A Deep Genetic Algorithm (Deep-Ga) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations

Abstract

Solving non-linear parabolic partial differential equations (PDEs) in high dimension becomes an interest for its curse of dimensionality problem. Recently, a deep learning method associated with a backward stochastic differential equation (deep-BSDE) to solve the PDEs draws intensive discussions for its remarkable results. In this paper we propose an alternative, namely the deep-GA method, to improve the performance of the former method. We embed the genetic algorithm (GA) into the BSDE solver to optimize the initial guess that results in a more efficient computation with higher accuracy. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes equation with default risk and Hamilton-Jacobi-Bellman equation. We compare the results of our method to the deep-BSDE and show that our method provides a better accuracy and efficiency than the deep-BSDE.

Keywords: high dimensionality, non-linear equations, Genetic algorithm, backward stochastic differential equation