conductor(score_dynamics_giorgini): Phase 3 OCR - 31 frames OCR'd via winsdk in 2.3s
This commit is contained in:
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# OCR Results
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## frame_00001.jpg
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```
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Introduction
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Many scientific problems—from climate dynamics and fluid
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turbulence to molecular systems and neuroscience—involve
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modeling high-dimensional, chaotic, multiscale dynamical
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systems observed only through a subset of variables.
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Ludovico Giorghi
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2
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```
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## frame_00002.jpg
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```
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e 5 of
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Introduction
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Many scientific problems—from climate dynamics and fluid
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turbulence to molecular systems and neuroscience—involve
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modeling high-dimensional, chaotic, multiscale dynamical
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systems observed only through a subset of variables.
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For these systems, trajectory prediction is often the wrong
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target: small errors amplify rapidly, so reproducing
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individual paths is neither realistic nor especially useful.
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The meaningful goal is instead to learn a data-driven
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reduced model that reproduces the relevant statistical and
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dynamical observables of the resolved variables.
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Ludovico Giorgt))
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Effective reduced
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model
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* = f (x) + g(x)
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x: resolved observed variables
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f: effective deterministic drift
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g(x) Q: unresolved fast fluctuations
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2
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```
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## frame_00004.jpg
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```
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e 7 of
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What We Aim For
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Ludovico Giorghi
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We want to develop a data-driven modeling method that scales well with the system's dimension and can handle
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realistic datasets, which may be unevenly sampled or restricted to a low sampling frequency. This requires
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searching for a modeling method with the following features:
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```
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## frame_00006.jpg
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```
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What We Aim For
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Ludovico Giorgini
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We want to develop a data-driven modeling method that scales well with the system's dimension and can handle
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realistic datasets, which may be unevenly sampled or restricted to a low sampling frequency. This requires
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searching for a modeling method with the following features:
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1
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No or very few model integrations. We must be able to infer the model from data without needing long and
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repeated forward simulations to calibrate candidate models.
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Avoid state-space clustering. We want to bypass state-space clustering techniques for reconstructing the
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velocity field, since these approaches suffer from the curse of dimensionality and deteriorate quickly.
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Avoid finite-difference estimation from trajectories. The method should not rely on short-time derivatives of
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the trajectories to estimate the drift. Finite differences are highly sensitive to noise and data sparsity, whereas
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stationary statistics and correlations are much more robust to estimate.
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3
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```
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## frame_00008.jpg
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```
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'age 15 7ij
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Two Complementary Directions
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Direction 1: calibration within a
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model ansatz
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We assume a parametric stochastic model is
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already available.
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The goal is to calibrate its coefficients so that
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selected stationary observables match the target
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data.
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The emphasis is on obtaining the required
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sensitivities while using as few model integrations
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as possible.
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Ludovico Giorgini
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5
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```
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## frame_00009.jpg
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```
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e 16 of
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Two Complementary Directions
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Direction 1: calibration within a
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model ansatz
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We assume a parametric stochastic model is
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already available.
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The goal is to calibrate its coefficients so that
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selected stationary observables match the target
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data.
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The emphasis is on obtaining the required
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sensitivities while using as few model integrations
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as possible.
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Ludovico Giorghi
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Direction 2: direct construction
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from observables
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We do not assume a prescribed functional form for
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the reduced model.
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Instead, we derive a class of stochastic models that
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reproduces the chosen observables by construction.
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The emphasis is on enforcing the constraints
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directly from data, without simulating candidate
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models during inference.
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5
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```
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## frame_00011.jpg
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```
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e 20 of
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Score Function as the Key Ingredient
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For both directions, the key object is the stationary score:
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S(X) = Vx log pss(x),
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which encodes the local geometry of the invariant measure.
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Knowing the score is often easier than trying to reconstruct the full density explicitly.
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Modern generative methods make score estimation feasible in high-dimensional, non-Gaussian settings.
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Denoising Score Matching (DSM)
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= x -k az for z N (0, l), and train a neural network ge to minimize the
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We perturb the data with small Gaussian noise, x
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Ludovico Giorghi
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denoising loss:
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CDSM = E [Ilge(xO)
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2
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ZII
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1
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s(x)
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DSM turns score estimation into a denoising regression problem, bypassing intractable normalization constants and scaling
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efficiently to high dimensions.
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6
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```
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## frame_00013.jpg
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```
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•age 24 of 7t
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Parameter Sensitivities as Linear Responses
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Consider the parametric stochastic model:
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dxt= F(xt; a) dt + E(xt; P) dWt,
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For target observables Om, calibration requires the statistical Jacobian:
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smj(0) —
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DOJ
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An infinitesimal parameter variation induces a perturbation of the dynamics:
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uj(x) = Da. F (x; a),
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vj(x) = DßJE(x; P).
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Key reduction
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Ludovico
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Computing parameter sensitivities is equivalent to estimating the linear response of (Om) to infinitesimal drift and diffusion
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perturbations.
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7
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```
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## frame_00014.jpg
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```
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age 25 of
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Estimating Parameter Sensitivities with the GFDT
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For a stationary Ito SDE and any observable A, the GFDT gives the first-order response:
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< A(xt) B(XT, T)) dT,
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B(x, t) = V • ZA(x, t) -k ZA(x, t) • s(x),
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where s(x) =
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V log p(x) is the stationary score and
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Q(x, t) —
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Ludovico Giorgnq
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8
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```
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## frame_00016.jpg
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```
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23 of 7?)
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Parameter Sensitivities as Linear Responses
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Consider the parametric stochastic model:
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dxt= F(xt; a) dt + E(xt; P) dWt,
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For target observables Om, calibration requires the statistical Jacobian:
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smj(0) —
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An infinitesimal parameter variation induces a perturbation of the dynamics:
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uj(x) = Da. F (x; a),
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= DßJE(x; P).
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Ludovico Giorgini
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7
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```
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## frame_00019.jpg
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```
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e 27 of
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Estimating Parameter Sensitivities with the GFDT
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For a stationary Ito SDE and any observable A, the GFDT gives the first-order response:
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Q(x, t) — Y(x) V(x,
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Ludovico Giorghi
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where s(x) =
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< A(xt) B(XT, T)) dT,
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V log p(x) is the stationary score and
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t) -k ZÅ(x, t) • s(x),
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t)
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Hence, specifying the perturbation vector gives:
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drift perturbation: V = u,
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diffusion
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perturbation:
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1
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2
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Choose A = Om, and use (u, V) = (uj, 0) or (0, Vj) according to the parameter being differentiated.
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o
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The resulting response coefficient is exactly the desired sensitivity Smj.
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o
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Therefore, one long unperturbed trajectory plus an estimate of the stationary score is enough to build the full
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o
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statistical Jacobian.
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Calibration step
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Estimate S(0n) with GFDT, then use it in a regularized Gauss—Newton update for the model parameters.
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8
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```
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## frame_00021.jpg
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```
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e 32 of
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Past Works on GFDT + Score Modeling
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Ludovico Gior hi
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In all these response formulas, the central missing quantity is the stationary score s(x) = V log PSS For
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high-dimensional systems, estimating this score has historically been the main bottleneck in applying the
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GFDT.
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A common workaround is to approximate the steady-state density with a multivariate Gaussian, which gives a
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cheap closed-form score. This approximation introduces strong biases in strongly non-Gaussian regimes, such
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as bimodal, intermittent, or turbulent dynamics.
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Recent neural score-estimation methods now recover accurate non-Gaussian scores efficiently in high
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dimension.
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Recent works showed accurate response estimation in high-dimensional stochastic PDEs (0(103) grid points),
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including Allen—Cahn reaction—diffusion dynamics and 2D turbulence, demonstrating the feasibility of this
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approach.
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References
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Giorgini et al., Response Theory via Generative Score Modeling, Physical Review Letters (2024)
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Giorgini et al., Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation
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Theorem and Generative Modeling, PNAS (2025)
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9
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```
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## frame_00023.jpg
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```
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Triad Calibration
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Results
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dtter•nces
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mismatch
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age 34 of
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GFOT.
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Observable deviations
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neratim
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Parameter deviations
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2
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3
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4
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5
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9
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Ludovico Giorghi
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10
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```
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## frame_00024.jpg
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```
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e 35 of
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Application: Two-Scale Lorenz—96 Closure Calibration
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Goal: calibrate a stochastic closure for Xk so the reduced model matches key statistics of the full two-scale system.
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Ludovica Giorghi
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Full two-scale model
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—Xk—l — Xk+l) — X k + F
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dXk =
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hc
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Yj,k cit + ax dWkX,
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hc
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dYj,k =
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with k = 1, .
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Reduced stochastic closure
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—cb cYj,k -k —Xk dt + ay dWY
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36, and J = IO.
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— Xk — I — 2 — Xk+l ) — + F — + I Xk + a 2 + + d VVk .
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dXk =
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(P(X), V (X), Sk(X), Ku(X), G , where p, V, Sk, Ku, Cl denote the spatial mean,
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Active observables +(X) =
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variance, skewness, excess kurtosis, and nearest-neighbor covariance.
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Calibrated parameters 0 = (ao, al, a2, a3,
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Starting from a representative two-scale trajectory, we estimate S = DO with GFDT and update 0 via regularized
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Gauss—Newton.
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12
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```
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## frame_00027.jpg
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```
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e 45 of
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Imposing the Stationary PDF
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Start from the Ito diffusion:
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dxt = F(xt) dt -k v6E(xt) dWt,
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If pss is the target stationary density and s = V log pss, the stationary Fokker—Planck equation implies:
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+ V. R(x) -k R(x) s(x),
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where R(x)T
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Equivalently:
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M(x) s(x) + V. M(x),
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M(x) —
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Symmetric part
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D = MT)
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diffusion tensor and fluctuation amplitude
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score-driven relaxation toward pss
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Ludovico Giorghi
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15
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```
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## frame_00028.jpg
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```
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46 of 7t
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Imposing the Stationary PDF
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Start from the Ito diffusion:
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dxt = -k v6E(xt) dWt,
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If pss is the target stationary density and s = V log pss, the stationary Fokker—Planck equation implies:
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+ V. R(x) -k R(x) s(x),
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where R(x)T
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Equivalently:
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M(x) s(x) + V. M(x),
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Ludovico Giorgini
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Symmetric part
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diffusion tensor and fluctuation amplitude
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score-driven relaxation toward pss
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Antisymmetric part
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R = — MT)
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rotational / circulatory transport
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changes kinetics without changing pss
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15
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```
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## frame_00039.jpg
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```
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58 of
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Example 1: Analytic Warmup with the CIR Diffusion
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Cox—Ingersoll—Ross (CIR) square-root diffusion:
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dXt =
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True mobility and stationary score:
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M(x) = —
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K (o — Xt ) dt -F 27 Xt dWt ,
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+ öM(x),
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97,
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PSS (x) ¯
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r(K9/7)
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Exact transition density and conditional score:
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p(xt I xo) —
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s(x) =
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¯ä¯
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xoxt
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e—Kt)
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'(1 — e—Kt)
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xt
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69/7—1
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e K t xoxt
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xoxt
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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
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Ca,l (t)
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Ludovico Giorghi
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19
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```
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## frame_00041.jpg
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```
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e 60 of
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Example 2: A 2D Nonequilibrium SDE
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We next consider a two-dimensional nonequilibrium overdamped Langevin diffusion with affine multiplicative noise.
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(X, y) T, with
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The resolved state is x =
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dxt = f(xt) dt+ B(xt) d Wt
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w = 11
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Ludovico Giorgini
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The drift combines confinement and irreversible rotation:
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f(x) = —VU(x) + wJx,
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o
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1
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122
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—1
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with quartic potential
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The multiplicative noise is affine in the state:
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= åx4 + äXY + ay + + F2Y
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No analytic score or conditional score is available. We therefore learn
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the stationary score s(x) with DSM,
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o
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the joint score of lagged pairs and hence the conditional score,
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o
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the correction field öM(x) with a neural network.
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o
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The mobility fit uses coordinate, quadratic, and cubic probes, so the inverse problem is constrained by a broad family of
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lagged correlations.
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21
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```
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## frame_00042.jpg
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```
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Example 2:
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too
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Learned Mobility Field
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00
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00
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as
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00
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os
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Ludovico Giorgini
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• 00
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```
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## frame_00043.jpg
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```
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Example 2: Forward Validation of Station&Y and Statistics
|
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Ludovico Giorgini
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```
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## frame_00047.jpg
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```
|
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Example 3: Partially Observed Kuramoto—Sivashinsky
|
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Ludovico Giorgini
|
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Hee we the 8M O, •o the ROM fuly and constant m•ttix O.
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Ibe fu' system is the equat•o
|
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Au — IV(u2),
|
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a •Landard mo&l Of chaos.
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Jntecrate the POE 912 modes. y'eUing a 1024-dimensional real state.
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one mode 32 a partially observed system; the unresolved modes act as an effectrve
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stochastc bxh.
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The surrogate is to *adow trajectories, but to reproduce the invariant measure and time
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"Ede
|
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PDF
|
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Langevin PDF
|
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Langevin PDF
|
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0.4
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0.3
|
||||
xli]
|
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Data PDF
|
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Data PDF
|
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02
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Data PDF
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```
|
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## frame_00049.jpg
|
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```
|
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Example 4: Cyclo-Stationary PlaSim Extension
|
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is not st.tbnyy the Imposes a strong peOc•iic
|
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forcing.
|
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as an autonomous ane a..rnüttirg the
|
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•tat. hematic
|
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•e (•irqz•t), r
|
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in the augmented sp•ce. we the Md infer
|
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constant ceerat<€ e•sxtjy in tie cap.
|
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The •ith the curutt
|
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it—teed into
|
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dft.i.,. — v'äE,i.,.
|
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Apr/Jc•ton: a reduced rode* for the c«nponeot$ of
|
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PlaSim sut€aee-eunpeatur• driven by the —nual
|
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Ludovico Giorgini
|
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```
|
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|
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## frame_00052.jpg
|
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|
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```
|
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Example 4:
|
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PlaSim Validation in Physical Space
|
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262 206
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Ludovico Giorgini
|
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08
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06
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0 02
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06
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oo
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04
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02
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os
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04
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02
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00
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298
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1.10, .16
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300
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30,
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o e 31.60, .12
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os
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02
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00
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299
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291 295
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0.05'
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000
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302 2S4
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oto•
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005:
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0004
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230
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02
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00
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04.
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02.
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278
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274
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2m zeo 282
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1.45,
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297
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296
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208
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300
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ooo
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sot
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301
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04
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02
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00
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296
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20
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00
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06
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03
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02
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06
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291
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oe
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04
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290
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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
|
||||
```
|
||||
@@ -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
|
||||
+5
@@ -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
|
||||
Reference in New Issue
Block a user