Deep learning based numerical approximation algorithms for stochastic partial differential equations
This work provides a new computational tool for researchers and practitioners who need to solve high-dimensional SPDEs, which are common in fields like finance and physics.
This paper introduces a deep learning algorithm to approximate solutions of stochastic partial differential equations (SPDEs) by using neural networks to model solutions along noise realizations. The method accurately solves stochastic heat equations, stochastic Black-Scholes equations, and Zakai equations in up to 100 space dimensions with short runtimes.
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.