# Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations

arXiv: https://arxiv.org/abs/2501.19351 · HTML: https://mlsanity.com/paper/2501.19351/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 31 Jan 2025.

## Builds on (most cited first, 13 of 13)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere](https://mlsanity.com/paper/1605.01799/) (2016)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [Convergence Rates for Stochastic Approximation: Biased Noise with Unbounded Variance, and Applications](https://mlsanity.com/paper/2312.02828/) (2023)
- [Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations](https://mlsanity.com/paper/1507.04769/) (2015)
- [Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations](https://mlsanity.com/paper/1711.03314/) (2017)
- [Algorithm for Overcoming the Curse of Dimensionality for State-dependent Hamilton-Jacobi equations](https://mlsanity.com/paper/1704.02524/) (2017)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems](https://mlsanity.com/paper/2110.02541/) (2021)

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

- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving](https://mlsanity.com/paper/2602.03816/) (2026)
- [Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations](https://mlsanity.com/paper/2512.23829/) (2025)
- [Implicit Neural Optimal Transport via Fixed-Point Optimization](https://mlsanity.com/paper/2605.10792/) (2026)

## Abstract (arXiv)

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.