Neural networks-based backward scheme for fully nonlinear PDEs

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient multi-layer neural networks, while Hessian is approximated automatic differentiation of at previous step. This methodology extends to case approach recently proposed in Huré et al. (Math Comput 89(324):1547–1579, 2020) semi-linear PDEs. Numerical tests illustrate performance accuracy our on several examples dimension with non-linearity term including linear quadratic control problem diffusion coefficient, Monge-Ampère equation Hamilton–Jacobi–Bellman portfolio optimization.