Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Latent-Variable Learning of SPDEs via Wiener Chaos

Sebastian Zeng, Andreas Petersson, Wolfgang Bock

TL;DR

The paper presents a structure-aware latent-variable framework for learning the law of linear SPDEs with additive Gaussian forcing from spatiotemporal solution observations alone. By coupling a spectral Galerkin projection with a truncated Wiener-Itô chaos expansion, it derives a finite set of propagator ODEs for latent temporal modes while representing stochasticity via chaos coordinates, both learned through variational inference without observing noise. The method yields a principled separation between deterministic evolution and stochastic forcing, enabling recovery of the underlying stochastic law and improving both trajectory accuracy and law-level statistics across unbounded and bounded 1D domains. It demonstrates state-of-the-art performance on synthetic data and outlines extensions to nonlinear dynamics, higher dimensions, and physics-informed priors, highlighting a pathway toward law-aware data-driven stochastic modeling.

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.

Latent-Variable Learning of SPDEs via Wiener Chaos

TL;DR

The paper presents a structure-aware latent-variable framework for learning the law of linear SPDEs with additive Gaussian forcing from spatiotemporal solution observations alone. By coupling a spectral Galerkin projection with a truncated Wiener-Itô chaos expansion, it derives a finite set of propagator ODEs for latent temporal modes while representing stochasticity via chaos coordinates, both learned through variational inference without observing noise. The method yields a principled separation between deterministic evolution and stochastic forcing, enabling recovery of the underlying stochastic law and improving both trajectory accuracy and law-level statistics across unbounded and bounded 1D domains. It demonstrates state-of-the-art performance on synthetic data and outlines extensions to nonlinear dynamics, higher dimensions, and physics-informed priors, highlighting a pathway toward law-aware data-driven stochastic modeling.

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.
Paper Structure (39 sections, 1 theorem, 44 equations, 3 figures, 4 tables)

This paper contains 39 sections, 1 theorem, 44 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Notation and problem setting
  4. Galerkin reduction in space
  5. Wiener--Itô chaos expansion
  6. Propagator equations
  7. Truncation and reconstruction
  8. Remark on Truncation and Approximation.
  9. A latent-variable model
  10. Latent-variable structure and factorization
  11. Priors and variational family
  12. Reconstruction operator and observation model
  13. Encoder architecture
  14. Experiments
  15. Experimental setup and regimes
  16. ...and 24 more sections

Key Result

Proposition 2.1

For linear SPDEs with additive Gaussian noise, the propagator coefficients $\widetilde{X}_t^{(\alpha)}$ vanish identically for all $|\alpha|\ge2$.

Figures (3)

  • Figure 1: Law-level evaluation in Regime A (unbounded domain).(a) Temporal evolution of spatially averaged variance $\mathbb{E}_x[\mathrm{Var}(X_t(x))]$ (mean $\pm$ std over 3 seeds). (b) Energy spectrum $\mathbb{E}[|c_n(t)|^2]$ at the final time (mean $\pm$ std over 3 seeds, log scale). (c) Learned eigenvalues $\lambda_n$ compared to ground truth.
  • Figure 2: Law-level evaluation in Regime B (bounded domain).(a) Temporal evolution of spatially averaged variance $\mathbb{E}_x[\mathrm{Var}(X_t(x))]$ (mean $\pm$ std over 3 seeds). (b) Energy spectrum $\mathbb{E}[|c_n(t)|^2]$ at the final time (log scale; mean $\pm$ std over 3 seeds). (c) Learned eigenvalues $\lambda_n$ compared to ground truth.
  • Figure 3: True and learned noise amplitudes $q_n$ shown on a log--log scale. In Regime A, the learned coefficients recover the correct scale and decay behavior. In Regime B, higher-mode noise amplitudes are suppressed.

Theorems & Definitions (1)

  • Proposition 2.1: First-order closure