Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling
This work addresses a domain-specific challenge in climate dynamics by enabling more accurate predictions of system responses, though it appears incremental as it builds on existing theories and methods.
The authors tackled the problem of predicting higher-order moment responses in nonlinear stochastic systems to small perturbations, combining the Generalized Fluctuation-Dissipation Theorem with score-based generative modeling to accurately capture non-Gaussian features, significantly outperforming traditional Gaussian approximations in validation on climate-relevant models.
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.