A Numerical Method Of High-Dimensional Nonlinear Partial Differential Equations Based On Deep Learning

Solving high-dimensional partial differential equations(PDEs)is one of the most challenging topics in Applied Mathematics.Currently,deep learning(DL)method is a research hotspot in solving high-dimensional PDEs.It is expected that deep learning method can overcome the curse of dimensionality problem of high-dimensional PDEs which cannot be solved by classical numerical methods.However,it is still a rather difficult task to transform the problem of solving PDEs into a learning problem and make the deep learning technology fit the specific problem of high-dimensional PDEs.In this paper,based on the backward stochastic differential equation(BSDE)representation of PDEs,we use deep neural network to estimate the solutions and gradients of the equations at the same time.By using the nonlinear Feynman-KAC formula,the solution of the high-dimensional PDEs can be expressed as the solution of the corresponding BSDE equations.Then the numerical problem is expressed as a stochastic control problem,and the gradient operator of the solution function is regarded as a policy function,and this policy function is approximated by a deep neural network,and then the numerical solution of the high-dimensional PDEs is obtained.In this paper,through the analysis of the existing neural network(NN)parameter optimization methods,a new improved optimization algorithm is proposed.In addition,the mini-batch sample extraction method is studied and a new sample extraction method is proposed.Finally,the proposed method is applied to several practical high-dimensional nonlinear PDEs,and the accuracy and efficiency of them are satisfactory.