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

arXiv: https://arxiv.org/abs/2406.10758 · HTML: https://mlsanity.com/paper/2406.10758/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2024.

## Built on by (most cited first, 4 of 4)

- [Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms](https://mlsanity.com/paper/2601.17562/) (2026)
- [Monotone Neural Policy Iteration for High-Dimensional First-Order Hamilton--Jacobi--Bellman Equations](https://mlsanity.com/paper/2605.07116/) (2026)
- [Viscosity-Informed Generative Actor-Critic for High-Dimensional Stochastic Optimal Control](https://mlsanity.com/paper/2605.15668/) (2026)
- [Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs](https://mlsanity.com/paper/2606.23827/) (2026)

## Abstract (arXiv)

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.