Deep learning based numerical approximation algorithms for stochastic partial differential equations
Christian Beck
Sebastian Becker
Patrick Cheridito
Arnulf Jentzen
Ariel Neufeld
Sep 2026
Machine Learning
Data Science
Abstract
In this article, we introduce 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.
