conductor(score_dynamics_giorgini): Phase 4 Synthesis - report.md (1325 lines, 93KB) + summary.md (354 words)

conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/summary.md

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+# Summary: Learning Dynamics from Statistics (Giorgini)
+
+**Source:** https://youtu.be/P75iVMmbqQk
+**Author:** Ludovico Giorgini
+**Track:** Child #4 of `video_analysis_campaign_20260621`
+**Cluster:** A (Math foundations)
+**Pass:** 1 of 3 (research-only deep-dive)
+
+---
+
+## One-paragraph synthesis
+
+Giorgini presents a framework for **data-driven reduced stochastic modeling** of high-dimensional, partially observed, multiscale chaotic systems, reproducing selected statistical and dynamical observables without trajectory shadowing. The central primitive is the **stationary score** s(x) = ∇ log p_ss(x), estimated via **Denoising Score Matching** (Vincent 2011) — a denoising regression that bypasses intractable partition functions and scales to O(10³) dimensions. The framework has two directions: **(1) ansatz calibration** uses the **Generalized Fluctuation-Dissipation Theorem** to compute parameter sensitivities from a single unperturbed trajectory plus the learned score, replacing finite-difference Jacobian estimation that requires 2P integrations per iteration with O(1) — demonstrated on a 12-parameter model converging in 5 iterations at 12× lower cost, and on a 5-parameter Lorenz-96 closure; **(2) direct construction** uses the drift decomposition F = M·s + ∇·M, where the score enforces the stationary density and the learned mobility tensor M encodes the dynamics — applied to a 2D nonequilibrium Langevin with quartic potential and multiplicative noise, a 1024-mode Kuramoto-Sivashinsky PDE observed at 32 modes, and a 2000-dimensional PlaSim SST dataset reduced via 20 principal components + harmonic augmentation for cyclo-stationarity. **Forward connections:** diffusion LMs (cs336), DDPM (creikey), speculative cross-modal score (kumar). **Backward connections:** score matching in EBMs (cs229), Kolmogorov extension and FP foundations (probability_logic), pointwise Shannon information (entropy_epiplexity).
+
+---
+
+## Three key takeaways
+
+1. **Score is the right object to learn** — a vector field (one output per dimension) that bypasses the partition function; DSM makes it learnable in high dimensions.
+2. **GFDT + DSM = O(1) cost per calibration iteration** — one unperturbed trajectory plus the learned score gives the full statistical Jacobian, replacing O(P) finite-difference integrations.
+3. **F = M·s + ∇·M parameterizes all stationary Langevin drifts** — the score enforces p_ss; the free mobility M encodes dynamics. M = I gives fastest convergence; non-identity M enables circulation without changing the measure.
+
+---
+
+*Pass 2 (de-obfuscation via user's mathematical encoding) to follow.*