Convolutional neural network based simulation and analysis for backward stochastic partial differential equations

Abstract

We develop a generic convolutional neural network (CNN) based numerical scheme to simulate the 2-tuple adapted strong solution to a unified system of backward stochastic partial differential equations (B-SPDEs) driven by Brownian motions, which can be used to model many real-world system dynamics such as optimal control and differential game problems. The dynamics of the scheme is modeled by a CNN through conditional expectation projection. It consists of two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with mean-square error estimation and almost sure (a.s.) convergence supported by both theoretical proof and numerical examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution to the system under both conventional and Malliavin derivatives with general local Lipschitz and linear growth conditions.