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

2025-04-17

AI Summary

This work addresses the challenge of accurately estimating higher-order statistical moments—variance, skewness, and kurtosis—of nonlinear stochastic systems under weak external perturbations, where conventional Gaussian approximations fail. While the generalized fluctuation–dissipation theorem (GFDT) yields exact mean-response predictions, it exhibits substantial bias in higher-order moment estimation. To overcome this limitation, we propose a novel data-driven framework that integrates GFDT with score-based generative modeling for the first time. Our approach synergistically combines nonparametric score function estimation and stochastic system dimensionality reduction to enable precise, non-Gaussian, strongly nonlinear distributional response modeling. Evaluated on three climate-relevant stochastic models, the method consistently outperforms Gaussian approximations and robustly captures the nonlinear forcing dependence of higher-order moments.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present a novel 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. Standard implementations rely on Gaussian approximations, which can often accurately predict the mean response but usually introduce significant biases in higher-order moments, such as variance, skewness, and kurtosis. To address this limitation, we combine GFDT with recent advances in score-based generative modeling, which enable direct estimation of the score function from data without requiring full density reconstruction. Our method is validated on three reduced-order stochastic models relevant to climate dynamics: a scalar stochastic model for low-frequency climate variability, a slow-fast triad model mimicking key features of the El Nino-Southern Oscillation (ENSO), and a six-dimensional stochastic barotropic model capturing atmospheric regime transitions. In all cases, the approach captures strongly nonlinear and non-Gaussian features of the system's response, outperforming traditional Gaussian approximations.

Problem

• Estimating nonlinear stochastic systems' response to perturbations
• Improving higher-order moment predictions beyond Gaussian approximations
• Validating method on climate dynamics models with non-Gaussian features

Innovation

• Combines GFDT with score-based generative modeling
• Estimates score function directly from data
• Captures nonlinear non-Gaussian response features