Response Theory via Generative Score Modeling

TL;DR

This paper presents a novel method combining score-based generative modeling with GFDT to analyze and predict responses of complex dynamical systems, including non-Gaussian cases, validated on various stochastic PDEs.

Contribution

It introduces a new approach that leverages score-based models and GFDT for accurate response estimation in complex systems, extending applicability beyond Gaussian assumptions.

Findings

Improved response prediction accuracy over traditional methods

Validated on stochastic PDEs of increasing complexity

Effective for systems with non-Gaussian statistics

Abstract

We introduce an approach for analyzing the responses of dynamical systems to external perturbations that combines score-based generative modeling with the Generalized Fluctuation-Dissipation Theorem (GFDT). The methodology enables accurate estimation of system responses, including those with non-Gaussian statistics. We numerically validate our approach using time-series data from three different stochastic partial differential equations of increasing complexity: an Ornstein-Uhlenbeck process with spatially correlated noise, a modified stochastic Allen-Cahn equation, and the 2D Navier-Stokes equations. We demonstrate the improved accuracy of the methodology over conventional methods and discuss its potential as a versatile tool for predicting the statistical behavior of complex dynamical systems.