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Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion

Ariel Neufeld, Philipp Schmocker

TL;DR

The paper addresses numerical solutions of semilinear SPDEs by truncating the Wiener chaos expansion and learning the chaos propagators with (possibly random) neural networks, leveraging universal approximation and Malliavin calculus. It proves universal approximation results for SPDEs using both deterministic and random networks and provides explicit terminal-time approximation rates that separate neural-network approximation error from chaos-truncation error. The authors demonstrate the approach on three SPDEs—the stochastic heat equation, the HJM forward-rate equation, and the Zakai equation—showing that deterministic and random networks in the Wiener chaos can effectively learn SPDE solutions, with random nets offering computational advantages. This framework offers a scalable, data-driven method for solving SPDEs with additive or multiplicative noise and highlights clear trade-offs between network size, chaos order, and basis truncations, with potential broad impact in finance, filtering, and physics-informed modeling.

Abstract

In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise. Finally, we apply our results in numerical examples to approximate the solution of three SPDEs: the stochastic heat equation, the Heath-Jarrow-Morton equation, and the Zakai equation.

Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion

TL;DR

The paper addresses numerical solutions of semilinear SPDEs by truncating the Wiener chaos expansion and learning the chaos propagators with (possibly random) neural networks, leveraging universal approximation and Malliavin calculus. It proves universal approximation results for SPDEs using both deterministic and random networks and provides explicit terminal-time approximation rates that separate neural-network approximation error from chaos-truncation error. The authors demonstrate the approach on three SPDEs—the stochastic heat equation, the HJM forward-rate equation, and the Zakai equation—showing that deterministic and random networks in the Wiener chaos can effectively learn SPDE solutions, with random nets offering computational advantages. This framework offers a scalable, data-driven method for solving SPDEs with additive or multiplicative noise and highlights clear trade-offs between network size, chaos order, and basis truncations, with potential broad impact in finance, filtering, and physics-informed modeling.

Abstract

In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise. Finally, we apply our results in numerical examples to approximate the solution of three SPDEs: the stochastic heat equation, the Heath-Jarrow-Morton equation, and the Zakai equation.

Paper Structure

This paper contains 29 sections, 16 theorems, 112 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Stochastic partial differential equations (SPDEs)
  5. Existence and uniqueness of mild solutions
  6. Wiener chaos expansion
  7. Universal approximation of SPDEs
  8. Deterministic neural networks
  9. Random neural networks
  10. Approximation rates for certain SPDEs at terminal time
  11. Deterministic neural networks
  12. Random neural networks
  13. Numerical Experiments
  14. Stochastic heat equation
  15. Heath-Jarrow-Morton (HJM) equation in interest rate theory
  16. ...and 14 more sections

Key Result

Lemma 2.2

The processes $(W^{(i)})_{i \in \mathbb{N}}$ defined in EqDefBMi are pairwise independent real-valued Brownian motions such that for every $t \in [0,T]$ it holds that where the sum converges with respect to $\Vert \cdot \Vert_{L^2(\Omega,\mathcal{F},\mathbb{P};Z)}$.

Figures (10)

  • Figure 1: Learning performance and computational time
  • Figure 2: Approximation for $m = 1$
  • Figure 3: Approximation for $m = 5$
  • Figure 4: Approximation for $m = 10$
  • Figure 6: Learning performance and computational time for $(J,K) \in \lbrace 1,3,5,7 \rbrace \times \lbrace 2 \rbrace$
  • ...and 5 more figures

Theorems & Definitions (51)

  • Definition 2.1
  • Lemma 2.2: daprato14
  • Definition 2.4
  • Proposition 2.5: Existence and uniqueness of mild solutions to \ref{['EqDefSPDE']}
  • Lemma 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 2.10
  • Theorem 2.11: Cameron-Martin
  • ...and 41 more