Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations

TL;DR

This paper introduces a residual minimization framework for solving Hamilton-Jacobi equations using monotone discretizations, providing error estimates and scalability to high-dimensional problems.

Contribution

It develops a novel residual minimization approach with error bounds and neural network warm-start strategies for Hamilton-Jacobi equations.

Findings

Critical points of the residual functional are the unique solutions of the discretized scheme.

Error bounds relate the residual magnitude to the approximation error with explicit constants.

The method scales effectively to high-dimensional Eikonal and game-theoretic equations.

Abstract

We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order elliptic and parabolic problems. We prove that, under suitable monotonicity conditions, every critical point of the residual loss functional is the unique global minimizer and coincides with the solution of the discrete scheme. We derive \emph{a~posteriori} error estimates that bound the approximation error by the magnitude of the residual with explicit, computable constants, and extend the full analysis to time-dependent problems with implicit discretization of the time derivatives. A spectral analysis of the linearized system shows that the condition number scales as $O(\Delta x^{-1})$ for proper schemes, and as $O(\exp(\Delta x^{-1}))$ under a uniform ellipticity condition. These results quantify the increasing difficulty of solving the optimization problem on finer meshes, and motivates a progressive multi-level warm-start strategy using Artificial Neural Networks. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee that the solutions obtained converge to the unique viscosity solution as the mesh is refined. Numerical experiments demonstrate the scalability of the approach to high-dimensional Eikonal equations, level-set problems, and Hamilton--Jacobi--Isaacs equations with genuine second-order diffusion arising from stochastic differential games.