Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion
This addresses numerical challenges in SPDEs for fields like physics and finance, but it appears incremental as it builds on existing Wiener chaos methods with neural networks.
The paper tackles solving stochastic partial differential equations (SPDEs) by using neural networks within a truncated Wiener chaos expansion framework, providing approximation rates and applying it to examples like the stochastic heat equation.
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.