Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations

We derive sufficient conditions under which residual minimization yields well-posed discrete solutions for nonlinear equations defined by monotone finite--difference discretizations. Our analysis is motivated by the challenge of solving fully nonlinear Hamilton--Jacobi (HJ) equations in high dimensions by means of a Neural Network, which is trained by minimizing residuals arising from monotone discretizations of the Hamiltonian. While classical theory ensures that consistency and monotonicity imply convergence to the viscosity solution, treating these discrete systems as optimization problems introduces new analytical hurdles: solvability and the uniqueness of local minima do not follow from monotonicity alone. By establishing the well--posedness of these optimization--based solvers, our framework enables the adaptation of Level Set Methods to high--dimensional settings, unlocking new capabilities in applications such as high--dimensional segmentation and interface tracking. Finally, we observe that these arguments extend almost directly to degenerate elliptic or parabolic PDEs on graphs equipped with monotone graph Laplacians.

Source: arXiv:2601.21764v1 - http://arxiv.org/abs/2601.21764v1 PDF: https://arxiv.org/pdf/2601.21764v1 Original Link: http://arxiv.org/abs/2601.21764v1