# Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling

arXiv: https://arxiv.org/abs/2504.13333 · HTML: https://mlsanity.com/paper/2504.13333/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 86 references that are in the map.
- First posted on arXiv 17 Apr 2025.

## Builds on (most cited first, 9 of 9)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025)
- [Unpaired Downscaling of Fluid Flows with Diffusion Bridges](https://mlsanity.com/paper/2305.01822/) (2023)
- [Scientific Machine Learning Based Reduced-Order Models for Plasma Turbulence Simulations](https://mlsanity.com/paper/2401.05972/) (2024)
- [Response Theory via Generative Score Modeling](https://mlsanity.com/paper/2402.01029/) (2024)

## Built on by (most cited first, 3 of 3)

- [Probing forced responses and causality in data-driven climate emulators: conceptual limitations and the role of reduced-order models](https://mlsanity.com/paper/2506.22552/) (2025)
- [Reduced-Order Modeling of Cyclo-Stationary Time Series Using Score-Based Generative Methods](https://mlsanity.com/paper/2508.19448/) (2025)
- [Conditional Score-Based Modeling of Effective Langevin Dynamics](https://mlsanity.com/paper/2604.23952/) (2026)

## Abstract (arXiv)

We present a novel and flexible data-driven framework for estimating the response of higher-order moments of nonlinear stochastic systems to small external perturbations. The classical Generalized Fluctuation--Dissipation Theorem (GFDT) links the unperturbed steady-state distribution to the system's linear response. While standard implementations relying on Gaussian approximations can predict the mean response, they often fail to capture changes in higher-order moments. To overcome this, we combine GFDT with score-based generative modeling to estimate the system's score function directly from data. We demonstrate the framework's versatility by employing two complementary score estimation techniques tailored to the system's characteristics: (i) a clustering-based algorithm (KGMM) for systems with low-dimensional effective dynamics, and (ii) a denoising score matching method implemented with a U-Net architecture for high-dimensional, spatially-extended systems where reduced-order modeling is not feasible. Our method is validated on several stochastic models relevant to climate dynamics: three reduced-order models of increasing complexity and a 2D Navier--Stokes model representing a turbulent flow with a localized perturbation. In all cases, the approach accurately captures strongly nonlinear and non-Gaussian features of the system's response, significantly outperforming traditional Gaussian approximations.