Response Theory via Generative Score Modeling

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.

Figures (1)

• Figure 1: Column 1: Example snapshots of the Three Systems. In the top row, Ornstein-Uhlenbeck PDE, in the middle row Allen-Cahn, in the bottom row Navier-Stokes. Columns 2-5: Response functions for Three Systems. The response to a perturbation at the pixel component $i=1$ is evaluated at various pixel components with coordinates indicated at each panel. All evaluated components align with the advection direction. Responses are computed by integrating the dynamical system (orange lines, "Dynamical") via Gaussian approximation (black lines, "Gaussian"), using the score function derived from generative modeling (red lines, "Generative"), and using the analytic score function (blue lines, "Analytic"). Columns 6-7: Response Function Errors for the Three Systems. Here, we show both the RMS error and the Infinity Norm error of the Response function as a function of time. Here we use the Dynamic Response function as the "ground truth". We see that the Generative response (red) generally outperforms the Gaussian response (blue) for the nonlinear cases (Allen-Cahn and Navier-Stokes), especially in the Infinity Norm, and performs similarly for the linear system (Ornstein-Uhlenbeck). The response function computed with the exact score does not have zero error relative to the dynamical response function because of errors introduced in numerical discretization and empirical averaging.