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

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# OCR Results
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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
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## frame_00002.jpg
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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
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## frame_00004.jpg
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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:
```
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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
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'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
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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
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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
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## frame_00013.jpg
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•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
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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
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## frame_00016.jpg
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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
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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
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Triad Calibration
Results
dtter•nces
mismatch
age 34 of
GFOT.
Observable deviations
neratim
Parameter deviations
2
3
4
5
9
Ludovico Giorghi
10
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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
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## frame_00027.jpg
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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
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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
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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
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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
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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
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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
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age 69 of
Conclusions
Ludovico Giorgini
For high-dimensional chaotic systems, reproducing statistically relevant observables is often more meaningful
than reproducing trajectories.
30
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## frame_00059.jpg
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Ludovico Giorgili
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## 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
```
@@ -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
@@ -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
@@ -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