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

Carlos Esteve-Yagüe, Richard Tsai, Alex Massucco

TL;DR

This work introduces a neural-network framework to approximate viscosity solutions of high-dimensional Hamilton-Jacobi equations by minimizing a least-squares loss built on a monotone, consistent finite-difference Hamiltonian, specifically the Lax-Friedrichs scheme. By proving that critical points of the FD-based loss converge to the viscosity solution under suitable choices of the diffusion parameter $\alpha$ and grid spacing $\delta$, the approach addresses non-uniqueness and instability issues typical of PINNs for HJ equations. The authors develop SGD-based training with domain resampling, analyze data-efficiency strategies, and demonstrate the method across Eikonal, time-dependent HJs, and differential-game problems (Reeds-Shepp car, pursuit-evasion), obtaining accurate approximations in dimensions up to $d=10$ with comparative runtimes on CPU. The combination of FD-induced regularisation, shifted-grid averaging, and supervised data offers a practical path toward scalable, accurate high-dimensional HJ solvers with potential impact on optimal control and differential games. The work also includes rigorous proofs of uniqueness and convergence, along with extensive numerical experiments highlighting data-distribution choices and resampling benefits.

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.

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

TL;DR

This work introduces a neural-network framework to approximate viscosity solutions of high-dimensional Hamilton-Jacobi equations by minimizing a least-squares loss built on a monotone, consistent finite-difference Hamiltonian, specifically the Lax-Friedrichs scheme. By proving that critical points of the FD-based loss converge to the viscosity solution under suitable choices of the diffusion parameter and grid spacing , the approach addresses non-uniqueness and instability issues typical of PINNs for HJ equations. The authors develop SGD-based training with domain resampling, analyze data-efficiency strategies, and demonstrate the method across Eikonal, time-dependent HJs, and differential-game problems (Reeds-Shepp car, pursuit-evasion), obtaining accurate approximations in dimensions up to with comparative runtimes on CPU. The combination of FD-induced regularisation, shifted-grid averaging, and supervised data offers a practical path toward scalable, accurate high-dimensional HJ solvers with potential impact on optimal control and differential games. The work also includes rigorous proofs of uniqueness and convergence, along with extensive numerical experiments highlighting data-distribution choices and resampling benefits.

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.
Paper Structure (31 sections, 6 theorems, 167 equations, 16 figures, 6 tables, 2 algorithms)

This paper contains 31 sections, 6 theorems, 167 equations, 16 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Example
  3. Deep Learning and Partial Differential Equations
  4. Why finite-difference approximations?
  5. Larger domain of dependence/influence.
  6. Simpler loss gradient evaluations.
  7. Convergence to the viscosity solutions.
  8. Finite difference residual functionals
  9. A sufficient condition for residuals to be zero
  10. Uniqueness of critical points by regularisation of the Hamiltonian
  11. The least square principle involving a numerical Hamiltonian and supervised data
  12. Incorporating additional labelled data
  13. Time dependent problems
  14. Numerics
  15. The proposed algorithms
  16. ...and 16 more sections

Key Result

Theorem 1

Let $H$ be a Hamiltonian satisfying cond H, and let $\Omega=(0,1)^d,$$d\in \mathbb{N}$. Consider the uniform Cartesian grid $\Omega_\delta := \delta \mathbb{Z}^d\cap \Omega$, for some $\delta=1/N$ and $N\in\mathbb{N}$, and the functional $\widehat{\mathcal{R}} (\cdot)$ defined in R hat delta. Let $u then

Figures (16)

  • Figure 1: Four weak solutions of the Eikonal equation $( \partial_x u )^2 -1 =0$ in $\Omega := (0, 1)$ with boundary condition $u(0) = u(1) = 0$, correspodning to four global minimizers to the functional \ref{['PINNs functional intro']}.
  • Figure 2: The numerical approximation of the Eikonal equation in $\Omega = (-3,3)^2$ constructed by Algorithm \ref{['alg: SGD']}. The top-left plot corresponds to the NN at initialisation. Then, from left to right and from top to bottom, each plot corresponds to the approximation after every iteration. The values of $\alpha$ and $\delta$ are taken as in \ref{['params alpha delta numerics']}.
  • Figure 3: (Eikonal equation in a square in $\mathbb{R}^2$) Evolution of the Mean Square Error and the $L^\infty$-Error after each round of Algorithm \ref{['alg: training']}, for the Eikonal equation in $(-3,3)^2$. The parameters $\alpha$ and $\delta$ in each of the five training rounds are taken as in \ref{['params alpha delta numerics']}. The experiment is repeated 10 independent times. The solid line represents the average error, and the shadowed area represents the interval within one standard deviation.
  • Figure 4: (Eikonal equation in a hypercube in $\mathbb{R}^8$) Evolution of the MSE (left) and the $L^\infty$-error (right) throughout the iterations of Algorithm \ref{['alg: training']}, for the Experiments 4, 5 and 6 reported in Table \ref{['tab: experiments d-dimensional cube']}. These correspond to the Eikonal equation in an 8-dimensional cube. Each experiment is repeated 10 times with different random seeds. The solid lines represent the average error over the 10 experiments, and the shadowed area represents the interval within one standard deviation.
  • Figure 5: (Eikonal equation in an annulus in $\mathbb{R}^{10}$) Evolution of the MSE (left) and the $L^\infty$-error (right) throughout the iterations of Algorithm \ref{['alg: training']}, for the experiments 4,5 and 6 reported in Table \ref{['tab: experiments d-dimensional annulus']}. These correspond to the Eikonal equation in a 10-dimensional annulus. Each experiment is repeated 10 times with different random seeds. The solid lines represent the average error over the 10 experiments, and the shadowed area represents the interval within one standard deviation.
  • ...and 11 more figures

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 3 more