conductor(score_dynamics_giorgini): Phase 3 OCR - 31 frames OCR'd via winsdk in 2.3s

conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/ocr.md

@@ -0,0 +1,693 @@
+# OCR Results
+
+## frame_00001.jpg
+
+```
+Introduction
+Many scientific problems—from climate dynamics and fluid
+turbulence to molecular systems and neuroscience—involve
+modeling high-dimensional, chaotic, multiscale dynamical
+systems observed only through a subset of variables.
+Ludovico Giorghi
+2
+```
+
+## frame_00002.jpg
+
+```
+e 5 of
+Introduction
+Many scientific problems—from climate dynamics and fluid
+turbulence to molecular systems and neuroscience—involve
+modeling high-dimensional, chaotic, multiscale dynamical
+systems observed only through a subset of variables.
+For these systems, trajectory prediction is often the wrong
+target: small errors amplify rapidly, so reproducing
+individual paths is neither realistic nor especially useful.
+The meaningful goal is instead to learn a data-driven
+reduced model that reproduces the relevant statistical and
+dynamical observables of the resolved variables.
+Ludovico Giorgt))
+Effective reduced
+model
+* = f (x) + g(x)
+x: resolved observed variables
+f: effective deterministic drift
+g(x) Q: unresolved fast fluctuations
+2
+```
+
+## frame_00004.jpg
+
+```
+e 7 of
+What We Aim For
+Ludovico Giorghi
+We want to develop a data-driven modeling method that scales well with the system's dimension and can handle
+realistic datasets, which may be unevenly sampled or restricted to a low sampling frequency. This requires
+searching for a modeling method with the following features:
+```
+
+## frame_00006.jpg
+
+```
+What We Aim For
+Ludovico Giorgini
+We want to develop a data-driven modeling method that scales well with the system's dimension and can handle
+realistic datasets, which may be unevenly sampled or restricted to a low sampling frequency. This requires
+searching for a modeling method with the following features:
+1
+No or very few model integrations. We must be able to infer the model from data without needing long and
+repeated forward simulations to calibrate candidate models.
+Avoid state-space clustering. We want to bypass state-space clustering techniques for reconstructing the
+velocity field, since these approaches suffer from the curse of dimensionality and deteriorate quickly.
+Avoid finite-difference estimation from trajectories. The method should not rely on short-time derivatives of
+the trajectories to estimate the drift. Finite differences are highly sensitive to noise and data sparsity, whereas
+stationary statistics and correlations are much more robust to estimate.
+3
+```
+
+## frame_00008.jpg
+
+```
+'age 15 7ij
+Two Complementary Directions
+Direction 1: calibration within a
+model ansatz
+We assume a parametric stochastic model is
+already available.
+The goal is to calibrate its coefficients so that
+selected stationary observables match the target
+data.
+The emphasis is on obtaining the required
+sensitivities while using as few model integrations
+as possible.
+Ludovico Giorgini
+5
+```
+
+## frame_00009.jpg
+
+```
+e 16 of
+Two Complementary Directions
+Direction 1: calibration within a
+model ansatz
+We assume a parametric stochastic model is
+already available.
+The goal is to calibrate its coefficients so that
+selected stationary observables match the target
+data.
+The emphasis is on obtaining the required
+sensitivities while using as few model integrations
+as possible.
+Ludovico Giorghi
+Direction 2: direct construction
+from observables
+We do not assume a prescribed functional form for
+the reduced model.
+Instead, we derive a class of stochastic models that
+reproduces the chosen observables by construction.
+The emphasis is on enforcing the constraints
+directly from data, without simulating candidate
+models during inference.
+5
+```
+
+## frame_00011.jpg
+
+```
+e 20 of
+Score Function as the Key Ingredient
+For both directions, the key object is the stationary score:
+S(X) = Vx log pss(x),
+which encodes the local geometry of the invariant measure.
+Knowing the score is often easier than trying to reconstruct the full density explicitly.
+Modern generative methods make score estimation feasible in high-dimensional, non-Gaussian settings.
+Denoising Score Matching (DSM)
+= x -k az for z N (0, l), and train a neural network ge to minimize the
+We perturb the data with small Gaussian noise, x
+Ludovico Giorghi
+denoising loss:
+CDSM = E [Ilge(xO)
+2
+ZII
+1
+s(x)
+DSM turns score estimation into a denoising regression problem, bypassing intractable normalization constants and scaling
+efficiently to high dimensions.
+6
+```
+
+## frame_00013.jpg
+
+```
+•age 24 of 7t
+Parameter Sensitivities as Linear Responses
+Consider the parametric stochastic model:
+dxt= F(xt; a) dt + E(xt; P) dWt,
+For target observables Om, calibration requires the statistical Jacobian:
+smj(0) —
+DOJ
+An infinitesimal parameter variation induces a perturbation of the dynamics:
+uj(x) = Da. F (x; a),
+vj(x) = DßJE(x; P).
+Key reduction
+Ludovico
+Computing parameter sensitivities is equivalent to estimating the linear response of (Om) to infinitesimal drift and diffusion
+perturbations.
+7
+```
+
+## frame_00014.jpg
+
+```
+age 25 of
+Estimating Parameter Sensitivities with the GFDT
+For a stationary Ito SDE and any observable A, the GFDT gives the first-order response:
+< A(xt) B(XT, T)) dT,
+B(x, t) = V • ZA(x, t) -k ZA(x, t) • s(x),
+where s(x) =
+V log p(x) is the stationary score and
+Q(x, t) —
+Ludovico Giorgnq
+8
+```
+
+## frame_00016.jpg
+
+```
+23 of 7?)
+Parameter Sensitivities as Linear Responses
+Consider the parametric stochastic model:
+dxt= F(xt; a) dt + E(xt; P) dWt,
+For target observables Om, calibration requires the statistical Jacobian:
+smj(0) —
+An infinitesimal parameter variation induces a perturbation of the dynamics:
+uj(x) = Da. F (x; a),
+= DßJE(x; P).
+Ludovico Giorgini
+7
+```
+
+## frame_00019.jpg
+
+```
+e 27 of
+Estimating Parameter Sensitivities with the GFDT
+For a stationary Ito SDE and any observable A, the GFDT gives the first-order response:
+Q(x, t) — Y(x) V(x,
+Ludovico Giorghi
+where s(x) =
+< A(xt) B(XT, T)) dT,
+V log p(x) is the stationary score and
+t) -k ZÅ(x, t) • s(x),
+t)
+Hence, specifying the perturbation vector gives:
+drift perturbation: V = u,
+diffusion
+perturbation:
+1
+2
+Choose A = Om, and use (u, V) = (uj, 0) or (0, Vj) according to the parameter being differentiated.
+o
+The resulting response coefficient is exactly the desired sensitivity Smj.
+o
+Therefore, one long unperturbed trajectory plus an estimate of the stationary score is enough to build the full
+o
+statistical Jacobian.
+Calibration step
+Estimate S(0n) with GFDT, then use it in a regularized Gauss—Newton update for the model parameters.
+8
+```
+
+## frame_00021.jpg
+
+```
+e 32 of
+Past Works on GFDT + Score Modeling
+Ludovico Gior hi
+In all these response formulas, the central missing quantity is the stationary score s(x) = V log PSS For
+high-dimensional systems, estimating this score has historically been the main bottleneck in applying the
+GFDT.
+A common workaround is to approximate the steady-state density with a multivariate Gaussian, which gives a
+cheap closed-form score. This approximation introduces strong biases in strongly non-Gaussian regimes, such
+as bimodal, intermittent, or turbulent dynamics.
+Recent neural score-estimation methods now recover accurate non-Gaussian scores efficiently in high
+dimension.
+Recent works showed accurate response estimation in high-dimensional stochastic PDEs (0(103) grid points),
+including Allen—Cahn reaction—diffusion dynamics and 2D turbulence, demonstrating the feasibility of this
+approach.
+References
+Giorgini et al., Response Theory via Generative Score Modeling, Physical Review Letters (2024)
+Giorgini et al., Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation
+Theorem and Generative Modeling, PNAS (2025)
+9
+```
+
+## frame_00023.jpg
+
+```
+Triad Calibration
+Results
+dtter•nces
+mismatch
+age 34 of
+GFOT.
+Observable deviations
+neratim
+Parameter deviations
+2
+3
+4
+5
+9
+Ludovico Giorghi
+10
+```
+
+## frame_00024.jpg
+
+```
+e 35 of
+Application: Two-Scale Lorenz—96 Closure Calibration
+Goal: calibrate a stochastic closure for Xk so the reduced model matches key statistics of the full two-scale system.
+Ludovica Giorghi
+Full two-scale model
+—Xk—l — Xk+l) — X k + F
+dXk =
+hc
+Yj,k cit + ax dWkX,
+hc
+dYj,k =
+with k = 1, .
+Reduced stochastic closure
+—cb cYj,k -k —Xk dt + ay dWY
+36, and J = IO.
+— Xk — I — 2 — Xk+l ) — + F — + I Xk + a 2 + + d VVk .
+dXk =
+(P(X), V (X), Sk(X), Ku(X), G , where p, V, Sk, Ku, Cl denote the spatial mean,
+Active observables +(X) =
+variance, skewness, excess kurtosis, and nearest-neighbor covariance.
+Calibrated parameters 0 = (ao, al, a2, a3,
+Starting from a representative two-scale trajectory, we estimate S = DO with GFDT and update 0 via regularized
+Gauss—Newton.
+12
+```
+
+## frame_00027.jpg
+
+```
+e 45 of
+Imposing the Stationary PDF
+Start from the Ito diffusion:
+dxt = F(xt) dt -k v6E(xt) dWt,
+If pss is the target stationary density and s = V log pss, the stationary Fokker—Planck equation implies:
++ V. R(x) -k R(x) s(x),
+where R(x)T
+Equivalently:
+M(x) s(x) + V. M(x),
+M(x) —
+Symmetric part
+D = MT)
+diffusion tensor and fluctuation amplitude
+score-driven relaxation toward pss
+Ludovico Giorghi
+15
+```
+
+## frame_00028.jpg
+
+```
+46 of 7t
+Imposing the Stationary PDF
+Start from the Ito diffusion:
+dxt = -k v6E(xt) dWt,
+If pss is the target stationary density and s = V log pss, the stationary Fokker—Planck equation implies:
++ V. R(x) -k R(x) s(x),
+where R(x)T
+Equivalently:
+M(x) s(x) + V. M(x),
+Ludovico Giorgini
+Symmetric part
+diffusion tensor and fluctuation amplitude
+score-driven relaxation toward pss
+Antisymmetric part
+R = — MT)
+rotational / circulatory transport
+changes kinetics without changing pss
+15
+```
+
+## frame_00039.jpg
+
+```
+58 of
+Example 1: Analytic Warmup with the CIR Diffusion
+Cox—Ingersoll—Ross (CIR) square-root diffusion:
+dXt =
+True mobility and stationary score:
+M(x) = —
+K (o — Xt ) dt -F 27 Xt dWt ,
++ öM(x),
+97,
+PSS (x) ¯
+r(K9/7)
+Exact transition density and conditional score:
+p(xt I xo) —
+s(x) =
+¯ä¯
+xoxt
+e—Kt)
+'(1 — e—Kt)
+xt
+69/7—1
+e K t xoxt
+xoxt
+Here r ( • ) is the Euler Gamma function and Iq(•) is the modified Bessel function of the first kind. Using (x) xa , the lagged correlation derivative is
+Ca,l (t)
+Ludovico Giorghi
+19
+```
+
+## frame_00041.jpg
+
+```
+e 60 of
+Example 2: A 2D Nonequilibrium SDE
+We next consider a two-dimensional nonequilibrium overdamped Langevin diffusion with affine multiplicative noise.
+(X, y) T, with
+The resolved state is x =
+dxt = f(xt) dt+ B(xt) d Wt
+w = 11
+Ludovico Giorgini
+The drift combines confinement and irreversible rotation:
+f(x) = —VU(x) + wJx,
+o
+1
+122
+—1
+with quartic potential
+The multiplicative noise is affine in the state:
+= åx4 + äXY + ay + + F2Y
+No analytic score or conditional score is available. We therefore learn
+the stationary score s(x) with DSM,
+o
+the joint score of lagged pairs and hence the conditional score,
+o
+the correction field öM(x) with a neural network.
+o
+The mobility fit uses coordinate, quadratic, and cubic probes, so the inverse problem is constrained by a broad family of
+lagged correlations.
+21
+```
+
+## frame_00042.jpg
+
+```
+Example 2:
+too
+Learned Mobility Field
+00
+00
+as
+00
+os
+Ludovico Giorgini
+• 00
+```
+
+## frame_00043.jpg
+
+```
+Example 2: Forward Validation of Station&Y and Statistics
+Ludovico Giorgini
+```
+
+## frame_00047.jpg
+
+```
+Example 3: Partially Observed Kuramoto—Sivashinsky
+Ludovico Giorgini
+Hee we the 8M O, •o the ROM fuly and constant m•ttix O.
+Ibe fu' system is the equat•o
+Au — IV(u2),
+a •Landard mo&l Of chaos.
+Jntecrate the POE 912 modes. y'eUing a 1024-dimensional real state.
+one mode 32 a partially observed system; the unresolved modes act as an effectrve
+stochastc bxh.
+The surrogate is to *adow trajectories, but to reproduce the invariant measure and time
+"Ede
+PDF
+Langevin PDF
+Langevin PDF
+0.4
+0.3
+xli]
+Data PDF
+Data PDF
+02
+Data PDF
+```
+
+## frame_00049.jpg
+
+```
+Example 4: Cyclo-Stationary PlaSim Extension
+is not st.tbnyy the Imposes a strong peOc•iic
+forcing.
+as an autonomous ane a..rnüttirg the
+•tat. hematic
+•e (•irqz•t), r
+in the augmented sp•ce. we the Md infer
+constant ceerat<€ e•sxtjy in tie cap.
+The •ith the curutt
+it—teed into
+dft.i.,. — v'äE,i.,.
+Apr/Jc•ton: a reduced rode* for the c«nponeot$ of
+PlaSim sut€aee-eunpeatur• driven by the —nual
+Ludovico Giorgini
+```
+
+## frame_00052.jpg
+
+```
+Example 4:
+PlaSim Validation in Physical Space
+262 206
+Ludovico Giorgini
+08
+06
+0 02
+06
+oo
+04
+02
+os
+04
+02
+00
+298
+1.10, .16
+300
+30,
+o e 31.60, .12
+os
+02
+00
+299
+291 295
+0.05'
+000
+302 2S4
+oto•
+005:
+0004
+230
+02
+00
+04.
+02.
+278
+274
+2m zeo 282
+1.45,
+297
+296
+208
+300
+ooo
+sot
+301
+04
+02
+00
+296
+20
+00
+06
+03
+02
+06
+291
+oe
+04
+02
+oo
+oo
+290
+2B2
+295
+297
+(K)
+296
+Ternswature (K)
+Ternperature (K)
+Temperature (K)
+(an PC') • 120 pcs) -Model pcs)
+```
+
+## frame_00055.jpg
+
+```
+age 69 of
+Conclusions
+Ludovico Giorgini
+For high-dimensional chaotic systems, reproducing statistically relevant observables is often more meaningful
+than reproducing trajectories.
+30
+```
+
+## frame_00059.jpg
+
+```
+Ludovico Giorgili
+```
+
+## frame_00080.jpg
+
+```
+Imposing the Stationary PDF
+Ludovico Giorgini
+Start from the ltd diffusion:
+Symmetric part
+dxt -k v'fiE(xt) d Wt,
+If pss is the target stationary density and s = V log pss, the stationary Fokker—Planck equation implies:
++ V. R(x) + R(x) s(x),
+where R x) T
+Equivalently:
+—R(x).
+Symmetric part
+diffusion tensor and fluctuation amplitude
+score-driven relaxation toward pss
+Antisymmetric part
+R = — MT)
+rotational / circulatory transport
+changes kinetics without changing pss
+15
+```
+
+## frame_00081.jpg
+
+```
+If i' the target *Mionary duuity S
+Imposing the Stationary PDF
+lt6 diffusion:
+Symmetric part
+where R(x)T
+log the statiat—y
+— + v. 00)
+ck rmpl—
+e stationary Fokker—Planck equation implies:
++ R(x) + R(x) s(x),
+and fluctuation amplitude
+score•driven relaxation towa/å)j M(x) + V' M(x), M(x) D(x) + R(x).
+Symmetric part
+diffusion tensor and fluctuation amplitude
+score-driven relaxation toward pss
+Antisymmetric part
+R = — MT)
+rotational / circulatory transport
+changes kinetics without changing pss
+Ludovico Giorgini
+15
+```
+
+## frame_00082.jpg
+
+```
+Start the
+If is the tatg•e dut•ity S A. , the station—y Fokker—Planck eeu•tsn imple
+Equivakntjy
+FIX)
+Symmetric part
+diffusion tensoe and fluctuation ampfitude
+wore-driven relaxation toward
+score-driven relaxation toward pss
+M(x) - DO) RIX).
+changes kinetics without changing pss
+Ludovico Giorgili
+15
+```
+
+## frame_00083.jpg
+
+```
+Imposing the PDF
+Start the It' d"won.
+dgt
+If is the duuity ard S
+T — -RIRI.
+eber• R(x)
+FIX)
+Symmetric part
+log , the Fokker—Planck impl—
+— m.) + 00)
+M(x) - DO) RIX).
+diffusion and fluctuation amplitude
+score•driven relaxation toward
+Ludovico Giorgili
+15
+```
+
+## frame_00087.jpg
+
+```
+e 45 of
+Start from the Ito diffusion:
+dxt = F(xt) dt -k VäE(xt) dWt,
+If pss is the target stationary density and s = V log pss, the stationary Fokker—Planck equation implies:
++ V. R(x) -k R(x) s(x),
+where R(x)T
+Equivalently:
+M(x) s(x) + V. M(x),
+Symmetric part
+diffusion tensor and fluctuation amplitude
+score-driven relaxation toward pss
+wore-driven relaxation toward
+M(x) —
+changes kinetics without changing
+Ludovico Giorgi-li
+15
+```
+
+## frame_00090.jpg
+
+```
+Page 45 of 77
+Equivalently
+Imposing the Stationry PDF
+Surt frem the It'S d4Son:
+dÅt
+If is the dÜt•ity j log A. , the •tatka•y implea
+Equiv•akntjy
+Ludovica Giorghi
+FIX)
+Symmetric part
+diffusion tensor and fluctuation ampGtude
+wore-driven relaxation toward
+MO) - DO) RIX).
+Antisymmetric part
+rotational / circulatory transport
+changes kinetics without changing
+```

conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/phase2.log

@@ -0,0 +1,2 @@
+Phase 2 Keyframes for C:\projects\manual_slop\conductor\tracks\video_analysis_score_dynamics_giorgini_20260621\artifacts\video.mp4
+  OK: kept 31 frames

conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/phase2_commit.log

@@ -0,0 +1,5 @@
+[master edd2f181] conductor(score_dynamics_giorgini): Phase 2 Keyframes - 31 unique frames from 91 raw (threshold 0.05)
+ 32 files changed, 39 insertions(+)
+ create mode 100644 conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/frames/extraction_meta.json
+ create mode 100644 conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/frames/frame_00001.jpg
+ create mode 100644 conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/frames/frame_00002.jpg

conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/artifacts/phase3.log

@@ -0,0 +1,2 @@
+Phase 3 OCR for C:\projects\manual_slop\conductor\tracks\video_analysis_score_dynamics_giorgini_20260621\artifacts\frames (winsdk)
+  OK: OCR'd 31 frames in 2.3s