A Deep Learning-based Method For Solving High-dimensional Partical Differential Equations

In this paper,the deep learning method is applied to solve the numerical solution of high-dimensional nonlinear parabolic partial differential equations.An deep learning-based numerical methods called deep BSDE algorithm is proposed.Through the nonlinear Feynman-Kac formula,the solutions of partial differential equations can be formulated equivalently by the solution to the related stochastic differential equation.The problem of solving the numerical solution can be regarded as a stochastic control problem.The gradient operator of the solution can be regarded as a strategy function,and the multi-feedback neural network is used to approximate the strategy function.In this paper,the deep learning method is used to train this neural network.The deep learning method we used is AdamW optimization algorithm.This paper is divided into four parts.The first part is the introduction,which mainly introduces the development history of partial differential equations and deep learning methods.As the same time the research progress of numerical solutions of partial differential equations is introduced.The second part lists some knowledge points that will be used below.The main ideas of the deep BSDE algorithm and the specific algorithm steps are expounded.The third part gives the numerical simulation results of two specific partial differential equations(ie Allen-Cahn equation and Hamilton-JacobiBellman equation)under the deep BSDE algorithm.The fourth part summarizes the main ideas and achievements of the proposed algorithm,and points out the highlights and shortcomings.Finally the direction of further research is given.