Latent-Variable Learning of SPDEs via Wiener Chaos

TL;DR

This paper introduces a novel latent-variable approach combining spectral Galerkin and Wiener chaos expansions to learn the stochastic dynamics of SPDEs from solution observations, without needing noise or initial condition data.

Contribution

It presents a structured method that separates deterministic and stochastic components, enabling efficient inference of SPDEs' stochastic structure solely from solution data.

Findings

Achieves state-of-the-art performance on synthetic data

Effectively captures intrinsic stochasticity without explicit noise observations

Applicable to both bounded and unbounded spatial domains

Abstract

We study the problem of learning the law of linear stochastic partial differential equations (SPDEs) with additive Gaussian forcing from spatiotemporal observations. Most existing deep learning approaches either assume access to the driving noise or initial condition, or rely on deterministic surrogate models that fail to capture intrinsic stochasticity. We propose a structured latent-variable formulation that requires only observations of solution realizations and learns the underlying randomly forced dynamics. Our approach combines a spectral Galerkin projection with a truncated Wiener chaos expansion, yielding a principled separation between deterministic evolution and stochastic forcing. This reduces the infinite-dimensional SPDE to a finite system of parametrized ordinary differential equations governing latent temporal dynamics. The latent dynamics and stochastic forcing are jointly inferred through variational learning, allowing recovery of stochastic structure without explicit observation or simulation of noise during training. Empirical evaluation on synthetic data demonstrates state-of-the-art performance under comparable modeling assumptions across bounded and unbounded one-dimensional spatial domains.