Deep backward schemes for high-dimensional nonlinear PDEs

Abstract

We propose new machine learning schemes for solving high dimensional\nnonlinear partial differential equations (PDEs). Relying on the classical\nbackward stochastic differential equation (BSDE) representation of PDEs, our\nalgorithms estimate simultaneously the solution and its gradient by deep neural\nnetworks. These approximations are performed at each time step from the\nminimization of loss functions defined recursively by backward induction. The\nmethodology is extended to variational inequalities arising in optimal stopping\nproblems. We analyze the convergence of the deep learning schemes and provide\nerror estimates in terms of the universal approximation of neural networks.\nNumerical results show that our algorithms give very good results till\ndimension 50 (and certainly above), for both PDEs and variational inequalities\nproblems. For the PDEs resolution, our results are very similar to those\nobtained by the recent method in \\cite{weinan2017deep} when the latter\nconverges to the right solution or does not diverge. Numerical tests indicate\nthat the proposed methods are not stuck in poor local minimaas it can be the\ncase with the algorithm designed in \\cite{weinan2017deep}, and no divergence is\nexperienced. The only limitation seems to be due to the inability of the\nconsidered deep neural networks to represent a solution with a too complex\nstructure in high dimension.\n