Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

Developing efficient numerical algorithms for high dimensional (say, hundreds of dimensions) partial differential equations (PDEs) has been one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the "curse of dimensionality" [1], namely, as the dimensionality grows, the complexity of the algorithms grows exponentially. For this reason, there are only a limited number of cases where practical 2 high dimensional algorithms have been developed. For linear parabolic PDEs, one can use the Feynman-Kac formula and Monte Carlo methods to develop efficient algorithms to evaluate solutions at any given space-time locations. For a class of inviscid Hamilton-Jacobi equations, Darbon & Osher have recently developed an algorithm which performs numerically well in the case of such high dimensional inviscid Hamilton-Jacobi equations; see [9]. Darbon & Osher's algorithm is based on results from compressed sensing and on the Hopf formulas for the Hamilton-Jacobi equations. A general algorithm for (nonlinear) parabolic PDEs based on the Feynman-Kac and Bismut-Elworthy-Li formula and a multilevel decomposition of Picard iteration was developed in [11] and has been shown to be quite efficient on a number examples in finance and physics.