# Neural networks-based backward scheme for fully nonlinear PDEs

## TL;DR

This paper introduces a neural network-based backward scheme for efficiently solving high-dimensional fully nonlinear PDEs, demonstrating high accuracy and applicability to complex problems like control and Monge-Ampère equations.

## Contribution

It extends neural network methods to fully nonlinear PDEs, enabling simultaneous estimation of solutions, gradients, and Hessians through automatic differentiation.

## Key findings

- Accurate solutions for high-dimensional nonlinear PDEs.
- Effective handling of nonlinearity in Hessian terms.
- Successful application to control and Monge-Ampère equations.

## 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, while the Hessian is approximated by automatic differentiation of the gradient at previous step. This methodology extends to the fully nonlinear case the approach recently proposed in \cite{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, Monge-Amp{\`e}re equation and Hamilton-Jacobi-Bellman equation in portfolio optimization.