Latent-Variable Learning of SPDEs via Wiener Chaos

2026-02-12

AI Summary

This work proposes a novel method for learning the statistical structure of linear stochastic partial differential equations (SPDEs) with additive Gaussian noise directly from spatiotemporal observational data, without requiring prior knowledge of the driving noise or initial conditions. By integrating spectral Galerkin projection with truncated Wiener chaos expansion, the SPDE is reduced to a finite-dimensional parametric system of ordinary differential equations. A structured latent variable model is introduced, enabling joint estimation of latent states and stochastic forcing terms through variational inference. This approach achieves, for the first time, end-to-end learning of the stochastic structure of SPDEs and theoretically disentangles deterministic dynamics from stochastic forcing. It attains state-of-the-art performance on synthetic data across both bounded and unbounded one-dimensional spatial domains, accurately recovering the underlying stochastic dynamical structure of the SPDEs.

Technology Category

• Machine Learning: Structured Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• stochastic partial differential equations
• latent-variable learning
• Wiener chaos
• spatiotemporal observations
• additive Gaussian forcing

Innovation

• latent-variable learning
• stochastic PDEs
• Wiener chaos expansion
• spectral Galerkin
• variational inference