Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks

TL;DR

This paper introduces a neural network-based finite difference method to efficiently approximate viscosity solutions of Hamilton-Jacobi equations, especially in higher dimensions where traditional methods struggle.

Contribution

It proposes a novel deep neural network approach combined with a least squares principle for solving HJ equations, overcoming grid limitations of classical methods.

Findings

Effective in higher dimensions where traditional methods fail

Converges to viscosity solutions under certain conditions

Demonstrated on various canonical HJ equations

Abstract

We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi (HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based method to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility.