Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations
Yesom Park, Stanley Osher
TL;DR
This work addresses solving high-dimensional Hamilton-Jacobi PDEs by introducing a novel implicit solution formula derived from the characteristic equations, which avoids Legendre transforms and explicit trajectory integration. A mesh-free neural solver based on implicit neural representations learns the solution as a Lipschitz function by minimizing the residual of the implicit formula over randomly sampled spacetime collocation points, enabling scalable inference up to 40 dimensions. The authors extend the framework to state-dependent Hamiltonians using a short-time, piecewise-linear characteristic approximation and a time-marching algorithm, achieving accurate viscosity solutions for both convex and nonconvex problems. Overall, the approach bridges the method of characteristics, Hopf-Lax/Bellman principles, and modern neural solvers to deliver a practical, scalable tool for high-dimensional HJ PDEs with broad potential in optimal control, level-set methods, and related dynamics.
Abstract
This paper presents an implicit solution formula for the Hamilton-Jacobi partial differential equation (HJ PDE). The formula is derived using the method of characteristics and is shown to coincide with the Hopf and Lax formulas in the case where either the Hamiltonian or the initial function is convex. It provides a simple and efficient numerical approach for computing the viscosity solution of HJ PDEs, bypassing the need for the Legendre transform of the Hamiltonian or the initial condition, and the explicit computation of individual characteristic trajectories. A deep learning-based methodology is proposed to learn this implicit solution formula, leveraging the mesh-free nature of deep learning to ensure scalability for high-dimensional problems. Building upon this framework, an algorithm is developed that approximates the characteristic curves piecewise linearly for state-dependent Hamiltonians. Extensive experimental results demonstrate that the proposed method delivers highly accurate solutions, even for nonconvex Hamiltonians, and exhibits remarkable scalability, achieving computational efficiency for problems up to 40 dimensions.