用深度学习实现高效求解哈密顿-雅可比方程的隐式解法。
Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations
- 基于特征线方法构造隐式解公式，避免复杂的变换和轨迹计算。
- 在40维问题上仍保持高精度与计算效率，适用于非凸哈密顿量。
- 适合高维动态规划、最优控制等需要快速求解PDE的场景。
本文提出一种哈密顿-雅可比偏微分方程（HJ PDE）的隐式解公式，该公式通过特征线方法推导，在哈密顿量或初值函数为凸时与霍普夫和拉克斯公式一致。该方法无需对哈密顿量或初值进行勒让德变换，也无需显式追踪特征轨迹，提供了一种简洁高效的数值求解黏性解的方式。进一步提出基于深度学习的方法来学习该隐式解公式，利用深度学习的无网格特性，确保在高维问题中的可扩展性。在此框架下，设计了一种算法，对状态依赖的哈密顿量以分段线性方式逼近特征曲线。大量实验表明，该方法即使在非凸哈密顿量下也能获得高精度解，并展现出卓越的可扩展性，可高效求解高达40维的问题。
原文摘要 · Abstract (English)
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.
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