FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs

Abstract

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial differential equations (PDEs), which is related to the FBSDEs from the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the path-wise difference between two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solution, respectively. The proposed algorithms are shown to generate DNN solution for a 100-dimensional Black–Scholes–Barenblatt equation, which is accurate in a finite region in the solution space, and has a convergence rate close to that of the Euler–Maruyama scheme used for discretizing the FBSDEs.