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Deep learning based numerical approximation algorithms for stochastic partial differential equations

Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld

TL;DR

This work develops a deep learning–based method for solving high-dimensional SPDEs by learning the solution along fixed realizations of the driving noise and aggregating over multiple noise trajectories to obtain empirical solution distributions. The core approach combines a splitting-up time discretization, a conditioned Feynman–Kac representation, and recursive $L^2$ minimization reformulations, with neural networks trained by stochastic gradient descent to approximate conditional expectations. The framework is instantiated in a special-case algorithm and a general framework that accommodates higher-order SDE schemes and ML enhancements, and is validated on four challenging examples (stochastic heat with additive and multiplicative noise, stochastic Black–Scholes with multiplicative noise, and Zakai equations), achieving accurate results and scalable performance up to 100 space dimensions. The method yields fast runtimes and distributional information (means, variances) crucial for applications in physics, engineering, and nonlinear filtering, demonstrating practical impact for high-dimensional SPDEs.

Abstract

In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Deep learning based numerical approximation algorithms for stochastic partial differential equations

TL;DR

This work develops a deep learning–based method for solving high-dimensional SPDEs by learning the solution along fixed realizations of the driving noise and aggregating over multiple noise trajectories to obtain empirical solution distributions. The core approach combines a splitting-up time discretization, a conditioned Feynman–Kac representation, and recursive minimization reformulations, with neural networks trained by stochastic gradient descent to approximate conditional expectations. The framework is instantiated in a special-case algorithm and a general framework that accommodates higher-order SDE schemes and ML enhancements, and is validated on four challenging examples (stochastic heat with additive and multiplicative noise, stochastic Black–Scholes with multiplicative noise, and Zakai equations), achieving accurate results and scalable performance up to 100 space dimensions. The method yields fast runtimes and distributional information (means, variances) crucial for applications in physics, engineering, and nonlinear filtering, demonstrating practical impact for high-dimensional SPDEs.

Abstract

In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Paper Structure

This paper contains 15 sections, 99 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Derivation of the proposed approximation algorithm
  3. Temporal approximations
  4. An approximate Feynman--Kac representation
  5. Formulation as iterative recursive minimization problems
  6. Deep neural network approximations
  7. Stochastic gradient descent based minimization
  8. Temporal discretization of the auxiliary stochastic process
  9. Description of the proposed approximation algorithm in a special case
  10. Description of the proposed approximation algorithm in the general case
  11. Examples
  12. Stochastic heat equation with additive noise
  13. Stochastic heat equation with multiplicative noise
  14. Stochastic Black--Scholes equation with multiplicative noise
  15. Zakai equations

Figures (1)

  • Figure 1: Histogram of 1000 approximate realizations of $X_1(0)$ for the stochastic heat equation with additive noise \ref{['eq:ex-heat-add']} in the case $d=1$.