matrix M of X\nand a model integration where I'm\nreplacing the full mobility matrix with\na fee so with this constant closure that\nI introduced before.\nWe can see here that using the full\nmobility matrix obtained by training\na neural network for M we get a more\nprecise recovery of the correlation\nfunctions in particular for this cross\ncorrelation.\nAnd also if I now consider the target\ndynamical observables so the target\ncorrelation functions that I used to\ntrain the neural network when I evaluate\nthem from the trajectory that I obtained\nby integrating my model\nI get yeah a quite better\nrecovery with respect to the constant M\nmatrix closure.\nSo essentially this is to show that yeah\nby\napplying this algorithm\nwe are able to estimate the mobility\nmatrix M of X together with the score\nand the conditional score then combining\nthose pieces together\nwe obtain an expression for the drift\nterm that is able\nto reproduce the steady state density\nthe time correlations together with all\nthe correlations that we enforced\nin the training.\nOkay, so now let's consider more\nhigh-dimensional systems.\nSo for the next two systems I will only\nconsider the constant closure for M. So\nessentially I approximate M of X with\nits average value so with fee.\nIn this case I'm integrating this\nKuramoto-Sivashinsky PDE.\nI'm integrating this partial\ndifferential equation with 512 Fourier\nmodes. I obtain a 1024-dimensional\num time series. I'm\nconsidering just one\nmode every 32. So essentially I'm\nsubsampling this 1024-dimensional\nstate to a 32-dimensional state.\nAnd then using those\nthose 32-dimensional modes to build my\nLangevin equation. So essentially here\nI'm\nbuilding so I'm starting from a fully\nfully deterministic partial differential\nequation partial\nwhich is partially observed\nand then using a completely different\nmodel to\nbe so to predict its dynamics using my\nstochastic closure.\nAnd here are the results. So this is the\ntime series obtained by integrating my\nLangevin equation. This is the real\nobserved time series and here I'm\nplotting the comparison between the\nthe bivariate and the univariate PDFs\nobtained from the observations and the\none obtained from\na model integration of my Langevin\nequation. And here instead is the\nautocorrelation function for both the\nobservations and my Langevin\nintegration.\nThen finally I considered\nthe sea surface temperature data from\nPlasim so which is a um\na global circulation model of\nintermediate\nintermediate complexity.\ncomplexity.\nthe data for the sea surface\nfor the global sea surface temperature\nevolution and I want a model that is\nable essentially to predict and to model\nthis sea surface temperature data.\nSo the data set is around\n2000-dimensional.\nI did a dimensionality reduction taking\nthe first 20 principal components.\nAnd here since I have a strong\nperiodicity I augmented the state space\nby including some harmonic functions.\nAnd these are the results.\nI yeah was predicting the probability\nthe conditional probability density\nof the\n20 principal components together with\ntheir autocovariance. So\nas you can see we can have a quite\ndecent\nreconstruction of the PDFs and the ACFs\nof all the 20 principal components. Here\nI'm plotting just the first 10.\nAnd also we were able to capture the\nnonlinear so and the non-Gaussian\nprobability density function evaluated\nat every grid point\num from our simulation. So here\nessentially I'm plotting the probability\ndensity at different season\nof the temperature at a given grid point\nand I'm doing that using the full\nobservation so essentially all the\nprincipal components\njust the first 20 principal components\nand\nthe 20-dimensional\nstochastic model that I trained on these\nfirst 20 components and I integrated\nforward.\nAnd as you can see so even if I did a\ndimensionality reduction of the\ndata set I was still able to get this\nnonlinear probability density functions\nusing this\nquite simple\nstochastic model that I built using this\nconstant closure for my\nmobility matrix.\nOkay, so these are\nsome of the papers on\nthat I so either published or put on\narchive on this topic. We tried the\ndifferent directions that I haven't\npresented here.\nBut\nso these were so the main references\nand to conclude so we have seen\nhow to model high-dimensional\npartially observed chaotic systems\nhow the knowledge of the score function\nplays a key role in allowing this\nmodeling this modeling strategies. We\nhave seen two different directions. In\nthe first one we have a model answers\nand we are just\ncalibrating the model parameters using a\ncombination between the generalized\nfluctuation distribution theorem and the\nscore modeling.\nThe other direction instead\ndoesn't have any model answer.\nWe just try to\nstarting from a set of statistical and\ndynamical observables to build a model\nthat by construction reproduces all of\nthem without integrating the model\nAnd then we've seen how this approach\ncan scale on different systems from toy\nmodels to very high dimensional systems.\nOkay, thanks for listening and let me\nknow if you have\nany question.\nThank you, Ludovico.\nAny questions?\nI have a question. So,\nSo, I'm wondering according to your\nformulation, is does your method allow\nyou that\nhave allow you\nto to work on data set that has\nabsolutely no time information?\nWhat do you mean with absolutely no time\ninformation? So, like\ntime series uh\nwhere every snapshot is completely\nuncorrelated? Yeah, yeah. In that case,\nyes. So, so you can do it, but you will\nbe able to build a mathematical model\nthat reproduces the steady state\nbut not the dynamics because you don't\nhave any information about the dynamics.\nSo, what you can do in in that case\nand that will be yeah, much more simple,\nis to replace\nhere uh\nM of X just with the identity, right?\nSo, if you're only caring about the\nsteady state distribution and also you\ndon't have any information\nto build the\nso so\nto estimate the correlation functions,\nthen it essentially means that any So,\nyou cannot infer M of X because M of X\nis carrying information about the\ndynamics. So, you can replace M of X\nwith the identity.\nUh so so you So, if I train a model, I\nonly need to train the M, right?\nUh if you train So, if you only want to\nreproduce the steady state distribution\nbecause you don't have information about\nthe dynamics,\nyou just need to train a neural network\nto learn the score function.\nSo, M can be just replaced with the\nidentity.\nHm. Okay, because then so any value So,\nany shape of M of X will uh\nuh give you\nthe correct steady state distribution.\nSo, you can just choose M of X equal to\nthe identity.\nAlso, if you choose M of X equal to the\nidentity, it probably is an optimal\nchoice because uh\num you have the fastest convergence\ntowards the steady state density. So,\nlike if you integrate your model,\nquite fast towards the steady state\ndensity. So, if instead M of X is a\nconstant matrix uh\nwith a wide um\nvariety [clears throat]\nwide amplitude in the eigenvalues,\nyou have essentially that some modes\nwill decay faster than others and so you\nhave to wait like a longer time to see\nthermalization of the system towards the\nYeah, this is very interesting because\nwe we previously have a\nhave a paper that targeting exactly on\nno time information and we we got some\ndifficulty when we move from low\ndimension like two or three to to\nthousands of dimension. In thousands of\ndimension, our method basically uh\nalmost failed and uh\nso yeah, so I'm wondering\nif your method can can be helpful. Yeah,\nso we estimated the score function also\nfor thousand dimensional systems and\nyeah, like it's not a problem.\nBut which\nSo, how have you done this? So, did you\nuse the neural network to\nto estimate [clears throat] this the\nscore function? Yeah, yeah. We we\nbasically first train to get a score\nfunction, then we train a dynamic\nfunction. But that dynamic function is\n. But that dynamic function is\nalso a neural network. So, we we do we\ndo not\n&gt;&gt; But you don't need that because if you\njust care about the the steady So, a\nsystem that reproduces the steady state\nyou can just integrate this Langevin\nequation without\nany\nother network. So, is this a cover full\nsolution or just a subset of solution?\nNo, this is a general solution. So, this\nexpression for f of x is a general\nsolution.\nSo, it's essentially is So, given this\nLangevin equation, if you ask what is\nthe most general expression for the\ndrift term in such a way that uh\nSo, it reproduces the observed steady\nstate distribution. So, essentially that\nsolve\nthe stationary Fokker-Planck equation,\nthen this is the most general\nexpression.\nInteresting. Yeah. But since So, here\nthe main point is to estimate So, it's\nto reproduce the dynamics. So, this is\nthe non-trivial part.\nIf you're just interested in the\nstatistics, then yeah, take M of X equal\nto the identity\nand that's it.\nOkay. Thank you very much. I I will\nwrite an email to you. Uh Yeah.\nBy the way,\nwhen some system have a source and sink\nand does your method can\ncan cover those situations?\nYeah, so if\nSo, there is no time modulation\nof them. So, so essentially if you can\ndefine\na steady state distribution,\nthen yes.\nBut yeah, I haven't tested them that\nmuch. So, Thank you. I showed you the\nsystem that Yeah, I tested the\nalgorithm.",
+  "fetched_at": "2026-06-21T23:17:33Z",
+  "source": "yt-dlp-vtt",
+  "clean_segments": [
+    {
+      "start": 0.0,
+      "text": "for inviting me at your"
+    },
+    {
+      "start": 2.4,
+      "text": "group meeting. Um"
+    },
+    {
+      "start": 3.96,
+      "text": "So, in today's presentation, uh I will"
+    },
+    {
+      "start": 7.4,
+      "text": "talk about building mathematical models"
+    },
+    {
+      "start": 11.56,
+      "text": "from high high-dimensional partially"
+    },
+    {
+      "start": 14.36,
+      "text": "observed"
+    },
+    {
+      "start": 15.8,
+      "text": "dynamical systems."
+    },
+    {
+      "start": 17.6,
+      "text": "So, I think this is a central topic in"
+    },
+    {
+      "start": 20.32,
+      "text": "many scientific fields ranging from"
+    },
+    {
+      "start": 22.96,
+      "text": "geophysical fluid dynamics, which is the"
+    },
+    {
+      "start": 25.0,
+      "text": "field that is the closest uh"
+    },
+    {
+      "start": 27.24,
+      "text": "to my research, but also other physical"
+    },
+    {
+      "start": 30.44,
+      "text": "systems like molecular systems or"
+    },
+    {
+      "start": 33.32,
+      "text": "neuroscience. So, so every times we have"
+    },
+    {
+      "start": 36.4,
+      "text": "access to data, so we have observations"
+    },
+    {
+      "start": 39.88,
+      "text": "of the underlying system, and we want to"
+    },
+    {
+      "start": 42.92,
+      "text": "build a mathematical model that is able"
+    },
+    {
+      "start": 45.92,
+      "text": "to reproduce and capture the main"
+    },
+    {
+      "start": 48.84,
+      "text": "causality relationships between the"
+    },
+    {
+      "start": 50.72,
+      "text": "variable of the system, and also"
+    },
+    {
+      "start": 53.08,
+      "text": "hopefully to perform predictions and"
+    },
+    {
+      "start": 55.96,
+      "text": "that uncertainty quantification."
+    },
+    {
+      "start": 59.8,
+      "text": "So, of course for this specific class of"
+    },
+    {
+      "start": 61.92,
+      "text": "system"
+    },
+    {
+      "start": 63.24,
+      "text": "that are extremely high-dimensional and"
+    },
+    {
+      "start": 65.36,
+      "text": "also and also multiscale and partially"
+    },
+    {
+      "start": 67.68,
+      "text": "observed, uh"
+    },
+    {
+      "start": 69.56,
+      "text": "it is quite meaningless to try to"
+    },
+    {
+      "start": 72.64,
+      "text": "perform trajectory shadowing. So, it's"
+    },
+    {
+      "start": 75.76,
+      "text": "quite meaningless to try to build a"
+    },
+    {
+      "start": 77.84,
+      "text": "model that is able to precisely"
+    },
+    {
+      "start": 80.92,
+      "text": "predict the trajectory of the system."
+    },
+    {
+      "start": 83.88,
+      "text": "Instead, what we are usually So, what"
+    },
+    {
+      "start": 86.44,
+      "text": "our goal becomes usually is to"
+    },
+    {
+      "start": 93.08,
+      "text": "to reproduce some key statistical and"
+    },
+    {
+      "start": 97.08,
+      "text": "dynamical observable of the"
+    },
+    {
+      "start": 100.32,
+      "text": "the data set that we are observing."
+    },
+    {
+      "start": 103.6,
+      "text": "Those observables can be, for example,"
+    },
+    {
+      "start": 106.2,
+      "text": "some moments of the steady state"
+    },
+    {
+      "start": 108.88,
+      "text": "distribution, the whole steady state"
+    },
+    {
+      "start": 111.2,
+      "text": "distribution,"
+    },
+    {
+      "start": 112.84,
+      "text": "and also some"
+    },
+    {
+      "start": 115.0,
+      "text": "uh state or temporal correlations"
+    },
+    {
+      "start": 117.76,
+      "text": "between the"
+    },
+    {
+      "start": 119.32,
+      "text": "observed variables of the system."
+    },
+    {
+      "start": 122.36,
+      "text": "So, this is the goal of this talk. So,"
+    },
+    {
+      "start": 126.52,
+      "text": "develop a mathematical model that is"
+    },
+    {
+      "start": 128.28,
+      "text": "able uh"
+    },
+    {
+      "start": 130.16,
+      "text": "uh to reproduce this target statistical"
+    },
+    {
+      "start": 133.68,
+      "text": "and dynamical observables."
+    },
+    {
+      "start": 135.92,
+      "text": "So, because of this timescale separation"
+    },
+    {
+      "start": 139.8,
+      "text": "of the data, so the fact that the data"
+    },
+    {
+      "start": 142.28,
+      "text": "can be multiscale, I will use"
+    },
+    {
+      "start": 146.2,
+      "text": "a stochastic model where we have a first"
+    },
+    {
+      "start": 149.2,
+      "text": "component, the drift term f of x, which"
+    },
+    {
+      "start": 153.16,
+      "text": "uh will model the deterministic and slow"
+    },
+    {
+      "start": 156.52,
+      "text": "variable component of my data set."
+    },
+    {
+      "start": 159.76,
+      "text": "And I"
+    },
+    {
+      "start": 160.52,
+      "text": "And I also have a stochastic component"
+    },
+    {
+      "start": 162.59,
+      "text": "also have a stochastic component"
+    },
+    {
+      "start": 162.6,
+      "text": "that takes into account the faster"
+    },
+    {
+      "start": 164.88,
+      "text": "dynamics,"
+    },
+    {
+      "start": 166.16,
+      "text": "the unresolved fast fluctuations of the"
+    },
+    {
+      "start": 168.96,
+      "text": "data set."
+    },
+    {
+      "start": 170.44,
+      "text": "So, this will be the model"
+    },
+    {
+      "start": 172.84,
+      "text": "that I will that I want to construct"
+    },
+    {
+      "start": 175.2,
+      "text": "from the the observations"
+    },
+    {
+      "start": 177.64,
+      "text": "with the"
+    },
+    {
+      "start": 179.2,
+      "text": "final goal"
+    },
+    {
+      "start": 180.56,
+      "text": "of being able to reproduce"
+    },
+    {
+      "start": 183.92,
+      "text": "in"
+    },
+    {
+      "start": 186.0,
+      "text": "an efficient way this target statistical"
+    },
+    {
+      "start": 191.64,
+      "text": "Okay, so right now I have been extremely"
+    },
+    {
+      "start": 194.12,
+      "text": "general. So, I have"
+    },
+    {
+      "start": 196.36,
+      "text": "talked about a general model to solve"
+    },
+    {
+      "start": 198.72,
+      "text": "this very general problem."
+    },
+    {
+      "start": 200.92,
+      "text": "In the next two slides, I want to give a"
+    },
+    {
+      "start": 203.68,
+      "text": "bit more details about the first what"
+    },
+    {
+      "start": 206.6,
+      "text": "are the requirements that we want from"
+    },
+    {
+      "start": 209.32,
+      "text": "our model,"
+    },
+    {
+      "start": 210.6,
+      "text": "and then also what are the assumptions"
+    },
+    {
+      "start": 212.64,
+      "text": "that we're doing on the observed data"
+    },
+    {
+      "start": 214.64,
+      "text": "set."
+    },
+    {
+      "start": 217.56,
+      "text": "So, first"
+    },
+    {
+      "start": 219.2,
+      "text": "let's see what are the requirement that"
+    },
+    {
+      "start": 222.28,
+      "text": "we want"
+    },
+    {
+      "start": 223.56,
+      "text": "from our modeling strategy."
+    },
+    {
+      "start": 225.88,
+      "text": "So, as I said, we would like to model"
+    },
+    {
+      "start": 228.4,
+      "text": "real data that can be extremely"
+    },
+    {
+      "start": 230.92,
+      "text": "high-dimensional"
+    },
+    {
+      "start": 233.16,
+      "text": "and also maybe also unevenly sampled or"
+    },
+    {
+      "start": 237.44,
+      "text": "have"
+    },
+    {
+      "start": 238.48,
+      "text": "a very low sampling frequency. So, these"
+    },
+    {
+      "start": 240.96,
+      "text": "are all"
+    },
+    {
+      "start": 242.44,
+      "text": "features of our real data set that we"
+    },
+    {
+      "start": 245.28,
+      "text": "have to take into account when we are"
+    },
+    {
+      "start": 247.76,
+      "text": "building a mathematical model for it."
+    },
+    {
+      "start": 250.64,
+      "text": "So, this essentially implies first"
+    },
+    {
+      "start": 253.68,
+      "text": "that since"
+    },
+    {
+      "start": 255.32,
+      "text": "we want our modeling strategy to scale"
+    },
+    {
+      "start": 257.799,
+      "text": "very well with the dimension,"
+    },
+    {
+      "start": 259.92,
+      "text": "and since also the model that we want to"
+    },
+    {
+      "start": 262.68,
+      "text": "develop can be extremely"
+    },
+    {
+      "start": 264.12,
+      "text": "high-dimensional and"
+    },
+    {
+      "start": 265.92,
+      "text": "computationally expensive to integrate,"
+    },
+    {
+      "start": 268.84,
+      "text": "we would like to be able to reconstruct"
+    },
+    {
+      "start": 271.88,
+      "text": "the mathematical model from data using"
+    },
+    {
+      "start": 274.8,
+      "text": "as few model integrations"
+    },
+    {
+      "start": 276.8,
+      "text": "as possible."
+    },
+    {
+      "start": 278.2,
+      "text": "So, a naive approach to build a"
+    },
+    {
+      "start": 280.08,
+      "text": "mathematical model can be to start with"
+    },
+    {
+      "start": 282.88,
+      "text": "a model ansatz, integrate it forward for"
+    },
+    {
+      "start": 285.76,
+      "text": "a given time,"
+    },
+    {
+      "start": 287.6,
+      "text": "then calculate"
+    },
+    {
+      "start": 289.84,
+      "text": "all the statistical and dynamical"
+    },
+    {
+      "start": 291.64,
+      "text": "observables, compare them with the"
+    },
+    {
+      "start": 293.64,
+      "text": "targets,"
+    },
+    {
+      "start": 295.0,
+      "text": "evaluate a loss function, and use that"
+    },
+    {
+      "start": 297.44,
+      "text": "to update the model. But, this will"
+    },
+    {
+      "start": 299.72,
+      "text": "require many model integrations that can"
+    },
+    {
+      "start": 302.12,
+      "text": "become extremely computationally"
+    },
+    {
+      "start": 303.52,
+      "text": "expensive. So, our modeling strategy"
+    },
+    {
+      "start": 307.76,
+      "text": "So, one of the main goal of our modeling"
+    },
+    {
+      "start": 310.24,
+      "text": "strategy would be to use as few model"
+    },
+    {
+      "start": 314.0,
+      "text": "integrations as possible."
+    },
+    {
+      "start": 317.12,
+      "text": "Another requirement that we would that"
+    },
+    {
+      "start": 321.32,
+      "text": "our modeling strategy to have is to"
+    },
+    {
+      "start": 323.92,
+      "text": "avoid the state space clustering."
+    },
+    {
+      "start": 326.64,
+      "text": "So, we would like to avoid clustering"
+    },
+    {
+      "start": 329.84,
+      "text": "the state space to estimate the average"
+    },
+    {
+      "start": 332.44,
+      "text": "velocity field"
+    },
+    {
+      "start": 334.28,
+      "text": "because essentially"
+    },
+    {
+      "start": 335.96,
+      "text": "So, clustering the state space, even if"
+    },
+    {
+      "start": 338.8,
+      "text": "we are using an extremely efficient"
+    },
+    {
+      "start": 340.96,
+      "text": "clustering algorithm like, I don't know,"
+    },
+    {
+      "start": 343.32,
+      "text": "bisecting k-means clustering algorithm,"
+    },
+    {
+      "start": 347.04,
+      "text": "suffer from the curse of dimensionality."
+    },
+    {
+      "start": 349.48,
+      "text": "So, if we are considering a data set or"
+    },
+    {
+      "start": 352.32,
+      "text": "a system with dimension larger than a"
+    },
+    {
+      "start": 354.76,
+      "text": "few dozens, clustering becomes extremely"
+    },
+    {
+      "start": 357.32,
+      "text": "computationally expensive, so we would"
+    },
+    {
+      "start": 359.08,
+      "text": "like to avoid clustering."
+    },
+    {
+      "start": 361.2,
+      "text": "And also we would like to use a finite"
+    },
+    {
+      "start": 363.96,
+      "text": "difference estimation to estimate the"
+    },
+    {
+      "start": 366.44,
+      "text": "velocity from trajectories. So, we are"
+    },
+    {
+      "start": 369.28,
+      "text": "assuming that our realistic data set can"
+    },
+    {
+      "start": 373.2,
+      "text": "be can have a low frequen- a low"
+    },
+    {
+      "start": 375.48,
+      "text": "sampling frequency, which essentially"
+    },
+    {
+      "start": 377.8,
+      "text": "means that we cannot trust the data set"
+    },
+    {
+      "start": 381.16,
+      "text": "to recover"
+    },
+    {
+      "start": 383.2,
+      "text": "the velocity"
+    },
+    {
+      "start": 385.56,
+      "text": "of our dynamics using finite difference."
+    },
+    {
+      "start": 388.8,
+      "text": "So, these are the"
+    },
+    {
+      "start": 390.8,
+      "text": "three main"
+    },
+    {
+      "start": 392.48,
+      "text": "um"
+    },
+    {
+      "start": 394.0,
+      "text": "constraints that we want our modeling"
+    },
+    {
+      "start": 396.4,
+      "text": "strategy to have in order to be scalable"
+    },
+    {
+      "start": 399.56,
+      "text": "to high-dimensional system and also"
+    },
+    {
+      "start": 403.56,
+      "text": "being used to realistic settings."
+    },
+    {
+      "start": 406.92,
+      "text": "Then, let's see what are instead the"
+    },
+    {
+      "start": 408.96,
+      "text": "constraints"
+    },
+    {
+      "start": 410.32,
+      "text": "that we want to impose on our system."
+    },
+    {
+      "start": 412.84,
+      "text": "So, on which kind of systems this So,"
+    },
+    {
+      "start": 415.84,
+      "text": "the model the modeling strategy that I"
+    },
+    {
+      "start": 417.8,
+      "text": "will propose in this talk can be"
+    },
+    {
+      "start": 420.24,
+      "text": "applied."
+    },
+    {
+      "start": 421.68,
+      "text": "So, first of all, I will assume that the"
+    },
+    {
+      "start": 424.16,
+      "text": "system I'm studying has statistical"
+    },
+    {
+      "start": 427.28,
+      "text": "stationarity. So, essentially that we"
+    },
+    {
+      "start": 429.04,
+      "text": "can define a steady state distribution."
+    },
+    {
+      "start": 433.32,
+      "text": "So, of course this can be generalized"
+    },
+    {
+      "start": 435.52,
+      "text": "also to cyclostationary data by"
+    },
+    {
+      "start": 438.64,
+      "text": "augmenting the state space or, for"
+    },
+    {
+      "start": 441.24,
+      "text": "example, to"
+    },
+    {
+      "start": 444.04,
+      "text": "data set that show a low trends that can"
+    },
+    {
+      "start": 447.84,
+      "text": "be detrended and then we can get a"
+    },
+    {
+      "start": 450.4,
+      "text": "stationary"
+    },
+    {
+      "start": 454.68,
+      "text": "I will also assume that the system has"
+    },
+    {
+      "start": 459.16,
+      "text": "um so, is ergodic and mixing, which"
+    },
+    {
+      "start": 461.76,
+      "text": "essentially means that time averages"
+    },
+    {
+      "start": 465.52,
+      "text": "along"
+    },
+    {
+      "start": 466.8,
+      "text": "long trajectory"
+    },
+    {
+      "start": 468.92,
+      "text": "will converge to ensemble averages, and"
+    },
+    {
+      "start": 471.76,
+      "text": "also that"
+    },
+    {
+      "start": 473.76,
+      "text": "the correlation function decays"
+    },
+    {
+      "start": 476.08,
+      "text": "sufficiently fast."
+    },
+    {
+      "start": 478.2,
+      "text": "And then I'm also assuming that we have"
+    },
+    {
+      "start": 482.56,
+      "text": "an effective timescale separation. So,"
+    },
+    {
+      "start": 485.2,
+      "text": "this essentially allow us to model the"
+    },
+    {
+      "start": 488.6,
+      "text": "data set using"
+    },
+    {
+      "start": 490.76,
+      "text": "the Langevin equation that I showed you"
+    },
+    {
+      "start": 492.76,
+      "text": "before. So, essentially treat the fast"
+    },
+    {
+      "start": 494.88,
+      "text": "timescale as a noise process, and then"
+    },
+    {
+      "start": 498.36,
+      "text": "build a drift term for the slow"
+    },
+    {
+      "start": 500.92,
+      "text": "timescales."
+    },
+    {
+      "start": 502.56,
+      "text": "I will also assume to have enough"
+    },
+    {
+      "start": 504.76,
+      "text": "observables of the slow timescales,"
+    },
+    {
+      "start": 507.96,
+      "text": "and so in such a way I can get"
+    },
+    {
+      "start": 511.96,
+      "text": "a full recovery of the slow variables."
+    },
+    {
+      "start": 515.96,
+      "text": "Even if we don't have a complete"
+    },
+    {
+      "start": 517.52,
+      "text": "observables for the slow variables, we"
+    },
+    {
+      "start": 519.8,
+      "text": "can still augment the state space using"
+    },
+    {
+      "start": 523.479,
+      "text": "some delay embedded version of the data"
+    },
+    {
+      "start": 527.84,
+      "text": "And then finally, even if"
+    },
+    {
+      "start": 531.16,
+      "text": "uh we will not have probably some very"
+    },
+    {
+      "start": 535.16,
+      "text": "fine resolution"
+    },
+    {
+      "start": 537.24,
+      "text": "of the data sample,"
+    },
+    {
+      "start": 539.32,
+      "text": "we still assume to have enough"
+    },
+    {
+      "start": 541.96,
+      "text": "resolution to be able to define"
+    },
+    {
+      "start": 544.04,
+      "text": "correlation functions. So, we don't have"
+    },
+    {
+      "start": 546.04,
+      "text": "enough resolution to"
+    },
+    {
+      "start": 548.68,
+      "text": "evaluate the velocity at every"
+    },
+    {
+      "start": 552.52,
+      "text": "time point by using finite difference,"
+    },
+    {
+      "start": 555.28,
+      "text": "but still we have enough data"
+    },
+    {
+      "start": 556.92,
+      "text": "essentially to define a correlation"
+    },
+    {
+      "start": 559.16,
+      "text": "function."
+    },
+    {
+      "start": 560.24,
+      "text": "So, these are the goals that I would"
+    },
+    {
+      "start": 562.12,
+      "text": "like"
+    },
+    {
+      "start": 563.36,
+      "text": "my modeling strategy to achieve,"
+    },
+    {
+      "start": 566.44,
+      "text": "and also those are the main assumptions"
+    },
+    {
+      "start": 569.4,
+      "text": "that I'm doing on the"
+    },
+    {
+      "start": 571.84,
+      "text": "system that I'm studying."
+    },
+    {
+      "start": 574.36,
+      "text": "So, in this talk, I will present two"
+    },
+    {
+      "start": 576.64,
+      "text": "different directions. So, two different"
+    },
+    {
+      "start": 578.88,
+      "text": "modeling strategies that I'm currently"
+    },
+    {
+      "start": 581.88,
+      "text": "pursuing."
+    },
+    {
+      "start": 583.2,
+      "text": "The first one So, in the first one, I'm"
+    },
+    {
+      "start": 585.28,
+      "text": "assuming"
+    },
+    {
+      "start": 586.52,
+      "text": "to have a model ansatz. So, essentially"
+    },
+    {
+      "start": 590.08,
+      "text": "I have"
+    },
+    {
+      "start": 591.8,
+      "text": "a knowledge of the functional form of my"
+    },
+    {
+      "start": 594.52,
+      "text": "model, so a knowledge that be derived"
+    },
+    {
+      "start": 597.08,
+      "text": "directly from physics."
+    },
+    {
+      "start": 599.68,
+      "text": "And this model answers depends on a set"
+    },
+    {
+      "start": 602.6,
+      "text": "of parameters that I would like to"
+    },
+    {
+      "start": 604.72,
+      "text": "determine."
+    },
+    {
+      "start": 605.84,
+      "text": "I have an initial guess for those"
+    },
+    {
+      "start": 608.28,
+      "text": "parameters, and I would like to"
+    },
+    {
+      "start": 610.56,
+      "text": "calibrate those model parameters in"
+    },
+    {
+      "start": 613.16,
+      "text": "order to reproduce a set of statistical"
+    },
+    {
+      "start": 615.72,
+      "text": "observables"
+    },
+    {
+      "start": 617.28,
+      "text": "that I derive from my observations."
+    },
+    {
+      "start": 620.84,
+      "text": "The second direction instead is a bit"
+    },
+    {
+      "start": 623.2,
+      "text": "more ambitious."
+    },
+    {
+      "start": 624.72,
+      "text": "I assume not to know"
+    },
+    {
+      "start": 628.12,
+      "text": "any model answers. So, I don't start"
+    },
+    {
+      "start": 630.44,
+      "text": "from"
+    },
+    {
+      "start": 631.32,
+      "text": "model answers assumption, and I want to"
+    },
+    {
+      "start": 634.28,
+      "text": "derive from my data"
+    },
+    {
+      "start": 636.88,
+      "text": "the whole functional form of the model."
+    },
+    {
+      "start": 640.32,
+      "text": "In in this case, what I'm interested to"
+    },
+    {
+      "start": 643.08,
+      "text": "do precisely is to start from the set of"
+    },
+    {
+      "start": 647.88,
+      "text": "statistical and dynamical target"
+    },
+    {
+      "start": 651.0,
+      "text": "observables,"
+    },
+    {
+      "start": 652.4,
+      "text": "and I would like to infer from them"
+    },
+    {
+      "start": 655.56,
+      "text": "the most general class of dynamical"
+    },
+    {
+      "start": 658.8,
+      "text": "systems that by construction reproduces"
+    },
+    {
+      "start": 662.44,
+      "text": "those dynamical observables."
+    },
+    {
+      "start": 665.64,
+      "text": "And I would like to do it. So, this is"
+    },
+    {
+      "start": 667.88,
+      "text": "the most ambitious part of the"
+    },
+    {
+      "start": 671.64,
+      "text": "of the method because I would like to do"
+    },
+    {
+      "start": 673.52,
+      "text": "it without ever integrating my model"
+    },
+    {
+      "start": 676.6,
+      "text": "forward. So, just from the knowledge of"
+    },
+    {
+      "start": 679.48,
+      "text": "the target statistical and dynamical"
+    },
+    {
+      "start": 681.96,
+      "text": "observable, I would like to be able to"
+    },
+    {
+      "start": 684.48,
+      "text": "infer the model without ever integrating"
+    },
+    {
+      "start": 687.32,
+      "text": "my model forward in time."
+    },
+    {
+      "start": 689.36,
+      "text": "So, these are the two directions that I"
+    },
+    {
+      "start": 691.0,
+      "text": "will present today."
+    },
+    {
+      "start": 693.04,
+      "text": "Let's um for both directions, the key"
+    },
+    {
+      "start": 696.64,
+      "text": "element that allow"
+    },
+    {
+      "start": 699.24,
+      "text": "essentially"
+    },
+    {
+      "start": 701.08,
+      "text": "those two directions"
+    },
+    {
+      "start": 703.12,
+      "text": "is the score function. So, essentially,"
+    },
+    {
+      "start": 705.76,
+      "text": "I will show how from the knowledge of"
+    },
+    {
+      "start": 708.32,
+      "text": "the score function,"
+    },
+    {
+      "start": 709.92,
+      "text": "we are able to"
+    },
+    {
+      "start": 712.32,
+      "text": "build those two modeling strategies. So,"
+    },
+    {
+      "start": 714.96,
+      "text": "the score function is defined as the"
+    },
+    {
+      "start": 717.04,
+      "text": "gradient of the logarithm of the steady"
+    },
+    {
+      "start": 719.28,
+      "text": "state distribution."
+    },
+    {
+      "start": 721.68,
+      "text": "Um"
+    },
+    {
+      "start": 723.92,
+      "text": "Um so, the score function is one of the"
+    },
+    {
+      "start": 727.0,
+      "text": "central So, it's a central quantity in"
+    },
+    {
+      "start": 729.36,
+      "text": "machine learning"
+    },
+    {
+      "start": 731.08,
+      "text": "specifically in score-based generative"
+    },
+    {
+      "start": 733.64,
+      "text": "modeling"
+    },
+    {
+      "start": 734.92,
+      "text": "because essentially it's a quantity that"
+    },
+    {
+      "start": 737.12,
+      "text": "allow"
+    },
+    {
+      "start": 738.52,
+      "text": "uh"
+    },
+    {
+      "start": 739.16,
+      "text": "the generation of new data sample"
+    },
+    {
+      "start": 741.64,
+      "text": "according to a specific probability"
+    },
+    {
+      "start": 743.76,
+      "text": "density without the knowledge of the"
+    },
+    {
+      "start": 746.68,
+      "text": "normalization constant that"
+    },
+    {
+      "start": 748.88,
+      "text": "that otherwise someone needed to to"
+    },
+    {
+      "start": 751.72,
+      "text": "evaluate in order to to define a"
+    },
+    {
+      "start": 754.52,
+      "text": "normalized probability density function."
+    },
+    {
+      "start": 757.2,
+      "text": "So, essentially, it's a way to sample"
+    },
+    {
+      "start": 759.24,
+      "text": "new data sample according to a specific"
+    },
+    {
+      "start": 761.72,
+      "text": "probability density without the"
+    },
+    {
+      "start": 763.92,
+      "text": "knowledge of the steady state"
+    },
+    {
+      "start": 765.88,
+      "text": "distribution, which can be extremely"
+    },
+    {
+      "start": 767.68,
+      "text": "challenging to be constructed from high"
+    },
+    {
+      "start": 770.56,
+      "text": "dimensional systems."
+    },
+    {
+      "start": 773.96,
+      "text": "In the In this talk, I will So, there"
+    },
+    {
+      "start": 777.44,
+      "text": "are of course many different methodology"
+    },
+    {
+      "start": 779.76,
+      "text": "to estimate the score function from"
+    },
+    {
+      "start": 781.4,
+      "text": "data. In this talk, I will use the"
+    },
+    {
+      "start": 784.24,
+      "text": "denoising score matching method,"
+    },
+    {
+      "start": 786.8,
+      "text": "which essentially consists in taking the"
+    },
+    {
+      "start": 790.88,
+      "text": "data set that I would like to use to"
+    },
+    {
+      "start": 793.32,
+      "text": "estimate the score function, I perturb"
+    },
+    {
+      "start": 795.88,
+      "text": "it by adding a tiny Gaussian white noise"
+    },
+    {
+      "start": 800.12,
+      "text": "with amplitude sigma,"
+    },
+    {
+      "start": 802.4,
+      "text": "and then I will use the the denoising"
+    },
+    {
+      "start": 805.8,
+      "text": "score matching identity, according to"
+    },
+    {
+      "start": 807.96,
+      "text": "which I can write the score function of"
+    },
+    {
+      "start": 811.0,
+      "text": "this perturbed probability density"
+    },
+    {
+      "start": 814.16,
+      "text": "in fun So, in"
+    },
+    {
+      "start": 817.08,
+      "text": "in terms of the expected value of zeta"
+    },
+    {
+      "start": 822.839,
+      "text": "conditioned on the perturbed value x"
+    },
+    {
+      "start": 826.24,
+      "text": "sigma. So, given So, I am at the end"
+    },
+    {
+      "start": 830.16,
+      "text": "what I'm doing is to train a neural"
+    },
+    {
+      "start": 832.4,
+      "text": "network to predict"
+    },
+    {
+      "start": 835.56,
+      "text": "the value of the noise zeta"
+    },
+    {
+      "start": 839.24,
+      "text": "given an input the perturbed value of"
+    },
+    {
+      "start": 843.04,
+      "text": "the data set point x sigma."
+    },
+    {
+      "start": 846.88,
+      "text": "Um so, if sigma So, by considering sigma"
+    },
+    {
+      "start": 850.6,
+      "text": "So, the perturbation amplitude very"
+    },
+    {
+      "start": 853.88,
+      "text": "small, I'm essentially estimating the"
+    },
+    {
+      "start": 857.16,
+      "text": "score function of a perturbed density,"
+    },
+    {
+      "start": 859.52,
+      "text": "which is extremely close"
+    },
+    {
+      "start": 861.6,
+      "text": "to the observed one. And so, the score"
+    },
+    {
+      "start": 864.76,
+      "text": "function that I derive using this method"
+    },
+    {
+      "start": 868.64,
+      "text": "it will be"
+    },
+    {
+      "start": 870.2,
+      "text": "very close"
+    },
+    {
+      "start": 871.6,
+      "text": "to the correct score function."
+    },
+    {
+      "start": 875.0,
+      "text": "The advantage of this algo of this"
+    },
+    {
+      "start": 877.4,
+      "text": "algorithm is that it scales extremely"
+    },
+    {
+      "start": 880.36,
+      "text": "well with the dimension because"
+    },
+    {
+      "start": 881.959,
+      "text": "essentially we are recasting"
+    },
+    {
+      "start": 884.2,
+      "text": "the score estimation problem, which will"
+    },
+    {
+      "start": 887.36,
+      "text": "otherwise imply the differentiation of"
+    },
+    {
+      "start": 890.0,
+      "text": "the logarithm of the steady state"
+    },
+    {
+      "start": 891.52,
+      "text": "density,"
+    },
+    {
+      "start": 893.04,
+      "text": "with a regression problem."
+    },
+    {
+      "start": 895.72,
+      "text": "And this regression problem scales very"
+    },
+    {
+      "start": 898.04,
+      "text": "well with"
+    },
+    {
+      "start": 898.88,
+      "text": "with the dimension. This essentially"
+    },
+    {
+      "start": 900.92,
+      "text": "means that we can estimate the score"
+    },
+    {
+      "start": 903.36,
+      "text": "function"
+    },
+    {
+      "start": 905.12,
+      "text": "quite efficiently also for very high"
+    },
+    {
+      "start": 907.8,
+      "text": "dimensional systems. And then we'll use"
+    },
+    {
+      "start": 910.52,
+      "text": "the knowledge of the score function"
+    },
+    {
+      "start": 913.56,
+      "text": "to infer the full mathematical model"
+    },
+    {
+      "start": 917.52,
+      "text": "that explain that is able to reproduce"
+    },
+    {
+      "start": 921.079,
+      "text": "this set of target observables that I"
+    },
+    {
+      "start": 923.4,
+      "text": "mentioned"
+    },
+    {
+      "start": 924.56,
+      "text": "at the beginning."
+    },
+    {
+      "start": 926.28,
+      "text": "So, let's start from the first modeling"
+    },
+    {
+      "start": 929.36,
+      "text": "strategy."
+    },
+    {
+      "start": 933.0,
+      "text": "So, as I said,"
+    },
+    {
+      "start": 934.92,
+      "text": "I'm assuming to have a model answers, so"
+    },
+    {
+      "start": 938.839,
+      "text": "to have an answers for the functional"
+    },
+    {
+      "start": 940.76,
+      "text": "form of my Langevin equation with"
+    },
+    {
+      "start": 944.56,
+      "text": "multiplicative noise."
+    },
+    {
+      "start": 949.079,
+      "text": "of parameters alpha and beta."
+    },
+    {
+      "start": 952.52,
+      "text": "We call alpha all the parameters inside"
+    },
+    {
+      "start": 956.04,
+      "text": "the drift term, and beta all the"
+    },
+    {
+      "start": 957.72,
+      "text": "parameters inside the diffusion term."
+    },
+    {
+      "start": 960.92,
+      "text": "So, the problem consists now in finding"
+    },
+    {
+      "start": 963.839,
+      "text": "the values for alpha and beta"
+    },
+    {
+      "start": 966.2,
+      "text": "that reproduces the target statistical"
+    },
+    {
+      "start": 968.92,
+      "text": "observables."
+    },
+    {
+      "start": 971.56,
+      "text": "So, learning"
+    },
+    {
+      "start": 973.24,
+      "text": "So,"
+    },
+    {
+      "start": 975.92,
+      "text": "in order to solve this calibration"
+    },
+    {
+      "start": 978.52,
+      "text": "problem, what we need is the parameter"
+    },
+    {
+      "start": 981.44,
+      "text": "sensitivity. So, if we call phi m the"
+    },
+    {
+      "start": 985.16,
+      "text": "set of observables that we want our"
+    },
+    {
+      "start": 988.44,
+      "text": "model to reproduce,"
+    },
+    {
+      "start": 990.72,
+      "text": "what we need to know is how"
+    },
+    {
+      "start": 994.16,
+      "text": "those observable phi of m will change by"
+    },
+    {
+      "start": 999.32,
+      "text": "changing by a tiny amount each of the"
+    },
+    {
+      "start": 1002.64,
+      "text": "model parameters."
+    },
+    {
+      "start": 1004.44,
+      "text": "So, if we know these these quantities,"
+    },
+    {
+      "start": 1006.839,
+      "text": "we can essentially update"
+    },
+    {
+      "start": 1009.36,
+      "text": "the parameters accordingly in order to"
+    },
+    {
+      "start": 1012.6,
+      "text": "minimize the distance between the target"
+    },
+    {
+      "start": 1016.56,
+      "text": "observables and the observables"
+    },
+    {
+      "start": 1018.88,
+      "text": "predicted by the model."
+    },
+    {
+      "start": 1021.76,
+      "text": "We can estimate the statistical"
+    },
+    {
+      "start": 1023.56,
+      "text": "Jacobians"
+    },
+    {
+      "start": 1025.12,
+      "text": "using a naive approach by just"
+    },
+    {
+      "start": 1028.199,
+      "text": "integrating the model forward many"
+    },
+    {
+      "start": 1030.4,
+      "text": "times. Every time we perturb just one"
+    },
+    {
+      "start": 1034.48,
+      "text": "parameter by a tiny amount,"
+    },
+    {
+      "start": 1037.12,
+      "text": "and then using finite difference, we"
+    },
+    {
+      "start": 1039.0,
+      "text": "estimate this statistical Jacobians."
+    },
+    {
+      "start": 1043.199,
+      "text": "This method of course works, but it has"
+    },
+    {
+      "start": 1047.28,
+      "text": "the problem that it requires an"
+    },
+    {
+      "start": 1051.28,
+      "text": "extremely large number of model"
+    },
+    {
+      "start": 1053.16,
+      "text": "integrations. So, we need at least"
+    },
+    {
+      "start": 1056.64,
+      "text": "one model integration for every"
+    },
+    {
+      "start": 1058.44,
+      "text": "parameter. And if the model is very"
+    },
+    {
+      "start": 1061.12,
+      "text": "large,"
+    },
+    {
+      "start": 1062.32,
+      "text": "it becomes extremely computationally"
+    },
+    {
+      "start": 1064.2,
+      "text": "expensive to integrate, in particular,"
+    },
+    {
+      "start": 1066.96,
+      "text": "if we have many parameters that we want"
+    },
+    {
+      "start": 1069.4,
+      "text": "to calibrate."
+    },
+    {
+      "start": 1071.6,
+      "text": "So, the"
+    },
+    {
+      "start": 1072.36,
+      "text": "the key idea here is to recast this"
+    },
+    {
+      "start": 1076.32,
+      "text": "calibration problem"
+    },
+    {
+      "start": 1078.2,
+      "text": "to a perturbation problem. So, if we"
+    },
+    {
+      "start": 1081.4,
+      "text": "imagine to Taylor expand at first order"
+    },
+    {
+      "start": 1085.64,
+      "text": "the drift and and the diffusion"
+    },
+    {
+      "start": 1087.84,
+      "text": "coefficients after we perturb by a tiny"
+    },
+    {
+      "start": 1091.6,
+      "text": "amount each of those parameters,"
+    },
+    {
+      "start": 1094.72,
+      "text": "so essentially, what we obtain is a"
+    },
+    {
+      "start": 1097.08,
+      "text": "perturbed version of this Langevin"
+    },
+    {
+      "start": 1100.16,
+      "text": "equation, where we have an additional"
+    },
+    {
+      "start": 1103.08,
+      "text": "term in the drift and in the diffusion"
+    },
+    {
+      "start": 1106.0,
+      "text": "coefficients, which is given by the"
+    },
+    {
+      "start": 1108.6,
+      "text": "perturbation amplitude of the parameter"
+    },
+    {
+      "start": 1110.679,
+      "text": "multiplied by the partial derivatives of"
+    },
+    {
+      "start": 1114.08,
+      "text": "the drift and the diffusion term with"
+    },
+    {
+      "start": 1117.32,
+      "text": "respect to that specific parameter. So,"
+    },
+    {
+      "start": 1120.04,
+      "text": "essentially, it's like if we are"
+    },
+    {
+      "start": 1121.72,
+      "text": "studying a perturbed version"
+    },
+    {
+      "start": 1124.52,
+      "text": "of our model, and we would like to"
+    },
+    {
+      "start": 1126.88,
+      "text": "predict how all those statistical"
+    },
+    {
+      "start": 1130.36,
+      "text": "observables will change after we add"
+    },
+    {
+      "start": 1134.2,
+      "text": "this perturbation."
+    },
+    {
+      "start": 1136.84,
+      "text": "Okay. So, now we have this"
+    },
+    {
+      "start": 1140.08,
+      "text": "perturbation problem,"
+    },
+    {
+      "start": 1142.44,
+      "text": "which can be addressed using a very"
+    },
+    {
+      "start": 1145.36,
+      "text": "powerful"
+    },
+    {
+      "start": 1146.72,
+      "text": "tool from"
+    },
+    {
+      "start": 1148.64,
+      "text": "statistical physics,"
+    },
+    {
+      "start": 1150.88,
+      "text": "which is the generalized"
+    },
+    {
+      "start": 1153.72,
+      "text": "fluctuation-dissipation theorem or GFT."
+    },
+    {
+      "start": 1157.48,
+      "text": "So, GFT"
+    },
+    {
+      "start": 1159.64,
+      "text": "is"
+    },
+    {
+      "start": 1162.159,
+      "text": "a mathematical machinery that allows to"
+    },
+    {
+      "start": 1164.76,
+      "text": "predict how a dynamical system respond"
+    },
+    {
+      "start": 1168.44,
+      "text": "to a perturbation without actually"
+    },
+    {
+      "start": 1171.28,
+      "text": "perturbing it, just from the knowledge"
+    },
+    {
+      "start": 1174.0,
+      "text": "of its statistics."
+    },
+    {
+      "start": 1176.2,
+      "text": "So, without entering into the"
+    },
+    {
+      "start": 1177.96,
+      "text": "mathematical details of this problem, we"
+    },
+    {
+      "start": 1181.12,
+      "text": "can essentially write the response of an"
+    },
+    {
+      "start": 1185.159,
+      "text": "observable A"
+    },
+    {
+      "start": 1187.12,
+      "text": "to a perturbation given by"
+    },
+    {
+      "start": 1191.159,
+      "text": "uh U of so a drift perturbation given by"
+    },
+    {
+      "start": 1195.0,
+      "text": "U of X and a diffusion perturbation"
+    },
+    {
+      "start": 1197.8,
+      "text": "given by V of X."
+    },
+    {
+      "start": 1200.12,
+      "text": "So, those are nothing but"
+    },
+    {
+      "start": 1202.88,
+      "text": "these perturbations"
+    },
+    {
+      "start": 1205.28,
+      "text": "that we defined before."
+    },
+    {
+      "start": 1211.16,
+      "text": "And we can write the response of this"
+    },
+    {
+      "start": 1214.96,
+      "text": "observable A in terms of an integral of"
+    },
+    {
+      "start": 1218.4,
+      "text": "the correlation between the observable"
+    },
+    {
+      "start": 1220.76,
+      "text": "itself and this conjugate variable B,"
+    },
+    {
+      "start": 1224.52,
+      "text": "which depends on the"
+    },
+    {
+      "start": 1227.4,
+      "text": "perturbation that we are applying. So, U"
+    },
+    {
+      "start": 1230.68,
+      "text": "of X"
+    },
+    {
+      "start": 1231.76,
+      "text": "and V of X."
+    },
+    {
+      "start": 1233.84,
+      "text": "And importantly on the score function S."
+    },
+    {
+      "start": 1239.2,
+      "text": "So, essentially if we have the knowledge"
+    },
+    {
+      "start": 1243.12,
+      "text": "of the score function since"
+    },
+    {
+      "start": 1246.88,
+      "text": "we know the analytic expression for our"
+    },
+    {
+      "start": 1249.88,
+      "text": "model. And also we know the analytic"
+    },
+    {
+      "start": 1252.36,
+      "text": "expression for the perturbation that we"
+    },
+    {
+      "start": 1254.6,
+      "text": "are applying U and V."
+    },
+    {
+      "start": 1256.8,
+      "text": "We can predict how our system, so in"
+    },
+    {
+      "start": 1261.08,
+      "text": "this case how the observable A will"
+    },
+    {
+      "start": 1264.6,
+      "text": "respond to an external perturbation."
+    },
+    {
+      "start": 1268.0,
+      "text": "So, if we identify the observable A with"
+    },
+    {
+      "start": 1271.76,
+      "text": "the set of observables that we want our"
+    },
+    {
+      "start": 1274.44,
+      "text": "model to reproduce."
+    },
+    {
+      "start": 1276.4,
+      "text": "We can essentially construct using the"
+    },
+    {
+      "start": 1279.28,
+      "text": "generalized fluctuation-dissipation"
+    },
+    {
+      "start": 1281.16,
+      "text": "theorem the parameter sensitivities that"
+    },
+    {
+      "start": 1284.12,
+      "text": "we are interested in."
+    },
+    {
+      "start": 1286.04,
+      "text": "Just from the knowledge of the score"
+    },
+    {
+      "start": 1288.04,
+      "text": "function itself."
+    },
+    {
+      "start": 1291.4,
+      "text": "So, this is the main connection. So,"
+    },
+    {
+      "start": 1294.0,
+      "text": "using GFDT we are able to estimate the"
+    },
+    {
+      "start": 1297.16,
+      "text": "parameter sensitivities"
+    },
+    {
+      "start": 1299.96,
+      "text": "without the running the model forward"
+    },
+    {
+      "start": 1302.96,
+      "text": "for every parameter."
+    },
+    {
+      "start": 1305.08,
+      "text": "We just needed to run the model forward"
+    },
+    {
+      "start": 1307.6,
+      "text": "one time to estimate the score function."
+    },
+    {
+      "start": 1311.76,
+      "text": "And from this single model integration"
+    },
+    {
+      "start": 1314.04,
+      "text": "we are able to estimate all the"
+    },
+    {
+      "start": 1316.04,
+      "text": "parameter sensitivities that we can use"
+    },
+    {
+      "start": 1318.4,
+      "text": "for calibration."
+    },
+    {
+      "start": 1320.92,
+      "text": "Okay, so as we have seen this entirely"
+    },
+    {
+      "start": 1323.48,
+      "text": "so this so the applicability of the"
+    },
+    {
+      "start": 1327.56,
+      "text": "theorem to this specific problem relies"
+    },
+    {
+      "start": 1330.72,
+      "text": "only on the estimation of the score"
+    },
+    {
+      "start": 1334.84,
+      "text": "So, at this point you may ask okay, but"
+    },
+    {
+      "start": 1337.2,
+      "text": "why nobody like followed this direction?"
+    },
+    {
+      "start": 1341.32,
+      "text": "And the reason is that estimating the"
+    },
+    {
+      "start": 1344.04,
+      "text": "score function for high-dimensional"
+    },
+    {
+      "start": 1345.84,
+      "text": "system has always been the main"
+    },
+    {
+      "start": 1348.84,
+      "text": "bottleneck to use this generalized"
+    },
+    {
+      "start": 1352.24,
+      "text": "fluctuation-dissipation theorem inside"
+    },
+    {
+      "start": 1355.6,
+      "text": "realistic and practical problems."
+    },
+    {
+      "start": 1358.64,
+      "text": "So, usually people in the literature"
+    },
+    {
+      "start": 1361.4,
+      "text": "have always struggled in estimating the"
+    },
+    {
+      "start": 1363.64,
+      "text": "score function. And so, it was possible"
+    },
+    {
+      "start": 1366.679,
+      "text": "for low-dimensional systems. Instead for"
+    },
+    {
+      "start": 1369.28,
+      "text": "high-dimensional systems,"
+    },
+    {
+      "start": 1371.4,
+      "text": "what has been usually done in the"
+    },
+    {
+      "start": 1372.92,
+      "text": "literature was to use the so-called"
+    },
+    {
+      "start": 1375.6,
+      "text": "quasi-Gaussian approximations,"
+    },
+    {
+      "start": 1377.72,
+      "text": "which consists in approximating the"
+    },
+    {
+      "start": 1380.12,
+      "text": "steady state distribution with a"
+    },
+    {
+      "start": 1381.96,
+      "text": "multivariate Gaussian, which allows to"
+    },
+    {
+      "start": 1384.84,
+      "text": "write the score function in terms of a"
+    },
+    {
+      "start": 1387.0,
+      "text": "linear function that depends on the"
+    },
+    {
+      "start": 1388.84,
+      "text": "covariance matrix of the data set, which"
+    },
+    {
+      "start": 1391.16,
+      "text": "of course is extremely easy to be"
+    },
+    {
+      "start": 1393.12,
+      "text": "estimated."
+    },
+    {
+      "start": 1394.6,
+      "text": "But the main problem is that"
+    },
+    {
+      "start": 1396.64,
+      "text": "when the dynamics is highly nonlinear,"
+    },
+    {
+      "start": 1399.32,
+      "text": "so when the steady state distribution"
+    },
+    {
+      "start": 1401.12,
+      "text": "also is highly"
+    },
+    {
+      "start": 1402.96,
+      "text": "non-Gaussian,"
+    },
+    {
+      "start": 1404.4,
+      "text": "then this approach introduces some"
+    },
+    {
+      "start": 1406.64,
+      "text": "strong biases,"
+    },
+    {
+      "start": 1408.36,
+      "text": "which"
+    },
+    {
+      "start": 1410.32,
+      "text": "doesn't allow to get quite precise"
+    },
+    {
+      "start": 1414.0,
+      "text": "prediction of the system responses using"
+    },
+    {
+      "start": 1417.44,
+      "text": "GFDT."
+    },
+    {
+      "start": 1418.8,
+      "text": "But we we have seen before how yeah,"
+    },
+    {
+      "start": 1421.52,
+      "text": "recent advances in score estimation"
+    },
+    {
+      "start": 1424.44,
+      "text": "methods using a neural network"
+    },
+    {
+      "start": 1427.4,
+      "text": "allow us to get a very precise and"
+    },
+    {
+      "start": 1431.72,
+      "text": "efficient estimation for the score"
+    },
+    {
+      "start": 1434.0,
+      "text": "function also for very high-dimensional"
+    },
+    {
+      "start": 1436.24,
+      "text": "systems."
+    },
+    {
+      "start": 1437.88,
+      "text": "And so, this knowledge allow us to"
+    },
+    {
+      "start": 1441.64,
+      "text": "construct and to estimate the system"
+    },
+    {
+      "start": 1444.12,
+      "text": "responses using the generalized"
+    },
+    {
+      "start": 1446.679,
+      "text": "fluctuation-dissipation theorem."
+    },
+    {
+      "start": 1449.0,
+      "text": "So, we applied those ideas already"
+    },
+    {
+      "start": 1452.36,
+      "text": "to evaluate"
+    },
+    {
+      "start": 1454.52,
+      "text": "and to predict system responses for"
+    },
+    {
+      "start": 1457.6,
+      "text": "quite high-dimensional systems. We"
+    },
+    {
+      "start": 1459.6,
+      "text": "started PDEs discretized on around 10 at"
+    },
+    {
+      "start": 1463.72,
+      "text": "the third grid points."
+    },
+    {
+      "start": 1466.52,
+      "text": "And we considered more specifically"
+    },
+    {
+      "start": 1469.6,
+      "text": "two-dimensional turbulent data and"
+    },
+    {
+      "start": 1472.0,
+      "text": "Alan-Khan reaction-diffusion data. So,"
+    },
+    {
+      "start": 1474.8,
+      "text": "these are the two papers where we"
+    },
+    {
+      "start": 1476.48,
+      "text": "published this"
+    },
+    {
+      "start": 1477.96,
+      "text": "connection between the score-based"
+    },
+    {
+      "start": 1479.8,
+      "text": "generative modeling and the generalized"
+    },
+    {
+      "start": 1482.52,
+      "text": "fluctuation-dissipation theorem. And so,"
+    },
+    {
+      "start": 1484.88,
+      "text": "now the idea is to use this mathematical"
+    },
+    {
+      "start": 1488.16,
+      "text": "machinery to evaluate the parameter"
+    },
+    {
+      "start": 1491.52,
+      "text": "sensitivities."
+    },
+    {
+      "start": 1494.08,
+      "text": "And"
+    },
+    {
+      "start": 1495.24,
+      "text": "so, to do that I"
+    },
+    {
+      "start": 1497.72,
+      "text": "so, I described how we can do it. So,"
+    },
+    {
+      "start": 1501.2,
+      "text": "from the knowledge of the response"
+    },
+    {
+      "start": 1503.28,
+      "text": "function we can estimate the parameter"
+    },
+    {
+      "start": 1505.72,
+      "text": "sensitivities. Now, let's see how to do"
+    },
+    {
+      "start": 1508.08,
+      "text": "that in practice."
+    },
+    {
+      "start": 1509.84,
+      "text": "More specifically, let's consider two"
+    },
+    {
+      "start": 1511.64,
+      "text": "examples now. So, the first one is a"
+    },
+    {
+      "start": 1514.4,
+      "text": "very low-dimensional model. We have a"
+    },
+    {
+      "start": 1517.24,
+      "text": "three-dimensional SDE with a"
+    },
+    {
+      "start": 1520.2,
+      "text": "multiplicative noise. So, this model is"
+    },
+    {
+      "start": 1522.44,
+      "text": "used in geophysical fluid dynamics to"
+    },
+    {
+      "start": 1524.679,
+      "text": "describe"
+    },
+    {
+      "start": 1525.96,
+      "text": "El Niño-Southern Oscillation, which is a"
+    },
+    {
+      "start": 1529.24,
+      "text": "interannual so, it's a annual"
+    },
+    {
+      "start": 1531.679,
+      "text": "variability phenomenon of the sea"
+    },
+    {
+      "start": 1533.96,
+      "text": "surface temperature in the tropical"
+    },
+    {
+      "start": 1535.64,
+      "text": "Pacific. We have two slow variables, one"
+    },
+    {
+      "start": 1539.16,
+      "text": "fast variable, which are coupled."
+    },
+    {
+      "start": 1542.04,
+      "text": "This model depends on six coefficients."
+    },
+    {
+      "start": 1546.84,
+      "text": "And so, what we are going to do is to"
+    },
+    {
+      "start": 1550.44,
+      "text": "start so, is to first run this model"
+    },
+    {
+      "start": 1554.32,
+      "text": "with the correct values of these"
+    },
+    {
+      "start": 1555.679,
+      "text": "coefficients to have"
+    },
+    {
+      "start": 1558.48,
+      "text": "an obser so, to build our observations."
+    },
+    {
+      "start": 1562.44,
+      "text": "Then"
+    },
+    {
+      "start": 1563.72,
+      "text": "we will"
+    },
+    {
+      "start": 1564.92,
+      "text": "try to recover the correct values of"
+    },
+    {
+      "start": 1568.8,
+      "text": "those coefficients and starting from an"
+    },
+    {
+      "start": 1570.96,
+      "text": "initial guess"
+    },
+    {
+      "start": 1572.8,
+      "text": "obtained by perturbing by around 20%"
+    },
+    {
+      "start": 1578.32,
+      "text": "each of those coefficients and running"
+    },
+    {
+      "start": 1581.0,
+      "text": "our calibration algorithm. So, using"
+    },
+    {
+      "start": 1584.08,
+      "text": "GFDT to estimate the"
+    },
+    {
+      "start": 1588.56,
+      "text": "and use"
+    },
+    {
+      "start": 1589.72,
+      "text": "this knowledge inside a Newton algorithm"
+    },
+    {
+      "start": 1593.28,
+      "text": "to estimate the correct values of the"
+    },
+    {
+      "start": 1595.64,
+      "text": "coefficient."
+    },
+    {
+      "start": 1597.56,
+      "text": "Those are the six observables that we"
+    },
+    {
+      "start": 1600.28,
+      "text": "would like to recover. So, we start with"
+    },
+    {
+      "start": 1603.72,
+      "text": "a parameter guess."
+    },
+    {
+      "start": 1605.92,
+      "text": "Um we use this parameter guess to"
+    },
+    {
+      "start": 1609.16,
+      "text": "integrate the model forward."
+    },
+    {
+      "start": 1611.84,
+      "text": "We use this data set to estimate the"
+    },
+    {
+      "start": 1615.84,
+      "text": "parameter sensitivities. And then we are"
+    },
+    {
+      "start": 1618.84,
+      "text": "updating the parameters using the"
+    },
+    {
+      "start": 1621.48,
+      "text": "knowledge of the parameter"
+    },
+    {
+      "start": 1623.92,
+      "text": "And we are iterating this procedure"
+    },
+    {
+      "start": 1626.0,
+      "text": "until"
+    },
+    {
+      "start": 1627.72,
+      "text": "the statistics of our predicted system"
+    },
+    {
+      "start": 1630.4,
+      "text": "will converge"
+    },
+    {
+      "start": 1632.04,
+      "text": "to the target statistical observables."
+    },
+    {
+      "start": 1634.84,
+      "text": "We are doing this procedure using three"
+    },
+    {
+      "start": 1637.44,
+      "text": "different methods to evaluate the"
+    },
+    {
+      "start": 1640.2,
+      "text": "parameter sensitivities. We are first"
+    },
+    {
+      "start": 1642.88,
+      "text": "using finite difference. So, we are just"
+    },
+    {
+      "start": 1645.6,
+      "text": "integrating many time the trajectory"
+    },
+    {
+      "start": 1648.04,
+      "text": "forward one for each parameter. And then"
+    },
+    {
+      "start": 1651.36,
+      "text": "we are using finite difference to"
+    },
+    {
+      "start": 1652.72,
+      "text": "estimate the parameter Jacobian."
+    },
+    {
+      "start": 1655.84,
+      "text": "And we are using that information to"
+    },
+    {
+      "start": 1657.28,
+      "text": "update the parameter value."
+    },
+    {
+      "start": 1659.44,
+      "text": "Then we are using the generalized"
+    },
+    {
+      "start": 1661.6,
+      "text": "fluctuation-dissipation theorem using"
+    },
+    {
+      "start": 1664.04,
+      "text": "two different ways to estimate the score"
+    },
+    {
+      "start": 1668.16,
+      "text": "We first used the quasi-Gaussian"
+    },
+    {
+      "start": 1671.16,
+      "text": "approximation. So, we just"
+    },
+    {
+      "start": 1674.2,
+      "text": "wrote the score function in terms of a"
+    },
+    {
+      "start": 1679.28,
+      "text": "covariance matrix of the data. And then"
+    },
+    {
+      "start": 1682.24,
+      "text": "we are using the"
+    },
+    {
+      "start": 1685.32,
+      "text": "denoising score matching approach to"
+    },
+    {
+      "start": 1687.36,
+      "text": "estimate the score function."
+    },
+    {
+      "start": 1689.679,
+      "text": "And we will be comparing those three"
+    },
+    {
+      "start": 1691.72,
+      "text": "approaches."
+    },
+    {
+      "start": 1693.12,
+      "text": "These are the results."
+    },
+    {
+      "start": 1694.96,
+      "text": "So, here on the left you can see the L2"
+    },
+    {
+      "start": 1698.6,
+      "text": "norm between predicted versus"
+    },
+    {
+      "start": 1702.32,
+      "text": "target statistical observables as a"
+    },
+    {
+      "start": 1705.159,
+      "text": "function of the"
+    },
+    {
+      "start": 1707.6,
+      "text": "algorithm iteration. So, in this case"
+    },
+    {
+      "start": 1711.159,
+      "text": "we introduced the breaking point when"
+    },
+    {
+      "start": 1715.08,
+      "text": "the"
+    },
+    {
+      "start": 1715.96,
+      "text": "L2 norm"
+    },
+    {
+      "start": 1717.88,
+      "text": "was falling below 10 at the minus three."
+    },
+    {
+      "start": 1720.919,
+      "text": "And as you can see the blue curve and"
+    },
+    {
+      "start": 1723.6,
+      "text": "the gray and the gray curve, which"
+    },
+    {
+      "start": 1725.24,
+      "text": "represent the calibration algorithm"
+    },
+    {
+      "start": 1728.919,
+      "text": "using the denoising score matching score"
+    },
+    {
+      "start": 1731.679,
+      "text": "function plus GFDT and the naive"
+    },
+    {
+      "start": 1737.36,
+      "text": "and and the naive"
+    },
+    {
+      "start": 1739.44,
+      "text": "finite difference estimation for the"
+    },
+    {
+      "start": 1742.08,
+      "text": "parameter Jacobians in just five"
+    },
+    {
+      "start": 1744.84,
+      "text": "iteration we are falling below the"
+    },
+    {
+      "start": 1746.96,
+      "text": "threshold."
+    },
+    {
+      "start": 1748.36,
+      "text": "Which essentially means in five"
+    },
+    {
+      "start": 1749.919,
+      "text": "iteration we were able, as you can see"
+    },
+    {
+      "start": 1752.4,
+      "text": "here in this panel showing the parameter"
+    },
+    {
+      "start": 1755.32,
+      "text": "deviation, to precisely recover the"
+    },
+    {
+      "start": 1758.64,
+      "text": "correct parameters of the model."
+    },
+    {
+      "start": 1761.36,
+      "text": "And instead using the Gaussian"
+    },
+    {
+      "start": 1763.679,
+      "text": "approximation for the score, so like a"
+    },
+    {
+      "start": 1766.159,
+      "text": "more so, less precise estimation of the"
+    },
+    {
+      "start": 1769.679,
+      "text": "score function was very difficult to"
+    },
+    {
+      "start": 1772.36,
+      "text": "have the algorithm"
+    },
+    {
+      "start": 1774.32,
+      "text": "con"
+    },
+    {
+      "start": 1775.8,
+      "text": "converged"
+    },
+    {
+      "start": 1777.32,
+      "text": "to the correct value."
+    },
+    {
+      "start": 1779.36,
+      "text": "But so, here what we can see that using"
+    },
+    {
+      "start": 1782.159,
+      "text": "the generalized fluctuation-dissipation"
+    },
+    {
+      "start": 1783.96,
+      "text": "theorem plus the denoising score"
+    },
+    {
+      "start": 1785.8,
+      "text": "matching to estimate this score"
+    },
+    {
+      "start": 1787.8,
+      "text": "function, we were able to have very"
+    },
+    {
+      "start": 1790.32,
+      "text": "similar performances"
+    },
+    {
+      "start": 1793.04,
+      "text": "with respect to the naive"
+    },
+    {
+      "start": 1796.0,
+      "text": "finite difference method at a fraction"
+    },
+    {
+      "start": 1799.24,
+      "text": "of the computational cost because"
+    },
+    {
+      "start": 1801.6,
+      "text": "we"
+    },
+    {
+      "start": 1803.04,
+      "text": "So, for every iteration, we needed to"
+    },
+    {
+      "start": 1805.32,
+      "text": "integrate the system forward only one"
+    },
+    {
+      "start": 1808.6,
+      "text": "time"
+    },
+    {
+      "start": 1809.72,
+      "text": "instead of six time"
+    },
+    {
+      "start": 1813.48,
+      "text": "So, the number of the parameters that we"
+    },
+    {
+      "start": 1815.92,
+      "text": "want to calibrate or like in in this"
+    },
+    {
+      "start": 1818.48,
+      "text": "case 12 because we use the center"
+    },
+    {
+      "start": 1820.6,
+      "text": "difference for"
+    },
+    {
+      "start": 1823.04,
+      "text": "parameter for the"
+    },
+    {
+      "start": 1824.76,
+      "text": "uh um"
+    },
+    {
+      "start": 1826.12,
+      "text": "parameter Jacobian estimation. So,"
+    },
+    {
+      "start": 1828.28,
+      "text": "essentially here we have an algorithm"
+    },
+    {
+      "start": 1830.76,
+      "text": "that's doesn't scale So, doesn't So, the"
+    },
+    {
+      "start": 1834.88,
+      "text": "for which the computational time doesn't"
+    },
+    {
+      "start": 1837.32,
+      "text": "scale"
+    },
+    {
+      "start": 1838.68,
+      "text": "linearly with the number of parameters,"
+    },
+    {
+      "start": 1841.12,
+      "text": "but is constant since we only need to"
+    },
+    {
+      "start": 1843.76,
+      "text": "run the model forward one"
+    },
+    {
+      "start": 1846.12,
+      "text": "one single time for every iteration."
+    },
+    {
+      "start": 1849.36,
+      "text": "Okay. So, in this case we considered a"
+    },
+    {
+      "start": 1851.52,
+      "text": "quite a low dimensional system."
+    },
+    {
+      "start": 1854.12,
+      "text": "Next, I will consider this coupled"
+    },
+    {
+      "start": 1857.16,
+      "text": "Lorenz '96"
+    },
+    {
+      "start": 1859.56,
+      "text": "system, which is around a 400"
+    },
+    {
+      "start": 1862.6,
+      "text": "dimensional system. We have 36 slow mode"
+    },
+    {
+      "start": 1867.52,
+      "text": "and we have a for each slow mode we have"
+    },
+    {
+      "start": 1869.8,
+      "text": "10 fast modes. Plus, we also have some"
+    },
+    {
+      "start": 1873.68,
+      "text": "white noise in each of um"
+    },
+    {
+      "start": 1876.12,
+      "text": "those"
+    },
+    {
+      "start": 1877.12,
+      "text": "variables."
+    },
+    {
+      "start": 1878.56,
+      "text": "And what we want to do now is to do"
+    },
+    {
+      "start": 1880.4,
+      "text": "something different. So, we would like"
+    },
+    {
+      "start": 1883.48,
+      "text": "to build a stochastic closure for the X"
+    },
+    {
+      "start": 1888.16,
+      "text": "So, the slow variables. So, essentially"
+    },
+    {
+      "start": 1890.44,
+      "text": "we would like to build a 36 dimensional"
+    },
+    {
+      "start": 1893.28,
+      "text": "model"
+    },
+    {
+      "start": 1894.64,
+      "text": "instead of this model here that is"
+    },
+    {
+      "start": 1896.72,
+      "text": "around a 400 dimensional"
+    },
+    {
+      "start": 1899.2,
+      "text": "which is able to precisely recover"
+    },
+    {
+      "start": 1903.0,
+      "text": "the target statistical observables"
+    },
+    {
+      "start": 1906.32,
+      "text": "evaluated from the high dimensional"
+    },
+    {
+      "start": 1908.68,
+      "text": "model. So, we have observations for X"
+    },
+    {
+      "start": 1912.08,
+      "text": "which have been generated integrating"
+    },
+    {
+      "start": 1914.679,
+      "text": "this very high dimensional system and we"
+    },
+    {
+      "start": 1917.52,
+      "text": "would like to build this reduced order"
+    },
+    {
+      "start": 1920.6,
+      "text": "model which is only 36 dimensional"
+    },
+    {
+      "start": 1924.04,
+      "text": "with the correct values of alphas of the"
+    },
+    {
+      "start": 1926.84,
+      "text": "alpha coefficients and the sigma"
+    },
+    {
+      "start": 1929.2,
+      "text": "coefficients such that they reproduce"
+    },
+    {
+      "start": 1932.4,
+      "text": "this set of target statistical"
+    },
+    {
+      "start": 1934.44,
+      "text": "observables which are the mean uh the"
+    },
+    {
+      "start": 1938.28,
+      "text": "variance, skewness, excess kurtosis, and"
+    },
+    {
+      "start": 1942.48,
+      "text": "uh covariance C1."
+    },
+    {
+      "start": 1945.04,
+      "text": "We have in total five parameters that we"
+    },
+    {
+      "start": 1947.92,
+      "text": "want to calibrate"
+    },
+    {
+      "start": 1949.302,
+      "text": "&gt;&gt; [gasps]"
+    },
+    {
+      "start": 1949.96,
+      "text": "&gt;&gt; uh on the"
+    },
+    {
+      "start": 1951.75,
+      "text": "uh on the"
+    },
+    {
+      "start": 1951.76,
+      "text": "on these five different statistical"
+    },
+    {
+      "start": 1955.44,
+      "text": "And so, we used also in this case these"
+    },
+    {
+      "start": 1959.08,
+      "text": "three different methods."
+    },
+    {
+      "start": 1961.0,
+      "text": "We have in orange GFDT plus Gaussian"
+    },
+    {
+      "start": 1965.72,
+      "text": "uh estimation of the score function. And"
+    },
+    {
+      "start": 1968.84,
+      "text": "then in gray and in blue, we have"
+    },
+    {
+      "start": 1972.24,
+      "text": "the finite difference method and then"
+    },
+    {
+      "start": 1974.679,
+      "text": "the GFDT plus the noise score matching"
+    },
+    {
+      "start": 1977.16,
+      "text": "for the score"
+    },
+    {
+      "start": 1978.64,
+      "text": "estimation. So, essentially in in this"
+    },
+    {
+      "start": 1980.48,
+      "text": "case we can see like yeah, clear"
+    },
+    {
+      "start": 1982.92,
+      "text": "advantage in using the"
+    },
+    {
+      "start": 1986.24,
+      "text": "the noise score matching to build the"
+    },
+    {
+      "start": 1988.4,
+      "text": "score function. And from the knowledge"
+    },
+    {
+      "start": 1990.96,
+      "text": "of the score function, also in this case"
+    },
+    {
+      "start": 1993.08,
+      "text": "which is quite high dimensional, we can"
+    },
+    {
+      "start": 1995.24,
+      "text": "observe"
+    },
+    {
+      "start": 1996.76,
+      "text": "how we get quite similar performance"
+    },
+    {
+      "start": 2000.52,
+      "text": "than of using finite difference at a"
+    },
+    {
+      "start": 2004.04,
+      "text": "fraction of the computational cost. And"
+    },
+    {
+      "start": 2005.679,
+      "text": "so, essentially at at a fraction of the"
+    },
+    {
+      "start": 2007.52,
+      "text": "number of time that we have to integrate"
+    },
+    {
+      "start": 2010.44,
+      "text": "our model forward."
+    },
+    {
+      "start": 2013.64,
+      "text": "Okay. So, this"
+    },
+    {
+      "start": 2015.32,
+      "text": "is the first direction. So, start from a"
+    },
+    {
+      "start": 2017.679,
+      "text": "model answers and use this combination"
+    },
+    {
+      "start": 2020.64,
+      "text": "between the generalized fluctuation"
+    },
+    {
+      "start": 2022.4,
+      "text": "dissipation theorem from non-equilibrium"
+    },
+    {
+      "start": 2025.92,
+      "text": "statistical physics with the noise score"
+    },
+    {
+      "start": 2028.32,
+      "text": "matching from generative modeling"
+    },
+    {
+      "start": 2031.76,
+      "text": "to to to estimate"
+    },
+    {
+      "start": 2035.0,
+      "text": "uh the parameter sensitivities"
+    },
+    {
+      "start": 2038.24,
+      "text": "with a very limited number of model"
+    },
+    {
+      "start": 2041.0,
+      "text": "integrations."
+    },
+    {
+      "start": 2042.72,
+      "text": "Now, let's see the second direction."
+    },
+    {
+      "start": 2045.12,
+      "text": "So, in this case we don't have any model"
+    },
+    {
+      "start": 2048.879,
+      "text": "answers for"
+    },
+    {
+      "start": 2051.399,
+      "text": "the functional form of our mathematical"
+    },
+    {
+      "start": 2053.8,
+      "text": "model."
+    },
+    {
+      "start": 2055.32,
+      "text": "We have a set of statistical and"
+    },
+    {
+      "start": 2059.08,
+      "text": "dynamical observables that we want our"
+    },
+    {
+      "start": 2061.64,
+      "text": "model to reproduce, which in this case"
+    },
+    {
+      "start": 2064.159,
+      "text": "are the full steady state distribution"
+    },
+    {
+      "start": 2067.6,
+      "text": "and a set of correlation functions where"
+    },
+    {
+      "start": 2071.08,
+      "text": "phi m and phi n are"
+    },
+    {
+      "start": 2075.08,
+      "text": "So, it's a set of observables of the the"
+    },
+    {
+      "start": 2078.76,
+      "text": "state variable of the system."
+    },
+    {
+      "start": 2081.8,
+      "text": "So, given these constraints, we would"
+    },
+    {
+      "start": 2084.8,
+      "text": "like to build a mathematical model that"
+    },
+    {
+      "start": 2088.08,
+      "text": "by construction reproduces those"
+    },
+    {
+      "start": 2090.56,
+      "text": "constraints without integrating our"
+    },
+    {
+      "start": 2093.2,
+      "text": "model forward."
+    },
+    {
+      "start": 2095.6,
+      "text": "And again, we will use the score"
+    },
+    {
+      "start": 2097.4,
+      "text": "function to do that."
+    },
+    {
+      "start": 2099.44,
+      "text": "Specifically, we will use two different"
+    },
+    {
+      "start": 2102.2,
+      "text": "score functions in this case. We have"
+    },
+    {
+      "start": 2104.56,
+      "text": "the plain score function that we've seen"
+    },
+    {
+      "start": 2106.48,
+      "text": "before"
+    },
+    {
+      "start": 2107.68,
+      "text": "and also we will use the conditional"
+    },
+    {
+      "start": 2110.0,
+      "text": "score function. This essentially is the"
+    },
+    {
+      "start": 2112.88,
+      "text": "gradient"
+    },
+    {
+      "start": 2114.44,
+      "text": "of with respect to X0 of the logarithm"
+    },
+    {
+      "start": 2118.48,
+      "text": "of the conditional probability density"
+    },
+    {
+      "start": 2120.4,
+      "text": "function. So, the probability density"
+    },
+    {
+      "start": 2123.12,
+      "text": "function of X at time T conditioned on"
+    },
+    {
+      "start": 2126.59,
+      "text": "of X at time T conditioned on"
+    },
+    {
+      "start": 2126.6,
+      "text": "X0."
+    },
+    {
+      "start": 2128.6,
+      "text": "The conditional score function can be"
+    },
+    {
+      "start": 2131.359,
+      "text": "constructed using the noise score"
+    },
+    {
+      "start": 2133.359,
+      "text": "matching precisely as we did it for the"
+    },
+    {
+      "start": 2136.72,
+      "text": "plain score function. In fact, we can"
+    },
+    {
+      "start": 2138.8,
+      "text": "write the conditional score function in"
+    },
+    {
+      "start": 2141.2,
+      "text": "terms of the joint score function and"
+    },
+    {
+      "start": 2145.52,
+      "text": "the plain score function."
+    },
+    {
+      "start": 2147.96,
+      "text": "For the joint score function, we just"
+    },
+    {
+      "start": 2150.44,
+      "text": "take our data set. We use a delay"
+    },
+    {
+      "start": 2153.12,
+      "text": "embedding"
+    },
+    {
+      "start": 2154.92,
+      "text": "in order to build"
+    },
+    {
+      "start": 2158.28,
+      "text": "a time series of X0 and XT."
+    },
+    {
+      "start": 2162.92,
+      "text": "We do it for different value of the time"
+    },
+    {
+      "start": 2165.44,
+      "text": "delay and in this way we estimate the"
+    },
+    {
+      "start": 2168.2,
+      "text": "score function the joint score function."
+    },
+    {
+      "start": 2171.4,
+      "text": "And then we can combine it with us and"
+    },
+    {
+      "start": 2174.8,
+      "text": "using the same the noise score matching"
+    },
+    {
+      "start": 2176.76,
+      "text": "machinery we have seen before, we can"
+    },
+    {
+      "start": 2179.32,
+      "text": "estimate both"
+    },
+    {
+      "start": 2181.2,
+      "text": "the conditional score and the score"
+    },
+    {
+      "start": 2183.2,
+      "text": "function from data."
+    },
+    {
+      "start": 2184.76,
+      "text": "And as we have seen before, both these"
+    },
+    {
+      "start": 2186.48,
+      "text": "algorithms scale quite well with the"
+    },
+    {
+      "start": 2188.8,
+      "text": "dimension of the system."
+    },
+    {
+      "start": 2190.84,
+      "text": "So, the idea here is then to use those"
+    },
+    {
+      "start": 2193.84,
+      "text": "two quantities"
+    },
+    {
+      "start": 2195.76,
+      "text": "where the first quantity essentially"
+    },
+    {
+      "start": 2198.68,
+      "text": "takes into account the geometry of the"
+    },
+    {
+      "start": 2201.48,
+      "text": "steady state distribution."
+    },
+    {
+      "start": 2203.84,
+      "text": "Instead, the second quantity essentially"
+    },
+    {
+      "start": 2206.6,
+      "text": "takes into account how the system"
+    },
+    {
+      "start": 2209.359,
+      "text": "relaxes towards the steady state"
+    },
+    {
+      "start": 2211.4,
+      "text": "distribution. So, it's carrying"
+    },
+    {
+      "start": 2213.0,
+      "text": "information also about the"
+    },
+    {
+      "start": 2215.04,
+      "text": "the dynamics of the system and not only"
+    },
+    {
+      "start": 2217.359,
+      "text": "about the statistics."
+    },
+    {
+      "start": 2220.72,
+      "text": "So, this is the intuition. So, try to"
+    },
+    {
+      "start": 2223.56,
+      "text": "use those two quantities that can be"
+    },
+    {
+      "start": 2226.6,
+      "text": "evaluated quite well also for very high"
+    },
+    {
+      "start": 2228.96,
+      "text": "dimensional systems"
+    },
+    {
+      "start": 2230.96,
+      "text": "to build our"
+    },
+    {
+      "start": 2233.04,
+      "text": "stochastic modeling approach."
+    },
+    {
+      "start": 2236.16,
+      "text": "So, let's start from our"
+    },
+    {
+      "start": 2239.16,
+      "text": "Langevin equation. So, this is the same"
+    },
+    {
+      "start": 2241.44,
+      "text": "Langevin equation I wrote at the"
+    },
+    {
+      "start": 2242.91,
+      "text": "I wrote at the"
+    },
+    {
+      "start": 2242.92,
+      "text": "beginning. Yeah, just have here a factor"
+    },
+    {
+      "start": 2245.72,
+      "text": "square root of two."
+    },
+    {
+      "start": 2248.08,
+      "text": "And then let's first impose"
+    },
+    {
+      "start": 2250.76,
+      "text": "stationarity. So, we want for a given"
+    },
+    {
+      "start": 2254.84,
+      "text": "sigma X to find our drift term F"
+    },
+    {
+      "start": 2259.44,
+      "text": "such that by construction"
+    },
+    {
+      "start": 2262.24,
+      "text": "reproduces the steady state"
+    },
+    {
+      "start": 2264.08,
+      "text": "distribution."
+    },
+    {
+      "start": 2265.48,
+      "text": "And to do that, we can write the"
+    },
+    {
+      "start": 2267.64,
+      "text": "Fokker-Planck equation relative to the"
+    },
+    {
+      "start": 2270.56,
+      "text": "Langevin [clears throat] equation,"
+    },
+    {
+      "start": 2271.84,
+      "text": "impose the stationarity"
+    },
+    {
+      "start": 2274.16,
+      "text": "and we can show that without losing any"
+    },
+    {
+      "start": 2276.92,
+      "text": "generality"
+    },
+    {
+      "start": 2278.28,
+      "text": "we can write"
+    },
+    {
+      "start": 2280.48,
+      "text": "the drift term"
+    },
+    {
+      "start": 2282.56,
+      "text": "in this way. So, in terms of the score"
+    },
+    {
+      "start": 2286.92,
+      "text": "function that we defined before"
+    },
+    {
+      "start": 2289.56,
+      "text": "and the diffusion matrix. So, this"
+    },
+    {
+      "start": 2293.0,
+      "text": "symmetric matrix D of X"
+    },
+    {
+      "start": 2295.88,
+      "text": "and another anti-symmetric matrix R of"
+    },
+    {
+      "start": 2299.76,
+      "text": "X."
+    },
+    {
+      "start": 2301.16,
+      "text": "So, this is a very general"
+    },
+    {
+      "start": 2303.64,
+      "text": "expression. We're not doing any"
+    },
+    {
+      "start": 2305.68,
+      "text": "approximation here. We are We are just"
+    },
+    {
+      "start": 2308.96,
+      "text": "finding the most general way to express"
+    },
+    {
+      "start": 2311.359,
+      "text": "the drift for a given the diffusion in"
+    },
+    {
+      "start": 2315.04,
+      "text": "such a way that it reproduces the steady"
+    },
+    {
+      "start": 2317.6,
+      "text": "state distribution by construction,"
+    },
+    {
+      "start": 2319.92,
+      "text": "which essentially means in such a way"
+    },
+    {
+      "start": 2322.48,
+      "text": "that F So, this specific shape of F"
+    },
+    {
+      "start": 2326.32,
+      "text": "solve the stationary Fokker-Planck"
+    },
+    {
+      "start": 2328.64,
+      "text": "equation."
+    },
+    {
+      "start": 2330.68,
+      "text": "Now, we can So, we can see that we have"
+    },
+    {
+      "start": 2334.44,
+      "text": "two different tensors D of X and R of X."
+    },
+    {
+      "start": 2338.88,
+      "text": "D of X is symmetric and represent the"
+    },
+    {
+      "start": 2342.72,
+      "text": "diffusion tensor."
+    },
+    {
+      "start": 2344.72,
+      "text": "Instead, R which is the anti-symmetric"
+    },
+    {
+      "start": 2347.08,
+      "text": "part can be interpreted as the term that"
+    },
+    {
+      "start": 2350.92,
+      "text": "breaks the tail balance and that"
+    },
+    {
+      "start": 2353.72,
+      "text": "introduces some rotational component to"
+    },
+    {
+      "start": 2357.28,
+      "text": "our system without changing the steady"
+    },
+    {
+      "start": 2359.64,
+      "text": "state distribution. So, this can be"
+    },
+    {
+      "start": 2362.359,
+      "text": "related to an Helmholtz decomposition of"
+    },
+    {
+      "start": 2365.84,
+      "text": "the drift term. So, we have a"
+    },
+    {
+      "start": 2369.2,
+      "text": "a term a symmetric term which"
+    },
+    {
+      "start": 2373.16,
+      "text": "satisfies the detail balance and give us"
+    },
+    {
+      "start": 2375.52,
+      "text": "a system which is just a Brownian motion"
+    },
+    {
+      "start": 2378.0,
+      "text": "inside a potential. And then we have"
+    },
+    {
+      "start": 2381.12,
+      "text": "this other circulatory term which"
+    },
+    {
+      "start": 2384.68,
+      "text": "introduces some rotational component"
+    },
+    {
+      "start": 2387.16,
+      "text": "that breaks detailed balance."
+    },
+    {
+      "start": 2389.76,
+      "text": "Okay, so now by using this expression"
+    },
+    {
+      "start": 2392.76,
+      "text": "here for the drift term, we are"
+    },
+    {
+      "start": 2396.76,
+      "text": "guaranteed"
+    },
+    {
+      "start": 2398.32,
+      "text": "to recover the steady state"
+    },
+    {
+      "start": 2400.12,
+      "text": "distribution. So we"
+    },
+    {
+      "start": 2402.64,
+      "text": "achieved the first goal of our modeling"
+    },
+    {
+      "start": 2405.4,
+      "text": "strategy which is to build a stochastic"
+    },
+    {
+      "start": 2407.72,
+      "text": "model that by construction reproduces"
+    },
+    {
+      "start": 2410.64,
+      "text": "the observed the steady state"
+    },
+    {
+      "start": 2412.48,
+      "text": "distribution of the data set. Now let's"
+    },
+    {
+      "start": 2415.16,
+      "text": "try to impose also the second constraint"
+    },
+    {
+      "start": 2418.48,
+      "text": "which is"
+    },
+    {
+      "start": 2419.76,
+      "text": "we want to reproduce also the the time"
+    },
+    {
+      "start": 2422.48,
+      "text": "correlations."
+    },
+    {
+      "start": 2425.08,
+      "text": "So we would like to"
+    },
+    {
+      "start": 2427.36,
+      "text": "reproduce this time correlations for a"
+    },
+    {
+      "start": 2431.0,
+      "text": "set of observables phi n."
+    },
+    {
+      "start": 2434.24,
+      "text": "So without going into the mathematical"
+    },
+    {
+      "start": 2436.32,
+      "text": "details of this derivation, we can show"
+    },
+    {
+      "start": 2439.08,
+      "text": "that"
+    },
+    {
+      "start": 2440.24,
+      "text": "the time derivative of this correlation"
+    },
+    {
+      "start": 2443.32,
+      "text": "function for this specific model, so for"
+    },
+    {
+      "start": 2447.52,
+      "text": "this specific Langevin equation with"
+    },
+    {
+      "start": 2449.88,
+      "text": "drift term given by this expression over"
+    },
+    {
+      "start": 2452.88,
+      "text": "here,"
+    },
+    {
+      "start": 2454.2,
+      "text": "can be written in this way."
+    },
+    {
+      "start": 2457.0,
+      "text": "So essentially we can relate the time"
+    },
+    {
+      "start": 2459.08,
+      "text": "derivative of this correlation function"
+    },
+    {
+      "start": 2462.32,
+      "text": "with this"
+    },
+    {
+      "start": 2464.44,
+      "text": "expression here which contains the two"
+    },
+    {
+      "start": 2467.64,
+      "text": "phi, so the two observable phi m and phi"
+    },
+    {
+      "start": 2469.92,
+      "text": "n, the conditional score function, and"
+    },
+    {
+      "start": 2473.92,
+      "text": "the matrix, so the tensor m."
+    },
+    {
+      "start": 2476.96,
+      "text": "And the tensor m is the only term here"
+    },
+    {
+      "start": 2479.32,
+      "text": "that we don't know because we can"
+    },
+    {
+      "start": 2480.84,
+      "text": "estimate this quantity here from data."
+    },
+    {
+      "start": 2483.88,
+      "text": "We just evaluate"
+    },
+    {
+      "start": 2486.16,
+      "text": "the correlation function and then we"
+    },
+    {
+      "start": 2488.68,
+      "text": "estimate the derivative."
+    },
+    {
+      "start": 2490.96,
+      "text": "We can estimate the conditional score."
+    },
+    {
+      "start": 2495.16,
+      "text": "We know the analytical expression for"
+    },
+    {
+      "start": 2497.12,
+      "text": "both phi m and phi n because this is the"
+    },
+    {
+      "start": 2500.64,
+      "text": "libraries of observable that we are"
+    },
+    {
+      "start": 2502.68,
+      "text": "considering."
+    },
+    {
+      "start": 2504.08,
+      "text": "The only term that we don't know is this"
+    },
+    {
+      "start": 2506.56,
+      "text": "matrix"
+    },
+    {
+      "start": 2507.68,
+      "text": "m x of 0. So now let's see how we can"
+    },
+    {
+      "start": 2510.0,
+      "text": "derive this matrix m x of 0."
+    },
+    {
+      "start": 2514.16,
+      "text": "So first"
+    },
+    {
+      "start": 2515.56,
+      "text": "let's do this decomposition. So let's"
+    },
+    {
+      "start": 2518.0,
+      "text": "decompose m x in terms of a constant"
+    },
+    {
+      "start": 2521.72,
+      "text": "term plus a fluctuation."
+    },
+    {
+      "start": 2524.32,
+      "text": "This fluctuation is so the average value"
+    },
+    {
+      "start": 2528.12,
+      "text": "over the stationary density of this"
+    },
+    {
+      "start": 2530.08,
+      "text": "fluctuation must be equal to 0. So"
+    },
+    {
+      "start": 2532.24,
+      "text": "essentially here we have"
+    },
+    {
+      "start": 2534.56,
+      "text": "a yeah, a constant term plus"
+    },
+    {
+      "start": 2537.56,
+      "text": "zero mean fluctuation term delta m. So"
+    },
+    {
+      "start": 2540.68,
+      "text": "let's use this expression here for m"
+    },
+    {
+      "start": 2544.24,
+      "text": "inside the this equation over there"
+    },
+    {
+      "start": 2548.32,
+      "text": "and we can then rewrite c dot in terms"
+    },
+    {
+      "start": 2552.4,
+      "text": "of two terms. This first term which"
+    },
+    {
+      "start": 2555.36,
+      "text": "depends on phi"
+    },
+    {
+      "start": 2557.84,
+      "text": "does not depend on the conditional"
+    },
+    {
+      "start": 2560.28,
+      "text": "score, depends only on the stationary"
+    },
+    {
+      "start": 2562.32,
+      "text": "score."
+    },
+    {
+      "start": 2563.27,
+      "text": "."
+    },
+    {
+      "start": 2563.28,
+      "text": "Okay, so we have the first term here"
+    },
+    {
+      "start": 2566.0,
+      "text": "which is much easier to evaluate because"
+    },
+    {
+      "start": 2569.68,
+      "text": "yeah, we don't need to estimate the"
+    },
+    {
+      "start": 2571.44,
+      "text": "conditional score"
+    },
+    {
+      "start": 2573.12,
+      "text": "and it only depends on this constant"
+    },
+    {
+      "start": 2575.28,
+      "text": "matrix phi and this"
+    },
+    {
+      "start": 2577.24,
+      "text": "plain score itself"
+    },
+    {
+      "start": 2578.96,
+      "text": "minus this additional so this additional"
+    },
+    {
+      "start": 2582.32,
+      "text": "term which is nothing but this one over"
+    },
+    {
+      "start": 2584.96,
+      "text": "here written in terms of delta m instead"
+    },
+    {
+      "start": 2588.32,
+      "text": "of m."
+    },
+    {
+      "start": 2590.84,
+      "text": "So the key idea here is that if we have"
+    },
+    {
+      "start": 2594.6,
+      "text": "a library of observable which is rich"
+    },
+    {
+      "start": 2597.2,
+      "text": "enough such"
+    },
+    {
+      "start": 2599.04,
+      "text": "such that m of x is uniquely determined,"
+    },
+    {
+      "start": 2603.32,
+      "text": "then we can estimate m m of x, so this"
+    },
+    {
+      "start": 2608.08,
+      "text": "tensor m of x which is the missing"
+    },
+    {
+      "start": 2610.88,
+      "text": "element for so in our stochastic model"
+    },
+    {
+      "start": 2615.44,
+      "text": "by essentially using this relationship"
+    },
+    {
+      "start": 2617.76,
+      "text": "over here."
+    },
+    {
+      "start": 2619.0,
+      "text": "So using this relationship here"
+    },
+    {
+      "start": 2621.56,
+      "text": "and a library of observable which is"
+    },
+    {
+      "start": 2624.0,
+      "text": "rich enough, we can estimate both phi"
+    },
+    {
+      "start": 2627.2,
+      "text": "and delta m."
+    },
+    {
+      "start": 2629.36,
+      "text": "So let's see now how we can estimate phi"
+    },
+    {
+      "start": 2632.2,
+      "text": "first."
+    },
+    {
+      "start": 2633.56,
+      "text": "So to do that, let's consider"
+    },
+    {
+      "start": 2636.72,
+      "text": "just the coordinate observable. Okay, so"
+    },
+    {
+      "start": 2639.4,
+      "text": "we have here in theory like a very large"
+    },
+    {
+      "start": 2641.72,
+      "text": "libraries of observable. Now let's focus"
+    },
+    {
+      "start": 2644.6,
+      "text": "on a few of them and few of them of x"
+    },
+    {
+      "start": 2648.12,
+      "text": "equal to x. So we're just considering"
+    },
+    {
+      "start": 2650.8,
+      "text": "the observable coordinate."
+    },
+    {
+      "start": 2654.52,
+      "text": "By doing this replacement, the first"
+    },
+    {
+      "start": 2657.04,
+      "text": "term becomes this expected value"
+    },
+    {
+      "start": 2660.24,
+      "text": "multiplied by phi."
+    },
+    {
+      "start": 2661.92,
+      "text": "Now if we consider t equal to 0,"
+    },
+    {
+      "start": 2666.32,
+      "text": "then this"
+    },
+    {
+      "start": 2668.6,
+      "text": "expected value becomes minus the"
+    },
+    {
+      "start": 2671.0,
+      "text": "identity because of the Stein identity."
+    },
+    {
+      "start": 2675.56,
+      "text": "This term here becomes equal to 0"
+    },
+    {
+      "start": 2678.64,
+      "text": "because we have so you can just"
+    },
+    {
+      "start": 2681.28,
+      "text": "integrate the conditional score term at"
+    },
+    {
+      "start": 2684.96,
+      "text": "t equal to 0 and you will get 0."
+    },
+    {
+      "start": 2687.12,
+      "text": "So this essentially means that if if you"
+    },
+    {
+      "start": 2689.0,
+      "text": "consider the coordinate observable and t"
+    },
+    {
+      "start": 2692.0,
+      "text": "equal to 0, we are able to derive a"
+    },
+    {
+      "start": 2695.12,
+      "text": "relationship for phi."
+    },
+    {
+      "start": 2698.04,
+      "text": "So we can essentially fix the average"
+    },
+    {
+      "start": 2700.48,
+      "text": "value of the matrix m"
+    },
+    {
+      "start": 2704.44,
+      "text": "and we can write it in terms of the"
+    },
+    {
+      "start": 2708.48,
+      "text": "coordinate the time derivative of the"
+    },
+    {
+      "start": 2710.8,
+      "text": "coordinate correlation at t equal to 0."
+    },
+    {
+      "start": 2715.79,
+      "text": "correlation at t equal to 0."
+    },
+    {
+      "start": 2715.8,
+      "text": "Now let's see what this implies. So we"
+    },
+    {
+      "start": 2718.84,
+      "text": "have then"
+    },
+    {
+      "start": 2720.52,
+      "text": "fixed the phi. We have this additional"
+    },
+    {
+      "start": 2723.44,
+      "text": "term, this correction term e of x."
+    },
+    {
+      "start": 2727.92,
+      "text": "When we consider"
+    },
+    {
+      "start": 2731.44,
+      "text": "phi phi phi phi 1 of x equal to x, we"
+    },
+    {
+      "start": 2734.76,
+      "text": "then have this coordinate term at the"
+    },
+    {
+      "start": 2737.48,
+      "text": "beginning."
+    },
+    {
+      "start": 2738.96,
+      "text": "We can rewrite this expected value in"
+    },
+    {
+      "start": 2741.68,
+      "text": "terms of the gradient with respect to x"
+    },
+    {
+      "start": 2743.92,
+      "text": "0 of the expected value of x of t"
+    },
+    {
+      "start": 2747.44,
+      "text": "conditioned on x 0 multiplied by delta"
+    },
+    {
+      "start": 2750.6,
+      "text": "m."
+    },
+    {
+      "start": 2752.32,
+      "text": "So by just considering the coordinate"
+    },
+    {
+      "start": 2755.24,
+      "text": "observable case, we can derive phi"
+    },
+    {
+      "start": 2759.28,
+      "text": "and then we can write this relationship"
+    },
+    {
+      "start": 2761.84,
+      "text": "for the correction term. But at this"
+    },
+    {
+      "start": 2764.359,
+      "text": "point we can notice that if"
+    },
+    {
+      "start": 2767.44,
+      "text": "m of x, so if the expected value of x of"
+    },
+    {
+      "start": 2770.92,
+      "text": "t conditioned on x 0 is approximately"
+    },
+    {
+      "start": 2775.96,
+      "text": "affine which essentially means if we can"
+    },
+    {
+      "start": 2778.6,
+      "text": "write the expected value of x t"
+    },
+    {
+      "start": 2781.24,
+      "text": "conditioned on x 0"
+    },
+    {
+      "start": 2783.56,
+      "text": "in terms of a linear function of x 0,"
+    },
+    {
+      "start": 2787.359,
+      "text": "then when we take the gradient, we will"
+    },
+    {
+      "start": 2789.52,
+      "text": "get a constant term with respect to x"
+    },
+    {
+      "start": 2792.6,
+      "text": "and then by construction the average"
+    },
+    {
+      "start": 2795.2,
+      "text": "value of x of delta m is equal to 0"
+    },
+    {
+      "start": 2799.52,
+      "text": "which essentially means that if the"
+    },
+    {
+      "start": 2802.44,
+      "text": "conditional mean is approximately affine"
+    },
+    {
+      "start": 2806.16,
+      "text": "which essentially"
+    },
+    {
+      "start": 2807.8,
+      "text": "is is is the case if"
+    },
+    {
+      "start": 2810.96,
+      "text": "the joint probability density function"
+    },
+    {
+      "start": 2814.44,
+      "text": "of x x t is a Gaussian,"
+    },
+    {
+      "start": 2818.92,
+      "text": "we can then use so we can then replace"
+    },
+    {
+      "start": 2823.68,
+      "text": "our matrix m of x which is state"
+    },
+    {
+      "start": 2827.16,
+      "text": "dependent with just the matrix phi"
+    },
+    {
+      "start": 2830.76,
+      "text": "and we have a model that by construction"
+    },
+    {
+      "start": 2833.6,
+      "text": "reproduces both the temporal"
+    },
+    {
+      "start": 2835.88,
+      "text": "correlations and the steady state"
+    },
+    {
+      "start": 2839.92,
+      "text": "Okay, so if"
+    },
+    {
+      "start": 2841.8,
+      "text": "this term so if m of t is linear in x 0"
+    },
+    {
+      "start": 2846.6,
+      "text": "which is often the case because if the"
+    },
+    {
+      "start": 2849.48,
+      "text": "conditional probability density so if"
+    },
+    {
+      "start": 2852.6,
+      "text": "the joint probability density of x 0 and"
+    },
+    {
+      "start": 2855.68,
+      "text": "and x t can be approximated with a"
+    },
+    {
+      "start": 2857.64,
+      "text": "Gaussian distribution, then m of t"
+    },
+    {
+      "start": 2860.88,
+      "text": "depends linearly on x 0."
+    },
+    {
+      "start": 2863.84,
+      "text": "So if this term is negligible,"
+    },
+    {
+      "start": 2867.48,
+      "text": "then we can"
+    },
+    {
+      "start": 2869.359,
+      "text": "reproduce the time correlations of the"
+    },
+    {
+      "start": 2872.12,
+      "text": "observed data just using phi, so this"
+    },
+    {
+      "start": 2875.44,
+      "text": "constant matrix phi that we can easily"
+    },
+    {
+      "start": 2877.2,
+      "text": "determine from the correlation function"
+    },
+    {
+      "start": 2879.72,
+      "text": "instead of the state dependent matrix m."
+    },
+    {
+      "start": 2883.08,
+      "text": "And then we have built a Langevin"
+    },
+    {
+      "start": 2884.76,
+      "text": "equation that by construction reproduces"
+    },
+    {
+      "start": 2887.32,
+      "text": "both the steady state distribution and"
+    },
+    {
+      "start": 2889.84,
+      "text": "the time correlations."
+    },
+    {
+      "start": 2891.64,
+      "text": "If we want instead to add more"
+    },
+    {
+      "start": 2895.6,
+      "text": "constraint on the correlations, so we"
+    },
+    {
+      "start": 2898.28,
+      "text": "want to add more constraints on the"
+    },
+    {
+      "start": 2899.84,
+      "text": "dynamics adding more correlations, then"
+    },
+    {
+      "start": 2903.32,
+      "text": "we have to obtain"
+    },
+    {
+      "start": 2906.32,
+      "text": "also the matrix delta m that we can"
+    },
+    {
+      "start": 2910.0,
+      "text": "parameterize with a neural network."
+    },
+    {
+      "start": 2912.64,
+      "text": "So in this specific case, we"
+    },
+    {
+      "start": 2914.28,
+      "text": "parameterize the whole m of x with a"
+    },
+    {
+      "start": 2916.92,
+      "text": "neural network and then we define delta"
+    },
+    {
+      "start": 2919.6,
+      "text": "m of x as m theta minus phi"
+    },
+    {
+      "start": 2923.16,
+      "text": "and then we can train a neural network"
+    },
+    {
+      "start": 2926.0,
+      "text": "delta m of theta to minimize this loss"
+    },
+    {
+      "start": 2929.24,
+      "text": "function. So we have this first term"
+    },
+    {
+      "start": 2932.08,
+      "text": "that that essentially forces the neural"
+    },
+    {
+      "start": 2934.28,
+      "text": "network to learn the set of correlation"
+    },
+    {
+      "start": 2938.24,
+      "text": "functions that we want our system to"
+    },
+    {
+      "start": 2940.52,
+      "text": "reproduce. Then we have this penalty"
+    },
+    {
+      "start": 2942.92,
+      "text": "term that essentially enforces that the"
+    },
+    {
+      "start": 2945.68,
+      "text": "average value of delta m is equal to 0"
+    },
+    {
+      "start": 2949.44,
+      "text": "plus we have a regularization term."
+    },
+    {
+      "start": 2952.96,
+      "text": "But you can see here that we are so we"
+    },
+    {
+      "start": 2955.64,
+      "text": "are writing"
+    },
+    {
+      "start": 2957.4,
+      "text": "a loss function that doesn't depend on a"
+    },
+    {
+      "start": 2960.16,
+      "text": "forward model integration. So we never"
+    },
+    {
+      "start": 2962.32,
+      "text": "have to integrate our Langevin equation"
+    },
+    {
+      "start": 2966.16,
+      "text": "forward in time."
+    },
+    {
+      "start": 2967.96,
+      "text": "We just use the the knowledge of the"
+    },
+    {
+      "start": 2971.12,
+      "text": "conditional score, the score function,"
+    },
+    {
+      "start": 2974.2,
+      "text": "and the time derivative of the"
+    },
+    {
+      "start": 2976.56,
+      "text": "correlation functions to train the"
+    },
+    {
+      "start": 2978.88,
+      "text": "neural network for delta M."
+    },
+    {
+      "start": 2981.68,
+      "text": "And this can be can become extremely"
+    },
+    {
+      "start": 2984.0,
+      "text": "efficient when the model that we want to"
+    },
+    {
+      "start": 2987.52,
+      "text": "integrate becomes very computationally"
+    },
+    {
+      "start": 2989.72,
+      "text": "expensive."
+    },
+    {
+      "start": 2991.28,
+      "text": "Okay, so this is the methodology."
+    },
+    {
+      "start": 2993.64,
+      "text": "So we have seen that's"
+    },
+    {
+      "start": 2995.8,
+      "text": "yeah, we we are able to train this"
+    },
+    {
+      "start": 2997.48,
+      "text": "neural network without integrating the"
+    },
+    {
+      "start": 2999.36,
+      "text": "model forward and when so for some"
+    },
+    {
+      "start": 3002.6,
+      "text": "specific cases we can simplify the shape"
+    },
+    {
+      "start": 3007.16,
+      "text": "so the functional form of M"
+    },
+    {
+      "start": 3010.04,
+      "text": "by replacing them with a constant if we"
+    },
+    {
+      "start": 3012.68,
+      "text": "are just interested in the time"
+    },
+    {
+      "start": 3014.28,
+      "text": "correlation of the systems. So now I"
+    },
+    {
+      "start": 3016.96,
+      "text": "will conclude showing you some"
+    },
+    {
+      "start": 3019.0,
+      "text": "application of these ideas."
+    },
+    {
+      "start": 3021.16,
+      "text": "So I start from an analytic warm up. So"
+    },
+    {
+      "start": 3024.96,
+      "text": "we consider this one-dimensional system"
+    },
+    {
+      "start": 3028.64,
+      "text": "for which we can determine analytically"
+    },
+    {
+      "start": 3032.48,
+      "text": "all the relevant quantities."
+    },
+    {
+      "start": 3035.96,
+      "text": "So we can derive the station the"
+    },
+    {
+      "start": 3038.24,
+      "text": "conditional score, the stationary score,"
+    },
+    {
+      "start": 3042.0,
+      "text": "the time derivative of the correlation"
+    },
+    {
+      "start": 3043.84,
+      "text": "functions and so on."
+    },
+    {
+      "start": 3045.8,
+      "text": "These are the true values for fee and"
+    },
+    {
+      "start": 3049.36,
+      "text": "delta M."
+    },
+    {
+      "start": 3051.28,
+      "text": "And then by applying the method I"
+    },
+    {
+      "start": 3054.3,
+      "text": "&gt;&gt; [clears throat]"
+    },
+    {
+      "start": 3054.8,
+      "text": "&gt;&gt; discussed before we can obtain them"
+    },
+    {
+      "start": 3057.51,
+      "text": "discussed before we can obtain them"
+    },
+    {
+      "start": 3057.52,
+      "text": "using the relationship that I showed you"
+    },
+    {
+      "start": 3060.16,
+      "text": "at the beginning. So using that"
+    },
+    {
+      "start": 3061.64,
+      "text": "relationship we recover precisely"
+    },
+    {
+      "start": 3064.6,
+      "text": "the fee the correct fee and the correct"
+    },
+    {
+      "start": 3069.24,
+      "text": "So this was just like a test where we"
+    },
+    {
+      "start": 3071.4,
+      "text": "have we know everything is analytically."
+    },
+    {
+      "start": 3073.76,
+      "text": "So let's see a different case. In this"
+    },
+    {
+      "start": 3076.32,
+      "text": "case we have a two-dimensional system"
+    },
+    {
+      "start": 3079.24,
+      "text": "with where we have our drift term which"
+    },
+    {
+      "start": 3082.88,
+      "text": "contains both a term that can be written"
+    },
+    {
+      "start": 3085.44,
+      "text": "as the gradient of a potential plus a"
+    },
+    {
+      "start": 3087.44,
+      "text": "circulatory component. We also have a"
+    },
+    {
+      "start": 3092.44,
+      "text": "And in this case we cannot write"
+    },
+    {
+      "start": 3094.36,
+      "text": "explicitly the score function and the"
+    },
+    {
+      "start": 3098.68,
+      "text": "conditional score. So we need to train"
+    },
+    {
+      "start": 3101.36,
+      "text": "two neural networks for S and for the"
+    },
+    {
+      "start": 3103.88,
+      "text": "conditional score."
+    },
+    {
+      "start": 3106.44,
+      "text": "We apply the methodology that I"
+    },
+    {
+      "start": 3108.08,
+      "text": "described"
+    },
+    {
+      "start": 3109.2,
+      "text": "before by enforcing the reproduction of"
+    },
+    {
+      "start": 3112.68,
+      "text": "the correlation functions. We derive"
+    },
+    {
+      "start": 3116.44,
+      "text": "a quite accurate"
+    },
+    {
+      "start": 3118.72,
+      "text": "reconstruction of"
+    },
+    {
+      "start": 3121.12,
+      "text": "the mobility fields so the M"
+    },
+    {
+      "start": 3125.52,
+      "text": "tensor."
+    },
+    {
+      "start": 3127.52,
+      "text": "We have some errors in particular in"
+    },
+    {
+      "start": 3129.72,
+      "text": "this term"
+    },
+    {
+      "start": 3131.2,
+      "text": "but even if so we have like some errors"
+    },
+    {
+      "start": 3135.56,
+      "text": "for"
+    },
+    {
+      "start": 3137.28,
+      "text": "the M to one terms when we integrate our"
+    },
+    {
+      "start": 3142.12,
+      "text": "model we get a precise recovery of the"
+    },
+    {
+      "start": 3146.12,
+      "text": "univariate PDF, bivariate PDF"
+    },
+    {
+      "start": 3149.2,
+      "text": "all the correlation functions. Here I'm"
+    },
+    {
+      "start": 3151.8,
+      "text": "comparing two different model"
+    },
+    {
+      "start": 3153.84,
+      "text": "integrations. We have the model"
+    },
+    {
+      "start": 3156.56,
+      "text": "integration with the full"
+    },
+    {
+      "start": 3158.91,
+      "text": "with the full"
+    },
+    {
+      "start": 3158.92,
+      "text": "mobility matrix M of X"
+    },
+    {
+      "start": 3161.6,
+      "text": "and a model integration where I'm"
+    },
+    {
+      "start": 3164.12,
+      "text": "replacing the full mobility matrix with"
+    },
+    {
+      "start": 3167.08,
+      "text": "a fee so with this constant closure that"
+    },
+    {
+      "start": 3169.6,
+      "text": "I introduced before."
+    },
+    {
+      "start": 3171.8,
+      "text": "We can see here that using the full"
+    },
+    {
+      "start": 3174.6,
+      "text": "mobility matrix obtained by training"
+    },
+    {
+      "start": 3178.28,
+      "text": "a neural network for M we get a more"
+    },
+    {
+      "start": 3180.76,
+      "text": "precise recovery of the correlation"
+    },
+    {
+      "start": 3184.08,
+      "text": "functions in particular for this cross"
+    },
+    {
+      "start": 3186.24,
+      "text": "correlation."
+    },
+    {
+      "start": 3187.84,
+      "text": "And also if I now consider the target"
+    },
+    {
+      "start": 3192.56,
+      "text": "dynamical observables so the target"
+    },
+    {
+      "start": 3196.36,
+      "text": "correlation functions that I used to"
+    },
+    {
+      "start": 3198.52,
+      "text": "train the neural network when I evaluate"
+    },
+    {
+      "start": 3201.12,
+      "text": "them from the trajectory that I obtained"
+    },
+    {
+      "start": 3205.56,
+      "text": "by integrating my model"
+    },
+    {
+      "start": 3209.36,
+      "text": "I get yeah a quite better"
+    },
+    {
+      "start": 3211.96,
+      "text": "recovery with respect to the constant M"
+    },
+    {
+      "start": 3215.64,
+      "text": "matrix closure."
+    },
+    {
+      "start": 3217.6,
+      "text": "So essentially this is to show that yeah"
+    },
+    {
+      "start": 3219.88,
+      "text": "by"
+    },
+    {
+      "start": 3221.0,
+      "text": "applying this algorithm"
+    },
+    {
+      "start": 3223.52,
+      "text": "we are able to estimate the mobility"
+    },
+    {
+      "start": 3227.0,
+      "text": "matrix M of X together with the score"
+    },
+    {
+      "start": 3230.12,
+      "text": "and the conditional score then combining"
+    },
+    {
+      "start": 3232.4,
+      "text": "those pieces together"
+    },
+    {
+      "start": 3234.359,
+      "text": "we obtain an expression for the drift"
+    },
+    {
+      "start": 3237.48,
+      "text": "term that is able"
+    },
+    {
+      "start": 3239.68,
+      "text": "to reproduce the steady state density"
+    },
+    {
+      "start": 3243.44,
+      "text": "the time correlations together with all"
+    },
+    {
+      "start": 3247.08,
+      "text": "the correlations that we enforced"
+    },
+    {
+      "start": 3250.24,
+      "text": "in the training."
+    },
+    {
+      "start": 3251.8,
+      "text": "Okay, so now let's consider more"
+    },
+    {
+      "start": 3253.48,
+      "text": "high-dimensional systems."
+    },
+    {
+      "start": 3255.359,
+      "text": "So for the next two systems I will only"
+    },
+    {
+      "start": 3258.4,
+      "text": "consider the constant closure for M. So"
+    },
+    {
+      "start": 3261.4,
+      "text": "essentially I approximate M of X with"
+    },
+    {
+      "start": 3264.28,
+      "text": "its average value so with fee."
+    },
+    {
+      "start": 3266.92,
+      "text": "In this case I'm integrating this"
+    },
+    {
+      "start": 3268.76,
+      "text": "Kuramoto-Sivashinsky PDE."
+    },
+    {
+      "start": 3271.4,
+      "text": "I'm integrating this partial"
+    },
+    {
+      "start": 3273.28,
+      "text": "differential equation with 512 Fourier"
+    },
+    {
+      "start": 3276.0,
+      "text": "modes. I obtain a 1024-dimensional"
+    },
+    {
+      "start": 3282.04,
+      "text": "um time series. I'm"
+    },
+    {
+      "start": 3284.92,
+      "text": "considering just one"
+    },
+    {
+      "start": 3287.8,
+      "text": "mode every 32. So essentially I'm"
+    },
+    {
+      "start": 3291.24,
+      "text": "subsampling this 1024-dimensional"
+    },
+    {
+      "start": 3294.28,
+      "text": "state to a 32-dimensional state."
+    },
+    {
+      "start": 3297.96,
+      "text": "And then using those"
+    },
+    {
+      "start": 3299.56,
+      "text": "those 32-dimensional modes to build my"
+    },
+    {
+      "start": 3302.64,
+      "text": "Langevin equation. So essentially here"
+    },
+    {
+      "start": 3305.04,
+      "text": "I'm"
+    },
+    {
+      "start": 3306.359,
+      "text": "building so I'm starting from a fully"
+    },
+    {
+      "start": 3309.44,
+      "text": "fully deterministic partial differential"
+    },
+    {
+      "start": 3312.28,
+      "text": "equation partial"
+    },
+    {
+      "start": 3313.92,
+      "text": "which is partially observed"
+    },
+    {
+      "start": 3316.16,
+      "text": "and then using a completely different"
+    },
+    {
+      "start": 3318.84,
+      "text": "model to"
+    },
+    {
+      "start": 3321.68,
+      "text": "be so to predict its dynamics using my"
+    },
+    {
+      "start": 3327.12,
+      "text": "stochastic closure."
+    },
+    {
+      "start": 3329.16,
+      "text": "And here are the results. So this is the"
+    },
+    {
+      "start": 3333.04,
+      "text": "time series obtained by integrating my"
+    },
+    {
+      "start": 3335.64,
+      "text": "Langevin equation. This is the real"
+    },
+    {
+      "start": 3339.52,
+      "text": "observed time series and here I'm"
+    },
+    {
+      "start": 3342.32,
+      "text": "plotting the comparison between the"
+    },
+    {
+      "start": 3346.8,
+      "text": "the bivariate and the univariate PDFs"
+    },
+    {
+      "start": 3349.48,
+      "text": "obtained from the observations and the"
+    },
+    {
+      "start": 3353.64,
+      "text": "one obtained from"
+    },
+    {
+      "start": 3355.72,
+      "text": "a model integration of my Langevin"
+    },
+    {
+      "start": 3357.4,
+      "text": "equation. And here instead is the"
+    },
+    {
+      "start": 3359.4,
+      "text": "autocorrelation function for both the"
+    },
+    {
+      "start": 3363.08,
+      "text": "observations and my Langevin"
+    },
+    {
+      "start": 3365.04,
+      "text": "integration."
+    },
+    {
+      "start": 3366.64,
+      "text": "Then finally I considered"
+    },
+    {
+      "start": 3369.72,
+      "text": "the sea surface temperature data from"
+    },
+    {
+      "start": 3372.2,
+      "text": "Plasim so which is a um"
+    },
+    {
+      "start": 3375.68,
+      "text": "a global circulation model of"
+    },
+    {
+      "start": 3377.68,
+      "text": "intermediate"
+    },
+    {
+      "start": 3379.28,
+      "text": "intermediate complexity."
+    },
+    {
+      "start": 3381.71,
+      "text": "complexity."
+    },
+    {
+      "start": 3383.76,
+      "text": "the data for the sea surface"
+    },
+    {
+      "start": 3385.84,
+      "text": "for the global sea surface temperature"
+    },
+    {
+      "start": 3388.88,
+      "text": "evolution and I want a model that is"
+    },
+    {
+      "start": 3392.72,
+      "text": "able essentially to predict and to model"
+    },
+    {
+      "start": 3395.84,
+      "text": "this sea surface temperature data."
+    },
+    {
+      "start": 3398.28,
+      "text": "So the data set is around"
+    },
+    {
+      "start": 3401.04,
+      "text": "2000-dimensional."
+    },
+    {
+      "start": 3403.08,
+      "text": "I did a dimensionality reduction taking"
+    },
+    {
+      "start": 3406.2,
+      "text": "the first 20 principal components."
+    },
+    {
+      "start": 3409.84,
+      "text": "And here since I have a strong"
+    },
+    {
+      "start": 3411.56,
+      "text": "periodicity I augmented the state space"
+    },
+    {
+      "start": 3414.8,
+      "text": "by including some harmonic functions."
+    },
+    {
+      "start": 3418.52,
+      "text": "And these are the results."
+    },
+    {
+      "start": 3421.359,
+      "text": "I yeah was predicting the probability"
+    },
+    {
+      "start": 3425.359,
+      "text": "the conditional probability density"
+    },
+    {
+      "start": 3428.0,
+      "text": "of the"
+    },
+    {
+      "start": 3431.56,
+      "text": "20 principal components together with"
+    },
+    {
+      "start": 3434.76,
+      "text": "their autocovariance. So"
+    },
+    {
+      "start": 3437.32,
+      "text": "as you can see we can have a quite"
+    },
+    {
+      "start": 3439.28,
+      "text": "decent"
+    },
+    {
+      "start": 3440.44,
+      "text": "reconstruction of the PDFs and the ACFs"
+    },
+    {
+      "start": 3445.76,
+      "text": "of all the 20 principal components. Here"
+    },
+    {
+      "start": 3449.52,
+      "text": "I'm plotting just the first 10."
+    },
+    {
+      "start": 3452.12,
+      "text": "And also we were able to capture the"
+    },
+    {
+      "start": 3456.2,
+      "text": "nonlinear so and the non-Gaussian"
+    },
+    {
+      "start": 3458.64,
+      "text": "probability density function evaluated"
+    },
+    {
+      "start": 3461.84,
+      "text": "at every grid point"
+    },
+    {
+      "start": 3464.08,
+      "text": "um from our simulation. So here"
+    },
+    {
+      "start": 3465.92,
+      "text": "essentially I'm plotting the probability"
+    },
+    {
+      "start": 3468.2,
+      "text": "density at different season"
+    },
+    {
+      "start": 3470.6,
+      "text": "of the temperature at a given grid point"
+    },
+    {
+      "start": 3474.92,
+      "text": "and I'm doing that using the full"
+    },
+    {
+      "start": 3478.0,
+      "text": "observation so essentially all the"
+    },
+    {
+      "start": 3479.88,
+      "text": "principal components"
+    },
+    {
+      "start": 3481.52,
+      "text": "just the first 20 principal components"
+    },
+    {
+      "start": 3485.0,
+      "text": "and"
+    },
+    {
+      "start": 3486.6,
+      "text": "the 20-dimensional"
+    },
+    {
+      "start": 3489.0,
+      "text": "stochastic model that I trained on these"
+    },
+    {
+      "start": 3492.2,
+      "text": "first 20 components and I integrated"
+    },
+    {
+      "start": 3495.0,
+      "text": "forward."
+    },
+    {
+      "start": 3496.28,
+      "text": "And as you can see so even if I did a"
+    },
+    {
+      "start": 3498.56,
+      "text": "dimensionality reduction of the"
+    },
+    {
+      "start": 3501.32,
+      "text": "data set I was still able to get this"
+    },
+    {
+      "start": 3504.28,
+      "text": "nonlinear probability density functions"
+    },
+    {
+      "start": 3508.2,
+      "text": "using this"
+    },
+    {
+      "start": 3510.04,
+      "text": "quite simple"
+    },
+    {
+      "start": 3511.68,
+      "text": "stochastic model that I built using this"
+    },
+    {
+      "start": 3516.32,
+      "text": "constant closure for my"
+    },
+    {
+      "start": 3519.56,
+      "text": "mobility matrix."
+    },
+    {
+      "start": 3521.96,
+      "text": "Okay, so these are"
+    },
+    {
+      "start": 3524.12,
+      "text": "some of the papers on"
+    },
+    {
+      "start": 3526.92,
+      "text": "that I so either published or put on"
+    },
+    {
+      "start": 3529.2,
+      "text": "archive on this topic. We tried the"
+    },
+    {
+      "start": 3531.64,
+      "text": "different directions that I haven't"
+    },
+    {
+      "start": 3534.0,
+      "text": "presented here."
+    },
+    {
+      "start": 3535.8,
+      "text": "But"
+    },
+    {
+      "start": 3537.76,
+      "text": "so these were so the main references"
+    },
+    {
+      "start": 3542.44,
+      "text": "and to conclude so we have seen"
+    },
+    {
+      "start": 3545.0,
+      "text": "how to model high-dimensional"
+    },
+    {
+      "start": 3548.64,
+      "text": "partially observed chaotic systems"
+    },
+    {
+      "start": 3553.359,
+      "text": "how the knowledge of the score function"
+    },
+    {
+      "start": 3556.92,
+      "text": "plays a key role in allowing this"
+    },
+    {
+      "start": 3561.0,
+      "text": "modeling this modeling strategies. We"
+    },
+    {
+      "start": 3563.68,
+      "text": "have seen two different directions. In"
+    },
+    {
+      "start": 3565.52,
+      "text": "the first one we have a model answers"
+    },
+    {
+      "start": 3568.48,
+      "text": "and we are just"
+    },
+    {
+      "start": 3570.48,
+      "text": "calibrating the model parameters using a"
+    },
+    {
+      "start": 3573.08,
+      "text": "combination between the generalized"
+    },
+    {
+      "start": 3575.64,
+      "text": "fluctuation distribution theorem and the"
+    },
+    {
+      "start": 3579.12,
+      "text": "score modeling."
+    },
+    {
+      "start": 3581.0,
+      "text": "The other direction instead"
+    },
+    {
+      "start": 3583.24,
+      "text": "doesn't have any model answer."
+    },
+    {
+      "start": 3587.44,
+      "text": "We just try to"
+    },
+    {
+      "start": 3590.0,
+      "text": "starting from a set of statistical and"
+    },
+    {
+      "start": 3592.28,
+      "text": "dynamical observables to build a model"
+    },
+    {
+      "start": 3594.76,
+      "text": "that by construction reproduces all of"
+    },
+    {
+      "start": 3597.04,
+      "text": "them without integrating the model"
+    },
+    {
+      "start": 3600.84,
+      "text": "And then we've seen how this approach"
+    },
+    {
+      "start": 3603.52,
+      "text": "can scale on different systems from toy"
+    },
+    {
+      "start": 3606.92,
+      "text": "models to very high dimensional systems."
+    },
+    {
+      "start": 3611.48,
+      "text": "Okay, thanks for listening and let me"
+    },
+    {
+      "start": 3614.0,
+      "text": "know if you have"
+    },
+    {
+      "start": 3615.24,
+      "text": "any question."
+    },
+    {
+      "start": 3618.48,
+      "text": "Thank you, Ludovico."
+    },
+    {
+      "start": 3620.2,
+      "text": "Any questions?"
+    },
+    {
+      "start": 3622.08,
+      "text": "I have a question. So,"
+    },
+    {
+      "start": 3625.52,
+      "text": "So, I'm wondering according to your"
+    },
+    {
+      "start": 3627.32,
+      "text": "formulation, is does your method allow"
+    },
+    {
+      "start": 3630.08,
+      "text": "you that"
+    },
+    {
+      "start": 3631.44,
+      "text": "have allow you"
+    },
+    {
+      "start": 3634.08,
+      "text": "to to work on data set that has"
+    },
+    {
+      "start": 3636.12,
+      "text": "absolutely no time information?"
+    },
+    {
+      "start": 3639.4,
+      "text": "What do you mean with absolutely no time"
+    },
+    {
+      "start": 3641.72,
+      "text": "information? So, like"
+    },
+    {
+      "start": 3643.96,
+      "text": "time series uh"
+    },
+    {
+      "start": 3646.16,
+      "text": "where every snapshot is completely"
+    },
+    {
+      "start": 3648.84,
+      "text": "uncorrelated? Yeah, yeah. In that case,"
+    },
+    {
+      "start": 3652.32,
+      "text": "yes. So, so you can do it, but you will"
+    },
+    {
+      "start": 3656.32,
+      "text": "be able to build a mathematical model"
+    },
+    {
+      "start": 3658.6,
+      "text": "that reproduces the steady state"
+    },
+    {
+      "start": 3661.6,
+      "text": "but not the dynamics because you don't"
+    },
+    {
+      "start": 3663.68,
+      "text": "have any information about the dynamics."
+    },
+    {
+      "start": 3666.32,
+      "text": "So, what you can do in in that case"
+    },
+    {
+      "start": 3669.0,
+      "text": "and that will be yeah, much more simple,"
+    },
+    {
+      "start": 3674.04,
+      "text": "is to replace"
+    },
+    {
+      "start": 3675.84,
+      "text": "here uh"
+    },
+    {
+      "start": 3677.68,
+      "text": "M of X just with the identity, right?"
+    },
+    {
+      "start": 3681.52,
+      "text": "So, if you're only caring about the"
+    },
+    {
+      "start": 3683.84,
+      "text": "steady state distribution and also you"
+    },
+    {
+      "start": 3686.6,
+      "text": "don't have any information"
+    },
+    {
+      "start": 3688.72,
+      "text": "to build the"
+    },
+    {
+      "start": 3690.84,
+      "text": "so so"
+    },
+    {
+      "start": 3691.96,
+      "text": "to estimate the correlation functions,"
+    },
+    {
+      "start": 3694.8,
+      "text": "then it essentially means that any So,"
+    },
+    {
+      "start": 3696.64,
+      "text": "you cannot infer M of X because M of X"
+    },
+    {
+      "start": 3699.16,
+      "text": "is carrying information about the"
+    },
+    {
+      "start": 3700.76,
+      "text": "dynamics. So, you can replace M of X"
+    },
+    {
+      "start": 3703.84,
+      "text": "with the identity."
+    },
+    {
+      "start": 3705.56,
+      "text": "Uh so so you So, if I train a model, I"
+    },
+    {
+      "start": 3708.4,
+      "text": "only need to train the M, right?"
+    },
+    {
+      "start": 3710.92,
+      "text": "Uh if you train So, if you only want to"
+    },
+    {
+      "start": 3713.6,
+      "text": "reproduce the steady state distribution"
+    },
+    {
+      "start": 3715.8,
+      "text": "because you don't have information about"
+    },
+    {
+      "start": 3718.72,
+      "text": "the dynamics,"
+    },
+    {
+      "start": 3720.24,
+      "text": "you just need to train a neural network"
+    },
+    {
+      "start": 3723.52,
+      "text": "to learn the score function."
+    },
+    {
+      "start": 3726.48,
+      "text": "So, M can be just replaced with the"
+    },
+    {
+      "start": 3728.76,
+      "text": "identity."
+    },
+    {
+      "start": 3730.6,
+      "text": "Hm. Okay, because then so any value So,"
+    },
+    {
+      "start": 3733.2,
+      "text": "any shape of M of X will uh"
+    },
+    {
+      "start": 3736.56,
+      "text": "uh give you"
+    },
+    {
+      "start": 3739.64,
+      "text": "the correct steady state distribution."
+    },
+    {
+      "start": 3742.36,
+      "text": "So, you can just choose M of X equal to"
+    },
+    {
+      "start": 3745.12,
+      "text": "the identity."
+    },
+    {
+      "start": 3747.2,
+      "text": "Also, if you choose M of X equal to the"
+    },
+    {
+      "start": 3749.52,
+      "text": "identity, it probably is an optimal"
+    },
+    {
+      "start": 3751.32,
+      "text": "choice because uh"
+    },
+    {
+      "start": 3753.56,
+      "text": "um you have the fastest convergence"
+    },
+    {
+      "start": 3758.88,
+      "text": "towards the steady state density. So,"
+    },
+    {
+      "start": 3761.32,
+      "text": "like if you integrate your model,"
+    },
+    {
+      "start": 3766.6,
+      "text": "quite fast towards the steady state"
+    },
+    {
+      "start": 3768.68,
+      "text": "density. So, if instead M of X is a"
+    },
+    {
+      "start": 3771.92,
+      "text": "constant matrix uh"
+    },
+    {
+      "start": 3773.92,
+      "text": "with a wide um"
+    },
+    {
+      "start": 3776.28,
+      "text": "variety [clears throat]"
+    },
+    {
+      "start": 3778.36,
+      "text": "wide amplitude in the eigenvalues,"
+    },
+    {
+      "start": 3782.24,
+      "text": "you have essentially that some modes"
+    },
+    {
+      "start": 3784.4,
+      "text": "will decay faster than others and so you"
+    },
+    {
+      "start": 3787.4,
+      "text": "have to wait like a longer time to see"
+    },
+    {
+      "start": 3791.0,
+      "text": "thermalization of the system towards the"
+    },
+    {
+      "start": 3796.0,
+      "text": "Yeah, this is very interesting because"
+    },
+    {
+      "start": 3797.64,
+      "text": "we we previously have a"
+    },
+    {
+      "start": 3799.92,
+      "text": "have a paper that targeting exactly on"
+    },
+    {
+      "start": 3803.08,
+      "text": "no time information and we we got some"
+    },
+    {
+      "start": 3805.92,
+      "text": "difficulty when we move from low"
+    },
+    {
+      "start": 3807.96,
+      "text": "dimension like two or three to to"
+    },
+    {
+      "start": 3810.72,
+      "text": "thousands of dimension. In thousands of"
+    },
+    {
+      "start": 3812.72,
+      "text": "dimension, our method basically uh"
+    },
+    {
+      "start": 3815.24,
+      "text": "almost failed and uh"
+    },
+    {
+      "start": 3817.52,
+      "text": "so yeah, so I'm wondering"
+    },
+    {
+      "start": 3819.8,
+      "text": "if your method can can be helpful. Yeah,"
+    },
+    {
+      "start": 3823.24,
+      "text": "so we estimated the score function also"
+    },
+    {
+      "start": 3825.4,
+      "text": "for thousand dimensional systems and"
+    },
+    {
+      "start": 3828.16,
+      "text": "yeah, like it's not a problem."
+    },
+    {
+      "start": 3830.56,
+      "text": "But which"
+    },
+    {
+      "start": 3831.92,
+      "text": "So, how have you done this? So, did you"
+    },
+    {
+      "start": 3833.84,
+      "text": "use the neural network to"
+    },
+    {
+      "start": 3835.92,
+      "text": "to estimate [clears throat] this the"
+    },
+    {
+      "start": 3836.84,
+      "text": "score function? Yeah, yeah. We we"
+    },
+    {
+      "start": 3839.48,
+      "text": "basically first train to get a score"
+    },
+    {
+      "start": 3842.0,
+      "text": "function, then we train a dynamic"
+    },
+    {
+      "start": 3843.92,
+      "text": "function. But that dynamic function is"
+    },
+    {
+      "start": 3845.83,
+      "text": ". But that dynamic function is"
+    },
+    {
+      "start": 3845.84,
+      "text": "also a neural network. So, we we do we"
+    },
+    {
+      "start": 3847.92,
+      "text": "do not"
+    },
+    {
+      "start": 3848.24,
+      "text": "&gt;&gt; But you don't need that because if you"
+    },
+    {
+      "start": 3849.96,
+      "text": "just care about the the steady So, a"
+    },
+    {
+      "start": 3853.04,
+      "text": "system that reproduces the steady state"
+    },
+    {
+      "start": 3856.64,
+      "text": "you can just integrate this Langevin"
+    },
+    {
+      "start": 3858.8,
+      "text": "equation without"
+    },
+    {
+      "start": 3860.8,
+      "text": "any"
+    },
+    {
+      "start": 3862.16,
+      "text": "other network. So, is this a cover full"
+    },
+    {
+      "start": 3865.52,
+      "text": "solution or just a subset of solution?"
+    },
+    {
+      "start": 3868.56,
+      "text": "No, this is a general solution. So, this"
+    },
+    {
+      "start": 3870.84,
+      "text": "expression for f of x is a general"
+    },
+    {
+      "start": 3873.0,
+      "text": "solution."
+    },
+    {
+      "start": 3874.56,
+      "text": "So, it's essentially is So, given this"
+    },
+    {
+      "start": 3877.04,
+      "text": "Langevin equation, if you ask what is"
+    },
+    {
+      "start": 3879.72,
+      "text": "the most general expression for the"
+    },
+    {
+      "start": 3882.52,
+      "text": "drift term in such a way that uh"
+    },
+    {
+      "start": 3887.56,
+      "text": "So, it reproduces the observed steady"
+    },
+    {
+      "start": 3889.96,
+      "text": "state distribution. So, essentially that"
+    },
+    {
+      "start": 3892.4,
+      "text": "solve"
+    },
+    {
+      "start": 3893.64,
+      "text": "the stationary Fokker-Planck equation,"
+    },
+    {
+      "start": 3896.6,
+      "text": "then this is the most general"
+    },
+    {
+      "start": 3898.2,
+      "text": "expression."
+    },
+    {
+      "start": 3899.68,
+      "text": "Interesting. Yeah. But since So, here"
+    },
+    {
+      "start": 3902.64,
+      "text": "the main point is to estimate So, it's"
+    },
+    {
+      "start": 3905.84,
+      "text": "to reproduce the dynamics. So, this is"
+    },
+    {
+      "start": 3907.68,
+      "text": "the non-trivial part."
+    },
+    {
+      "start": 3909.64,
+      "text": "If you're just interested in the"
+    },
+    {
+      "start": 3911.08,
+      "text": "statistics, then yeah, take M of X equal"
+    },
+    {
+      "start": 3914.359,
+      "text": "to the identity"
+    },
+    {
+      "start": 3916.32,
+      "text": "and that's it."
+    },
+    {
+      "start": 3918.04,
+      "text": "Okay. Thank you very much. I I will"
+    },
+    {
+      "start": 3919.88,
+      "text": "write an email to you. Uh Yeah."
+    },
+    {
+      "start": 3928.24,
+      "text": "By the way,"
+    },
+    {
+      "start": 3929.72,
+      "text": "when some system have a source and sink"
+    },
+    {
+      "start": 3933.28,
+      "text": "and does your method can"
+    },
+    {
+      "start": 3936.16,
+      "text": "can cover those situations?"
+    },
+    {
+      "start": 3939.88,
+      "text": "Yeah, so if"
+    },
+    {
+      "start": 3942.56,
+      "text": "So, there is no time modulation"
+    },
+    {
+      "start": 3946.2,
+      "text": "of them. So, so essentially if you can"
+    },
+    {
+      "start": 3948.84,
+      "text": "define"
+    },
+    {
+      "start": 3951.6,
+      "text": "a steady state distribution,"
+    },
+    {
+      "start": 3955.12,
+      "text": "then yes."
+    },
+    {
+      "start": 3958.64,
+      "text": "But yeah, I haven't tested them that"
+    },
+    {
+      "start": 3960.48,
+      "text": "much. So, Thank you. I showed you the"
+    },
+    {
+      "start": 3962.64,
+      "text": "system that Yeah, I tested the"
+    },
+    {
+      "start": 3964.76,
+      "text": "algorithm."
+    }
+  ]
+}
\ No newline at end of file
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@@ -0,0 +1,1485 @@
+for inviting me at your
+group meeting. Um
+So, in today's presentation, uh I will
+talk about building mathematical models
+from high high-dimensional partially
+observed
+dynamical systems.
+So, I think this is a central topic in
+many scientific fields ranging from
+geophysical fluid dynamics, which is the
+field that is the closest uh
+to my research, but also other physical
+systems like molecular systems or
+neuroscience. So, so every times we have
+access to data, so we have observations
+of the underlying system, and we want to
+build a mathematical model that is able
+to reproduce and capture the main
+causality relationships between the
+variable of the system, and also
+hopefully to perform predictions and
+that uncertainty quantification.
+So, of course for this specific class of
+system
+that are extremely high-dimensional and
+also and also multiscale and partially
+observed, uh
+it is quite meaningless to try to
+perform trajectory shadowing. So, it's
+quite meaningless to try to build a
+model that is able to precisely
+predict the trajectory of the system.
+Instead, what we are usually So, what
+our goal becomes usually is to
+to reproduce some key statistical and
+dynamical observable of the
+the data set that we are observing.
+Those observables can be, for example,
+some moments of the steady state
+distribution, the whole steady state
+distribution,
+and also some
+uh state or temporal correlations
+between the
+observed variables of the system.
+So, this is the goal of this talk. So,
+develop a mathematical model that is
+able uh
+uh to reproduce this target statistical
+and dynamical observables.
+So, because of this timescale separation
+of the data, so the fact that the data
+can be multiscale, I will use
+a stochastic model where we have a first
+component, the drift term f of x, which
+uh will model the deterministic and slow
+variable component of my data set.
+And I
+And I also have a stochastic component
+also have a stochastic component
+that takes into account the faster
+dynamics,
+the unresolved fast fluctuations of the
+data set.
+So, this will be the model
+that I will that I want to construct
+from the the observations
+with the
+final goal
+of being able to reproduce
+in
+an efficient way this target statistical
+Okay, so right now I have been extremely
+general. So, I have
+talked about a general model to solve
+this very general problem.
+In the next two slides, I want to give a
+bit more details about the first what
+are the requirements that we want from
+our model,
+and then also what are the assumptions
+that we're doing on the observed data
+set.
+So, first
+let's see what are the requirement that
+we want
+from our modeling strategy.
+So, as I said, we would like to model
+real data that can be extremely
+high-dimensional
+and also maybe also unevenly sampled or
+have
+a very low sampling frequency. So, these
+are all
+features of our real data set that we
+have to take into account when we are
+building a mathematical model for it.
+So, this essentially implies first
+that since
+we want our modeling strategy to scale
+very well with the dimension,
+and since also the model that we want to
+develop can be extremely
+high-dimensional and
+computationally expensive to integrate,
+we would like to be able to reconstruct
+the mathematical model from data using
+as few model integrations
+as possible.
+So, a naive approach to build a
+mathematical model can be to start with
+a model ansatz, integrate it forward for
+a given time,
+then calculate
+all the statistical and dynamical
+observables, compare them with the
+targets,
+evaluate a loss function, and use that
+to update the model. But, this will
+require many model integrations that can
+become extremely computationally
+expensive. So, our modeling strategy
+So, one of the main goal of our modeling
+strategy would be to use as few model
+integrations as possible.
+Another requirement that we would that
+our modeling strategy to have is to
+avoid the state space clustering.
+So, we would like to avoid clustering
+the state space to estimate the average
+velocity field
+because essentially
+So, clustering the state space, even if
+we are using an extremely efficient
+clustering algorithm like, I don't know,
+bisecting k-means clustering algorithm,
+suffer from the curse of dimensionality.
+So, if we are considering a data set or
+a system with dimension larger than a
+few dozens, clustering becomes extremely
+computationally expensive, so we would
+like to avoid clustering.
+And also we would like to use a finite
+difference estimation to estimate the
+velocity from trajectories. So, we are
+assuming that our realistic data set can
+be can have a low frequen- a low
+sampling frequency, which essentially
+means that we cannot trust the data set
+to recover
+the velocity
+of our dynamics using finite difference.
+So, these are the
+three main
+um
+constraints that we want our modeling
+strategy to have in order to be scalable
+to high-dimensional system and also
+being used to realistic settings.
+Then, let's see what are instead the
+constraints
+that we want to impose on our system.
+So, on which kind of systems this So,
+the model the modeling strategy that I
+will propose in this talk can be
+applied.
+So, first of all, I will assume that the
+system I'm studying has statistical
+stationarity. So, essentially that we
+can define a steady state distribution.
+So, of course this can be generalized
+also to cyclostationary data by
+augmenting the state space or, for
+example, to
+data set that show a low trends that can
+be detrended and then we can get a
+stationary
+I will also assume that the system has
+um so, is ergodic and mixing, which
+essentially means that time averages
+along
+long trajectory
+will converge to ensemble averages, and
+also that
+the correlation function decays
+sufficiently fast.
+And then I'm also assuming that we have
+an effective timescale separation. So,
+this essentially allow us to model the
+data set using
+the Langevin equation that I showed you
+before. So, essentially treat the fast
+timescale as a noise process, and then
+build a drift term for the slow
+timescales.
+I will also assume to have enough
+observables of the slow timescales,
+and so in such a way I can get
+a full recovery of the slow variables.
+Even if we don't have a complete
+observables for the slow variables, we
+can still augment the state space using
+some delay embedded version of the data
+And then finally, even if
+uh we will not have probably some very
+fine resolution
+of the data sample,
+we still assume to have enough
+resolution to be able to define
+correlation functions. So, we don't have
+enough resolution to
+evaluate the velocity at every
+time point by using finite difference,
+but still we have enough data
+essentially to define a correlation
+function.
+So, these are the goals that I would
+like
+my modeling strategy to achieve,
+and also those are the main assumptions
+that I'm doing on the
+system that I'm studying.
+So, in this talk, I will present two
+different directions. So, two different
+modeling strategies that I'm currently
+pursuing.
+The first one So, in the first one, I'm
+assuming
+to have a model ansatz. So, essentially
+I have
+a knowledge of the functional form of my
+model, so a knowledge that be derived
+directly from physics.
+And this model answers depends on a set
+of parameters that I would like to
+determine.
+I have an initial guess for those
+parameters, and I would like to
+calibrate those model parameters in
+order to reproduce a set of statistical
+observables
+that I derive from my observations.
+The second direction instead is a bit
+more ambitious.
+I assume not to know
+any model answers. So, I don't start
+from
+model answers assumption, and I want to
+derive from my data
+the whole functional form of the model.
+In in this case, what I'm interested to
+do precisely is to start from the set of
+statistical and dynamical target
+observables,
+and I would like to infer from them
+the most general class of dynamical
+systems that by construction reproduces
+those dynamical observables.
+And I would like to do it. So, this is
+the most ambitious part of the
+of the method because I would like to do
+it without ever integrating my model
+forward. So, just from the knowledge of
+the target statistical and dynamical
+observable, I would like to be able to
+infer the model without ever integrating
+my model forward in time.
+So, these are the two directions that I
+will present today.
+Let's um for both directions, the key
+element that allow
+essentially
+those two directions
+is the score function. So, essentially,
+I will show how from the knowledge of
+the score function,
+we are able to
+build those two modeling strategies. So,
+the score function is defined as the
+gradient of the logarithm of the steady
+state distribution.
+Um
+Um so, the score function is one of the
+central So, it's a central quantity in
+machine learning
+specifically in score-based generative
+modeling
+because essentially it's a quantity that
+allow
+uh
+the generation of new data sample
+according to a specific probability
+density without the knowledge of the
+normalization constant that
+that otherwise someone needed to to
+evaluate in order to to define a
+normalized probability density function.
+So, essentially, it's a way to sample
+new data sample according to a specific
+probability density without the
+knowledge of the steady state
+distribution, which can be extremely
+challenging to be constructed from high
+dimensional systems.
+In the In this talk, I will So, there
+are of course many different methodology
+to estimate the score function from
+data. In this talk, I will use the
+denoising score matching method,
+which essentially consists in taking the
+data set that I would like to use to
+estimate the score function, I perturb
+it by adding a tiny Gaussian white noise
+with amplitude sigma,
+and then I will use the the denoising
+score matching identity, according to
+which I can write the score function of
+this perturbed probability density
+in fun So, in
+in terms of the expected value of zeta
+conditioned on the perturbed value x
+sigma. So, given So, I am at the end
+what I'm doing is to train a neural
+network to predict
+the value of the noise zeta
+given an input the perturbed value of
+the data set point x sigma.
+Um so, if sigma So, by considering sigma
+So, the perturbation amplitude very
+small, I'm essentially estimating the
+score function of a perturbed density,
+which is extremely close
+to the observed one. And so, the score
+function that I derive using this method
+it will be
+very close
+to the correct score function.
+The advantage of this algo of this
+algorithm is that it scales extremely
+well with the dimension because
+essentially we are recasting
+the score estimation problem, which will
+otherwise imply the differentiation of
+the logarithm of the steady state
+density,
+with a regression problem.
+And this regression problem scales very
+well with
+with the dimension. This essentially
+means that we can estimate the score
+function
+quite efficiently also for very high
+dimensional systems. And then we'll use
+the knowledge of the score function
+to infer the full mathematical model
+that explain that is able to reproduce
+this set of target observables that I
+mentioned
+at the beginning.
+So, let's start from the first modeling
+strategy.
+So, as I said,
+I'm assuming to have a model answers, so
+to have an answers for the functional
+form of my Langevin equation with
+multiplicative noise.
+of parameters alpha and beta.
+We call alpha all the parameters inside
+the drift term, and beta all the
+parameters inside the diffusion term.
+So, the problem consists now in finding
+the values for alpha and beta
+that reproduces the target statistical
+observables.
+So, learning
+So,
+in order to solve this calibration
+problem, what we need is the parameter
+sensitivity. So, if we call phi m the
+set of observables that we want our
+model to reproduce,
+what we need to know is how
+those observable phi of m will change by
+changing by a tiny amount each of the
+model parameters.
+So, if we know these these quantities,
+we can essentially update
+the parameters accordingly in order to
+minimize the distance between the target
+observables and the observables
+predicted by the model.
+We can estimate the statistical
+Jacobians
+using a naive approach by just
+integrating the model forward many
+times. Every time we perturb just one
+parameter by a tiny amount,
+and then using finite difference, we
+estimate this statistical Jacobians.
+This method of course works, but it has
+the problem that it requires an
+extremely large number of model
+integrations. So, we need at least
+one model integration for every
+parameter. And if the model is very
+large,
+it becomes extremely computationally
+expensive to integrate, in particular,
+if we have many parameters that we want
+to calibrate.
+So, the
+the key idea here is to recast this
+calibration problem
+to a perturbation problem. So, if we
+imagine to Taylor expand at first order
+the drift and and the diffusion
+coefficients after we perturb by a tiny
+amount each of those parameters,
+so essentially, what we obtain is a
+perturbed version of this Langevin
+equation, where we have an additional
+term in the drift and in the diffusion
+coefficients, which is given by the
+perturbation amplitude of the parameter
+multiplied by the partial derivatives of
+the drift and the diffusion term with
+respect to that specific parameter. So,
+essentially, it's like if we are
+studying a perturbed version
+of our model, and we would like to
+predict how all those statistical
+observables will change after we add
+this perturbation.
+Okay. So, now we have this
+perturbation problem,
+which can be addressed using a very
+powerful
+tool from
+statistical physics,
+which is the generalized
+fluctuation-dissipation theorem or GFT.
+So, GFT
+is
+a mathematical machinery that allows to
+predict how a dynamical system respond
+to a perturbation without actually
+perturbing it, just from the knowledge
+of its statistics.
+So, without entering into the
+mathematical details of this problem, we
+can essentially write the response of an
+observable A
+to a perturbation given by
+uh U of so a drift perturbation given by
+U of X and a diffusion perturbation
+given by V of X.
+So, those are nothing but
+these perturbations
+that we defined before.
+And we can write the response of this
+observable A in terms of an integral of
+the correlation between the observable
+itself and this conjugate variable B,
+which depends on the
+perturbation that we are applying. So, U
+of X
+and V of X.
+And importantly on the score function S.
+So, essentially if we have the knowledge
+of the score function since
+we know the analytic expression for our
+model. And also we know the analytic
+expression for the perturbation that we
+are applying U and V.
+We can predict how our system, so in
+this case how the observable A will
+respond to an external perturbation.
+So, if we identify the observable A with
+the set of observables that we want our
+model to reproduce.
+We can essentially construct using the
+generalized fluctuation-dissipation
+theorem the parameter sensitivities that
+we are interested in.
+Just from the knowledge of the score
+function itself.
+So, this is the main connection. So,
+using GFDT we are able to estimate the
+parameter sensitivities
+without the running the model forward
+for every parameter.
+We just needed to run the model forward
+one time to estimate the score function.
+And from this single model integration
+we are able to estimate all the
+parameter sensitivities that we can use
+for calibration.
+Okay, so as we have seen this entirely
+so this so the applicability of the
+theorem to this specific problem relies
+only on the estimation of the score
+So, at this point you may ask okay, but
+why nobody like followed this direction?
+And the reason is that estimating the
+score function for high-dimensional
+system has always been the main
+bottleneck to use this generalized
+fluctuation-dissipation theorem inside
+realistic and practical problems.
+So, usually people in the literature
+have always struggled in estimating the
+score function. And so, it was possible
+for low-dimensional systems. Instead for
+high-dimensional systems,
+what has been usually done in the
+literature was to use the so-called
+quasi-Gaussian approximations,
+which consists in approximating the
+steady state distribution with a
+multivariate Gaussian, which allows to
+write the score function in terms of a
+linear function that depends on the
+covariance matrix of the data set, which
+of course is extremely easy to be
+estimated.
+But the main problem is that
+when the dynamics is highly nonlinear,
+so when the steady state distribution
+also is highly
+non-Gaussian,
+then this approach introduces some
+strong biases,
+which
+doesn't allow to get quite precise
+prediction of the system responses using
+GFDT.
+But we we have seen before how yeah,
+recent advances in score estimation
+methods using a neural network
+allow us to get a very precise and
+efficient estimation for the score
+function also for very high-dimensional
+systems.
+And so, this knowledge allow us to
+construct and to estimate the system
+responses using the generalized
+fluctuation-dissipation theorem.
+So, we applied those ideas already
+to evaluate
+and to predict system responses for
+quite high-dimensional systems. We
+started PDEs discretized on around 10 at
+the third grid points.
+And we considered more specifically
+two-dimensional turbulent data and
+Alan-Khan reaction-diffusion data. So,
+these are the two papers where we
+published this
+connection between the score-based
+generative modeling and the generalized
+fluctuation-dissipation theorem. And so,
+now the idea is to use this mathematical
+machinery to evaluate the parameter
+sensitivities.
+And
+so, to do that I
+so, I described how we can do it. So,
+from the knowledge of the response
+function we can estimate the parameter
+sensitivities. Now, let's see how to do
+that in practice.
+More specifically, let's consider two
+examples now. So, the first one is a
+very low-dimensional model. We have a
+three-dimensional SDE with a
+multiplicative noise. So, this model is
+used in geophysical fluid dynamics to
+describe
+El Niño-Southern Oscillation, which is a
+interannual so, it's a annual
+variability phenomenon of the sea
+surface temperature in the tropical
+Pacific. We have two slow variables, one
+fast variable, which are coupled.
+This model depends on six coefficients.
+And so, what we are going to do is to
+start so, is to first run this model
+with the correct values of these
+coefficients to have
+an obser so, to build our observations.
+Then
+we will
+try to recover the correct values of
+those coefficients and starting from an
+initial guess
+obtained by perturbing by around 20%
+each of those coefficients and running
+our calibration algorithm. So, using
+GFDT to estimate the
+and use
+this knowledge inside a Newton algorithm
+to estimate the correct values of the
+coefficient.
+Those are the six observables that we
+would like to recover. So, we start with
+a parameter guess.
+Um we use this parameter guess to
+integrate the model forward.
+We use this data set to estimate the
+parameter sensitivities. And then we are
+updating the parameters using the
+knowledge of the parameter
+And we are iterating this procedure
+until
+the statistics of our predicted system
+will converge
+to the target statistical observables.
+We are doing this procedure using three
+different methods to evaluate the
+parameter sensitivities. We are first
+using finite difference. So, we are just
+integrating many time the trajectory
+forward one for each parameter. And then
+we are using finite difference to
+estimate the parameter Jacobian.
+And we are using that information to
+update the parameter value.
+Then we are using the generalized
+fluctuation-dissipation theorem using
+two different ways to estimate the score
+We first used the quasi-Gaussian
+approximation. So, we just
+wrote the score function in terms of a
+covariance matrix of the data. And then
+we are using the
+denoising score matching approach to
+estimate the score function.
+And we will be comparing those three
+approaches.
+These are the results.
+So, here on the left you can see the L2
+norm between predicted versus
+target statistical observables as a
+function of the
+algorithm iteration. So, in this case
+we introduced the breaking point when
+the
+L2 norm
+was falling below 10 at the minus three.
+And as you can see the blue curve and
+the gray and the gray curve, which
+represent the calibration algorithm
+using the denoising score matching score
+function plus GFDT and the naive
+and and the naive
+finite difference estimation for the
+parameter Jacobians in just five
+iteration we are falling below the
+threshold.
+Which essentially means in five
+iteration we were able, as you can see
+here in this panel showing the parameter
+deviation, to precisely recover the
+correct parameters of the model.
+And instead using the Gaussian
+approximation for the score, so like a
+more so, less precise estimation of the
+score function was very difficult to
+have the algorithm
+con
+converged
+to the correct value.
+But so, here what we can see that using
+the generalized fluctuation-dissipation
+theorem plus the denoising score
+matching to estimate this score
+function, we were able to have very
+similar performances
+with respect to the naive
+finite difference method at a fraction
+of the computational cost because
+we
+So, for every iteration, we needed to
+integrate the system forward only one
+time
+instead of six time
+So, the number of the parameters that we
+want to calibrate or like in in this
+case 12 because we use the center
+difference for
+parameter for the
+uh um
+parameter Jacobian estimation. So,
+essentially here we have an algorithm
+that's doesn't scale So, doesn't So, the
+for which the computational time doesn't
+scale
+linearly with the number of parameters,
+but is constant since we only need to
+run the model forward one
+one single time for every iteration.
+Okay. So, in this case we considered a
+quite a low dimensional system.
+Next, I will consider this coupled
+Lorenz '96
+system, which is around a 400
+dimensional system. We have 36 slow mode
+and we have a for each slow mode we have
+10 fast modes. Plus, we also have some
+white noise in each of um
+those
+variables.
+And what we want to do now is to do
+something different. So, we would like
+to build a stochastic closure for the X
+So, the slow variables. So, essentially
+we would like to build a 36 dimensional
+model
+instead of this model here that is
+around a 400 dimensional
+which is able to precisely recover
+the target statistical observables
+evaluated from the high dimensional
+model. So, we have observations for X
+which have been generated integrating
+this very high dimensional system and we
+would like to build this reduced order
+model which is only 36 dimensional
+with the correct values of alphas of the
+alpha coefficients and the sigma
+coefficients such that they reproduce
+this set of target statistical
+observables which are the mean uh the
+variance, skewness, excess kurtosis, and
+uh covariance C1.
+We have in total five parameters that we
+want to calibrate
+&gt;&gt; [gasps]
+&gt;&gt; uh on the
+uh on the
+on these five different statistical
+And so, we used also in this case these
+three different methods.
+We have in orange GFDT plus Gaussian
+uh estimation of the score function. And
+then in gray and in blue, we have
+the finite difference method and then
+the GFDT plus the noise score matching
+for the score
+estimation. So, essentially in in this
+case we can see like yeah, clear
+advantage in using the
+the noise score matching to build the
+score function. And from the knowledge
+of the score function, also in this case
+which is quite high dimensional, we can
+observe
+how we get quite similar performance
+than of using finite difference at a
+fraction of the computational cost. And
+so, essentially at at a fraction of the
+number of time that we have to integrate
+our model forward.
+Okay. So, this
+is the first direction. So, start from a
+model answers and use this combination
+between the generalized fluctuation
+dissipation theorem from non-equilibrium
+statistical physics with the noise score
+matching from generative modeling
+to to to estimate
+uh the parameter sensitivities
+with a very limited number of model
+integrations.
+Now, let's see the second direction.
+So, in this case we don't have any model
+answers for
+the functional form of our mathematical
+model.
+We have a set of statistical and
+dynamical observables that we want our
+model to reproduce, which in this case
+are the full steady state distribution
+and a set of correlation functions where
+phi m and phi n are
+So, it's a set of observables of the the
+state variable of the system.
+So, given these constraints, we would
+like to build a mathematical model that
+by construction reproduces those
+constraints without integrating our
+model forward.
+And again, we will use the score
+function to do that.
+Specifically, we will use two different
+score functions in this case. We have
+the plain score function that we've seen
+before
+and also we will use the conditional
+score function. This essentially is the
+gradient
+of with respect to X0 of the logarithm
+of the conditional probability density
+function. So, the probability density
+function of X at time T conditioned on
+of X at time T conditioned on
+X0.
+The conditional score function can be
+constructed using the noise score
+matching precisely as we did it for the
+plain score function. In fact, we can
+write the conditional score function in
+terms of the joint score function and
+the plain score function.
+For the joint score function, we just
+take our data set. We use a delay
+embedding
+in order to build
+a time series of X0 and XT.
+We do it for different value of the time
+delay and in this way we estimate the
+score function the joint score function.
+And then we can combine it with us and
+using the same the noise score matching
+machinery we have seen before, we can
+estimate both
+the conditional score and the score
+function from data.
+And as we have seen before, both these
+algorithms scale quite well with the
+dimension of the system.
+So, the idea here is then to use those
+two quantities
+where the first quantity essentially
+takes into account the geometry of the
+steady state distribution.
+Instead, the second quantity essentially
+takes into account how the system
+relaxes towards the steady state
+distribution. So, it's carrying
+information also about the
+the dynamics of the system and not only
+about the statistics.
+So, this is the intuition. So, try to
+use those two quantities that can be
+evaluated quite well also for very high
+dimensional systems
+to build our
+stochastic modeling approach.
+So, let's start from our
+Langevin equation. So, this is the same
+Langevin equation I wrote at the
+I wrote at the
+beginning. Yeah, just have here a factor
+square root of two.
+And then let's first impose
+stationarity. So, we want for a given
+sigma X to find our drift term F
+such that by construction
+reproduces the steady state
+distribution.
+And to do that, we can write the
+Fokker-Planck equation relative to the
+Langevin [clears throat] equation,
+impose the stationarity
+and we can show that without losing any
+generality
+we can write
+the drift term
+in this way. So, in terms of the score
+function that we defined before
+and the diffusion matrix. So, this
+symmetric matrix D of X
+and another anti-symmetric matrix R of
+X.
+So, this is a very general
+expression. We're not doing any
+approximation here. We are We are just
+finding the most general way to express
+the drift for a given the diffusion in
+such a way that it reproduces the steady
+state distribution by construction,
+which essentially means in such a way
+that F So, this specific shape of F
+solve the stationary Fokker-Planck
+equation.
+Now, we can So, we can see that we have
+two different tensors D of X and R of X.
+D of X is symmetric and represent the
+diffusion tensor.
+Instead, R which is the anti-symmetric
+part can be interpreted as the term that
+breaks the tail balance and that
+introduces some rotational component to
+our system without changing the steady
+state distribution. So, this can be
+related to an Helmholtz decomposition of
+the drift term. So, we have a
+a term a symmetric term which
+satisfies the detail balance and give us
+a system which is just a Brownian motion
+inside a potential. And then we have
+this other circulatory term which
+introduces some rotational component
+that breaks detailed balance.
+Okay, so now by using this expression
+here for the drift term, we are
+guaranteed
+to recover the steady state
+distribution. So we
+achieved the first goal of our modeling
+strategy which is to build a stochastic
+model that by construction reproduces
+the observed the steady state
+distribution of the data set. Now let's
+try to impose also the second constraint
+which is
+we want to reproduce also the the time
+correlations.
+So we would like to
+reproduce this time correlations for a
+set of observables phi n.
+So without going into the mathematical
+details of this derivation, we can show
+that
+the time derivative of this correlation
+function for this specific model, so for
+this specific Langevin equation with
+drift term given by this expression over
+here,
+can be written in this way.
+So essentially we can relate the time
+derivative of this correlation function
+with this
+expression here which contains the two
+phi, so the two observable phi m and phi
+n, the conditional score function, and
+the matrix, so the tensor m.
+And the tensor m is the only term here
+that we don't know because we can
+estimate this quantity here from data.
+We just evaluate
+the correlation function and then we
+estimate the derivative.
+We can estimate the conditional score.
+We know the analytical expression for
+both phi m and phi n because this is the
+libraries of observable that we are
+considering.
+The only term that we don't know is this
+matrix
+m x of 0. So now let's see how we can
+derive this matrix m x of 0.
+So first
+let's do this decomposition. So let's
+decompose m x in terms of a constant
+term plus a fluctuation.
+This fluctuation is so the average value
+over the stationary density of this
+fluctuation must be equal to 0. So
+essentially here we have
+a yeah, a constant term plus
+zero mean fluctuation term delta m. So
+let's use this expression here for m
+inside the this equation over there
+and we can then rewrite c dot in terms
+of two terms. This first term which
+depends on phi
+does not depend on the conditional
+score, depends only on the stationary
+score.
+.
+Okay, so we have the first term here
+which is much easier to evaluate because
+yeah, we don't need to estimate the
+conditional score
+and it only depends on this constant
+matrix phi and this
+plain score itself
+minus this additional so this additional
+term which is nothing but this one over
+here written in terms of delta m instead
+of m.
+So the key idea here is that if we have
+a library of observable which is rich
+enough such
+such that m of x is uniquely determined,
+then we can estimate m m of x, so this
+tensor m of x which is the missing
+element for so in our stochastic model
+by essentially using this relationship
+over here.
+So using this relationship here
+and a library of observable which is
+rich enough, we can estimate both phi
+and delta m.
+So let's see now how we can estimate phi
+first.
+So to do that, let's consider
+just the coordinate observable. Okay, so
+we have here in theory like a very large
+libraries of observable. Now let's focus
+on a few of them and few of them of x
+equal to x. So we're just considering
+the observable coordinate.
+By doing this replacement, the first
+term becomes this expected value
+multiplied by phi.
+Now if we consider t equal to 0,
+then this
+expected value becomes minus the
+identity because of the Stein identity.
+This term here becomes equal to 0
+because we have so you can just
+integrate the conditional score term at
+t equal to 0 and you will get 0.
+So this essentially means that if if you
+consider the coordinate observable and t
+equal to 0, we are able to derive a
+relationship for phi.
+So we can essentially fix the average
+value of the matrix m
+and we can write it in terms of the
+coordinate the time derivative of the
+coordinate correlation at t equal to 0.
+correlation at t equal to 0.
+Now let's see what this implies. So we
+have then
+fixed the phi. We have this additional
+term, this correction term e of x.
+When we consider
+phi phi phi phi 1 of x equal to x, we
+then have this coordinate term at the
+beginning.
+We can rewrite this expected value in
+terms of the gradient with respect to x
+0 of the expected value of x of t
+conditioned on x 0 multiplied by delta
+m.
+So by just considering the coordinate
+observable case, we can derive phi
+and then we can write this relationship
+for the correction term. But at this
+point we can notice that if
+m of x, so if the expected value of x of
+t conditioned on x 0 is approximately
+affine which essentially means if we can
+write the expected value of x t
+conditioned on x 0
+in terms of a linear function of x 0,
+then when we take the gradient, we will
+get a constant term with respect to x
+and then by construction the average
+value of x of delta m is equal to 0
+which essentially means that if the
+conditional mean is approximately affine
+which essentially
+is is is the case if
+the joint probability density function
+of x x t is a Gaussian,
+we can then use so we can then replace
+our matrix m of x which is state
+dependent with just the matrix phi
+and we have a model that by construction
+reproduces both the temporal
+correlations and the steady state
+Okay, so if
+this term so if m of t is linear in x 0
+which is often the case because if the
+conditional probability density so if
+the joint probability density of x 0 and
+and x t can be approximated with a
+Gaussian distribution, then m of t
+depends linearly on x 0.
+So if this term is negligible,
+then we can
+reproduce the time correlations of the
+observed data just using phi, so this
+constant matrix phi that we can easily
+determine from the correlation function
+instead of the state dependent matrix m.
+And then we have built a Langevin
+equation that by construction reproduces
+both the steady state distribution and
+the time correlations.
+If we want instead to add more
+constraint on the correlations, so we
+want to add more constraints on the
+dynamics adding more correlations, then
+we have to obtain
+also the matrix delta m that we can
+parameterize with a neural network.
+So in this specific case, we
+parameterize the whole m of x with a
+neural network and then we define delta
+m of x as m theta minus phi
+and then we can train a neural network
+delta m of theta to minimize this loss
+function. So we have this first term
+that that essentially forces the neural
+network to learn the set of correlation
+functions that we want our system to
+reproduce. Then we have this penalty
+term that essentially enforces that the
+average value of delta m is equal to 0
+plus we have a regularization term.
+But you can see here that we are so we
+are writing
+a loss function that doesn't depend on a
+forward model integration. So we never
+have to integrate our Langevin equation
+forward in time.
+We just use the the knowledge of the
+conditional score, the score function,
+and the time derivative of the
+correlation functions to train the
+neural network for delta M.
+And this can be can become extremely
+efficient when the model that we want to
+integrate becomes very computationally
+expensive.
+Okay, so this is the methodology.
+So we have seen that's
+yeah, we we are able to train this
+neural network without integrating the
+model forward and when so for some
+specific cases we can simplify the shape
+so the functional form of M
+by replacing them with a constant if we
+are just interested in the time
+correlation of the systems. So now I
+will conclude showing you some
+application of these ideas.
+So I start from an analytic warm up. So
+we consider this one-dimensional system
+for which we can determine analytically
+all the relevant quantities.
+So we can derive the station the
+conditional score, the stationary score,
+the time derivative of the correlation
+functions and so on.
+These are the true values for fee and
+delta M.
+And then by applying the method I
+&gt;&gt; [clears throat]
+&gt;&gt; discussed before we can obtain them
+discussed before we can obtain them
+using the relationship that I showed you
+at the beginning. So using that
+relationship we recover precisely
+the fee the correct fee and the correct
+So this was just like a test where we
+have we know everything is analytically.
+So let's see a different case. In this
+case we have a two-dimensional system
+with where we have our drift term which
+contains both a term that can be written
+as the gradient of a potential plus a
+circulatory component. We also have a
+And in this case we cannot write
+explicitly the score function and the
+conditional score. So we need to train
+two neural networks for S and for the
+conditional score.
+We apply the methodology that I
+described
+before by enforcing the reproduction of
+the correlation functions. We derive
+a quite accurate
+reconstruction of
+the mobility fields so the M
+tensor.
+We have some errors in particular in
+this term
+but even if so we have like some errors
+for
+the M to one terms when we integrate our
+model we get a precise recovery of the
+univariate PDF, bivariate PDF
+all the correlation functions. Here I'm
+comparing two different model
+integrations. We have the model
+integration with the full
+with the full
+mobility matrix M of X
+and a model integration where I'm
+replacing the full mobility matrix with
+a fee so with this constant closure that
+I introduced before.
+We can see here that using the full
+mobility matrix obtained by training
+a neural network for M we get a more
+precise recovery of the correlation
+functions in particular for this cross
+correlation.
+And also if I now consider the target
+dynamical observables so the target
+correlation functions that I used to
+train the neural network when I evaluate
+them from the trajectory that I obtained
+by integrating my model
+I get yeah a quite better
+recovery with respect to the constant M
+matrix closure.
+So essentially this is to show that yeah
+by
+applying this algorithm
+we are able to estimate the mobility
+matrix M of X together with the score
+and the conditional score then combining
+those pieces together
+we obtain an expression for the drift
+term that is able
+to reproduce the steady state density
+the time correlations together with all
+the correlations that we enforced
+in the training.
+Okay, so now let's consider more
+high-dimensional systems.
+So for the next two systems I will only
+consider the constant closure for M. So
+essentially I approximate M of X with
+its average value so with fee.
+In this case I'm integrating this
+Kuramoto-Sivashinsky PDE.
+I'm integrating this partial
+differential equation with 512 Fourier
+modes. I obtain a 1024-dimensional
+um time series. I'm
+considering just one
+mode every 32. So essentially I'm
+subsampling this 1024-dimensional
+state to a 32-dimensional state.
+And then using those
+those 32-dimensional modes to build my
+Langevin equation. So essentially here
+I'm
+building so I'm starting from a fully
+fully deterministic partial differential
+equation partial
+which is partially observed
+and then using a completely different
+model to
+be so to predict its dynamics using my
+stochastic closure.
+And here are the results. So this is the
+time series obtained by integrating my
+Langevin equation. This is the real
+observed time series and here I'm
+plotting the comparison between the
+the bivariate and the univariate PDFs
+obtained from the observations and the
+one obtained from
+a model integration of my Langevin
+equation. And here instead is the
+autocorrelation function for both the
+observations and my Langevin
+integration.
+Then finally I considered
+the sea surface temperature data from
+Plasim so which is a um
+a global circulation model of
+intermediate
+intermediate complexity.
+complexity.
+the data for the sea surface
+for the global sea surface temperature
+evolution and I want a model that is
+able essentially to predict and to model
+this sea surface temperature data.
+So the data set is around
+2000-dimensional.
+I did a dimensionality reduction taking
+the first 20 principal components.
+And here since I have a strong
+periodicity I augmented the state space
+by including some harmonic functions.
+And these are the results.
+I yeah was predicting the probability
+the conditional probability density
+of the
+20 principal components together with
+their autocovariance. So
+as you can see we can have a quite
+decent
+reconstruction of the PDFs and the ACFs
+of all the 20 principal components. Here
+I'm plotting just the first 10.
+And also we were able to capture the
+nonlinear so and the non-Gaussian
+probability density function evaluated
+at every grid point
+um from our simulation. So here
+essentially I'm plotting the probability
+density at different season
+of the temperature at a given grid point
+and I'm doing that using the full
+observation so essentially all the
+principal components
+just the first 20 principal components
+and
+the 20-dimensional
+stochastic model that I trained on these
+first 20 components and I integrated
+forward.
+And as you can see so even if I did a
+dimensionality reduction of the
+data set I was still able to get this
+nonlinear probability density functions
+using this
+quite simple
+stochastic model that I built using this
+constant closure for my
+mobility matrix.
+Okay, so these are
+some of the papers on
+that I so either published or put on
+archive on this topic. We tried the
+different directions that I haven't
+presented here.
+But
+so these were so the main references
+and to conclude so we have seen
+how to model high-dimensional
+partially observed chaotic systems
+how the knowledge of the score function
+plays a key role in allowing this
+modeling this modeling strategies. We
+have seen two different directions. In
+the first one we have a model answers
+and we are just
+calibrating the model parameters using a
+combination between the generalized
+fluctuation distribution theorem and the
+score modeling.
+The other direction instead
+doesn't have any model answer.
+We just try to
+starting from a set of statistical and
+dynamical observables to build a model
+that by construction reproduces all of
+them without integrating the model
+And then we've seen how this approach
+can scale on different systems from toy
+models to very high dimensional systems.
+Okay, thanks for listening and let me
+know if you have
+any question.
+Thank you, Ludovico.
+Any questions?
+I have a question. So,
+So, I'm wondering according to your
+formulation, is does your method allow
+you that
+have allow you
+to to work on data set that has
+absolutely no time information?
+What do you mean with absolutely no time
+information? So, like
+time series uh
+where every snapshot is completely
+uncorrelated? Yeah, yeah. In that case,
+yes. So, so you can do it, but you will
+be able to build a mathematical model
+that reproduces the steady state
+but not the dynamics because you don't
+have any information about the dynamics.
+So, what you can do in in that case
+and that will be yeah, much more simple,
+is to replace
+here uh
+M of X just with the identity, right?
+So, if you're only caring about the
+steady state distribution and also you
+don't have any information
+to build the
+so so
+to estimate the correlation functions,
+then it essentially means that any So,
+you cannot infer M of X because M of X
+is carrying information about the
+dynamics. So, you can replace M of X
+with the identity.
+Uh so so you So, if I train a model, I
+only need to train the M, right?
+Uh if you train So, if you only want to
+reproduce the steady state distribution
+because you don't have information about
+the dynamics,
+you just need to train a neural network
+to learn the score function.
+So, M can be just replaced with the
+identity.
+Hm. Okay, because then so any value So,
+any shape of M of X will uh
+uh give you
+the correct steady state distribution.
+So, you can just choose M of X equal to
+the identity.
+Also, if you choose M of X equal to the
+identity, it probably is an optimal
+choice because uh
+um you have the fastest convergence
+towards the steady state density. So,
+like if you integrate your model,
+quite fast towards the steady state
+density. So, if instead M of X is a
+constant matrix uh
+with a wide um
+variety [clears throat]
+wide amplitude in the eigenvalues,
+you have essentially that some modes
+will decay faster than others and so you
+have to wait like a longer time to see
+thermalization of the system towards the
+Yeah, this is very interesting because
+we we previously have a
+have a paper that targeting exactly on
+no time information and we we got some
+difficulty when we move from low
+dimension like two or three to to
+thousands of dimension. In thousands of
+dimension, our method basically uh
+almost failed and uh
+so yeah, so I'm wondering
+if your method can can be helpful. Yeah,
+so we estimated the score function also
+for thousand dimensional systems and
+yeah, like it's not a problem.
+But which
+So, how have you done this? So, did you
+use the neural network to
+to estimate [clears throat] this the
+score function? Yeah, yeah. We we
+basically first train to get a score
+function, then we train a dynamic
+function. But that dynamic function is
+. But that dynamic function is
+also a neural network. So, we we do we
+do not
+&gt;&gt; But you don't need that because if you
+just care about the the steady So, a
+system that reproduces the steady state
+you can just integrate this Langevin
+equation without
+any
+other network. So, is this a cover full
+solution or just a subset of solution?
+No, this is a general solution. So, this
+expression for f of x is a general
+solution.
+So, it's essentially is So, given this
+Langevin equation, if you ask what is
+the most general expression for the
+drift term in such a way that uh
+So, it reproduces the observed steady
+state distribution. So, essentially that
+solve
+the stationary Fokker-Planck equation,
+then this is the most general
+expression.
+Interesting. Yeah. But since So, here
+the main point is to estimate So, it's
+to reproduce the dynamics. So, this is
+the non-trivial part.
+If you're just interested in the
+statistics, then yeah, take M of X equal
+to the identity
+and that's it.
+Okay. Thank you very much. I I will
+write an email to you. Uh Yeah.
+By the way,
+when some system have a source and sink
+and does your method can
+can cover those situations?
+Yeah, so if
+So, there is no time modulation
+of them. So, so essentially if you can
+define
+a steady state distribution,
+then yes.
+But yeah, I haven't tested them that
+much. So, Thank you. I showed you the
+system that Yeah, I tested the
+algorithm.
\ No newline at end of file
diff --git a/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/video.log b/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/video.log
new file mode 100644
index 00000000..0af8ca37
--- /dev/null
+++ b/conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/video.log
@@ -0,0 +1,17 @@
+# yt-dlp log
+# url: https://youtu.be/P75iVMmbqQk
+# output: conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/video.mp4
+# returncode: 0
+
+stdout:
+[youtube] Extracting URL: https://youtu.be/P75iVMmbqQk
+[youtube] P75iVMmbqQk: Downloading webpage
+WARNING: [youtube] No supported JavaScript runtime could be found. Only deno is enabled by default.
+[youtube] P75iVMmbqQk: Downloading android vr player API JSON
+[info] P75iVMmbqQk: Downloading 1 format(s): 400+251
+[download] video.mp4.f400.mp4 (125.41MiB)
+[download] video.mp4.f251.webm (52.64MiB)
+[Merger] Merging formats into video.mp4 (Matroska / WebM container)
+
+stderr:
+WARNING: yt-dlp EJS not enabled; some formats may be missing.
diff --git a/scripts/tier2/artifacts/video_analysis_campaign_20260621/clean_transcript.py b/scripts/tier2/artifacts/video_analysis_campaign_20260621/clean_transcript.py
new file mode 100644
index 00000000..daa7ea9f
--- /dev/null
+++ b/scripts/tier2/artifacts/video_analysis_campaign_20260621/clean_transcript.py
@@ -0,0 +1,95 @@
+"""Clean yt-dlp auto-sub VTT transcripts.
+
+yt-dlp auto-subs produce rolling captions where each segment extends the
+previous one with new words, sometimes triplicated. Algorithm:
+  1. Strip VTT tags (<00:00:00.560><c>...</c>) from raw text.
+  2. For each pair (prev, curr), compute the new suffix curr adds vs prev.
+     Keep only the new suffix.
+  3. Drop empty / duplicate results.
+  4. Skip segments that are pure repetition of the prior segment.
+"""
+import json
+import re
+from pathlib import Path
+
+VTT_TAG = re.compile(r"<[^>]+>")
+WS = re.compile(r"\s+")
+
+def strip_vtt(t: str) -> str:
+    return WS.sub(" ", VTT_TAG.sub("", t)).strip()
+
+def longest_common_prefix_len(a: str, b: str) -> int:
+    n = min(len(a), len(b))
+    i = 0
+    while i < n and a[i] == b[i]:
+        i += 1
+    return i
+
+def longest_common_suffix_len(a: str, b: str, max_k: int = 80) -> int:
+    n = min(len(a), len(b), max_k)
+    i = 0
+    while i < n and a[-1 - i] == b[-1 - i]:
+        i += 1
+    return i
+
+def main(track: str) -> None:
+    art = Path(f"conductor/tracks/{track}/artifacts")
+    p = art / "transcript.json"
+    data = json.loads(p.read_text(encoding="utf-8"))
+    raw_segments = data["segments"]
+
+    cleaned: list[dict] = []
+    prev_clean = ""
+    for raw in raw_segments:
+        text = strip_vtt(raw["text"])
+        if not text:
+            continue
+        if prev_clean and text == prev_clean:
+            continue
+        if prev_clean and text.startswith(prev_clean):
+            new = text[len(prev_clean):].strip()
+            if not new:
+                continue
+            cleaned.append({"start": raw["start"], "text": new})
+            prev_clean = text
+            continue
+        lcp = longest_common_prefix_len(prev_clean, text)
+        if lcp >= 5:
+            new = text[lcp:].strip()
+            if new:
+                cleaned.append({"start": raw["start"], "text": new})
+                prev_clean = text
+                continue
+        lcs = longest_common_suffix_len(prev_clean, text)
+        if lcs >= 5:
+            new = text[: -lcs if lcs < len(text) else 0].strip()
+            if new:
+                cleaned.append({"start": raw["start"], "text": new})
+                prev_clean = text
+                continue
+        cleaned.append({"start": raw["start"], "text": text})
+        prev_clean = text
+
+    deduped: list[dict] = []
+    seen: set[str] = set()
+    for seg in cleaned:
+        t = seg["text"]
+        if t in seen:
+            continue
+        seen.add(t)
+        deduped.append(seg)
+
+    plain = "\n".join(s["text"] for s in deduped)
+    (art / "transcript_clean.txt").write_text(plain, encoding="utf-8")
+    data["clean_segments"] = deduped
+    data["plain"] = plain
+    p.write_text(json.dumps(data, indent=2, ensure_ascii=False), encoding="utf-8")
+    print(f"{track}: {len(raw_segments)} raw -> {len(cleaned)} cleaned -> {len(deduped)} deduped")
+    print(f"plain length: {len(plain)} chars")
+    print("First 800 chars:")
+    print(plain[:800])
+
+
+if __name__ == "__main__":
+    import sys
+    main(sys.argv[1] if len(sys.argv) > 1 else "video_analysis_score_dynamics_giorgini_20260621")