Neural networks-based backward scheme for fully nonlinear PDEs

Neural networks-based backward scheme for fully nonlinear PDEs Huyˆ enPham † Xavier Warin ‡ August 2, 2019 Abstract 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 by multi-layer neural networks, through a sequence of learning problems obtained from the minimization of suitable quadratic loss functions and training simulations. This methodology extends to the fully nonlinear case the approach recently proposed in [HPW19] for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient. MSC Classification: 60H35, 65C20, 65M12. 1 Introduction This paper is devoted to the resolution in high dimension of fully nonlinear parabolic partial differential equations (PDEs) of the form null tu f ( .,.,u,D xu,D 2 xu) 0, on [0,T) R d, u(T,.)