Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations

2025-01-31

AI Summary

Traditional methods for solving high-dimensional Hamilton–Jacobi (HJ) equations suffer from grid dependence, restrictions to convex Hamiltonians, and requirements on initial-value regularity. To address these limitations, this work proposes a learnable implicit solution formula grounded in the method of characteristics. By approximating characteristic curves with piecewise-linear segments and jointly modeling the implicit solution via deep neural networks, our approach circumvents both Legendre transformation and explicit trajectory integration—enabling accurate learning of viscosity solutions for non-convex Hamiltonians and non-smooth initial data. The resulting framework is mesh-free, scalable to 40 dimensions, and preserves theoretical consistency with the underlying PDE. It achieves substantial gains in computational efficiency and generalization capability over prior approaches. This work establishes a new paradigm for high-dimensional nonlinear optimal control and differential games.

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• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds

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• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
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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.

Problem

• Hamilton-Jacobi equations
• high-dimensional problems
• efficient solution methods

Innovation

• Neural Networks
• Hamilton-Jacobi PDE
• Deep Learning