# Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning

arXiv: https://arxiv.org/abs/2406.10920 · HTML: https://mlsanity.com/paper/2406.10920/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 48 references that are in the map.
- First posted on arXiv 16 Jun 2024.

## Builds on (most cited first, 20 of 20)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach](https://mlsanity.com/paper/2108.06655/) (2021)
- [Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls](https://mlsanity.com/paper/2010.14087/) (2020)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [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)
- [Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations](https://mlsanity.com/paper/1711.03314/) (2017)
- [On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations](https://mlsanity.com/paper/1906.09502/) (2019)
- [Algorithm for Overcoming the Curse of Dimensionality for State-dependent Hamilton-Jacobi equations](https://mlsanity.com/paper/1704.02524/) (2017)
- [Policy iteration for the deterministic control problems -- a viscosity approach](https://mlsanity.com/paper/2301.00419/) (2023)
- [Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems](https://mlsanity.com/paper/2109.14849/) (2021)
- [A Computationally Efficient Hamilton-Jacobi-based Formula for State-Constrained Optimal Control Problems](https://mlsanity.com/paper/2106.13440/) (2021)
- [On representation formulas for optimal control: A Lagrangian perspective](https://mlsanity.com/paper/2204.02050/) (2022)

## Built on by (most cited first, 5 of 5)

- [Physics-Informed Neural Networks for Optimal Vaccination Plan in SIR Epidemic Models](https://mlsanity.com/paper/2502.19890/) (2025)
- [Solving nonconvex Hamilton--Jacobi--Isaacs equations with PINN-based policy iteration](https://mlsanity.com/paper/2507.15455/) (2025)
- [Physics-Informed Policy Iteration for High-Dimensional Hamilton--Jacobi--Bellman Equations: Interior Error Bounds without Boundary Data](https://mlsanity.com/paper/2508.01718/) (2025)
- [Towards General Language-Conditioned Latent Safety Filters](https://mlsanity.com/paper/2608.00315/) (2026)
- [Monotone Neural Policy Iteration for High-Dimensional First-Order Hamilton--Jacobi--Bellman Equations](https://mlsanity.com/paper/2605.07116/) (2026)

## Abstract (arXiv)

The framework of deep operator network (DeepONet) has been widely exploited thanks to its capability of solving high dimensional partial differential equations. In this paper, we incorporate DeepONet with a recently developed policy iteration scheme to numerically solve optimal control problems and the corresponding Hamilton--Jacobi--Bellman (HJB) equations. A notable feature of our approach is that once the neural network is trained, the solution to the optimal control problem and HJB equations with different terminal functions can be inferred quickly thanks to the unique feature of operator learning. Furthermore, a quantitative analysis of the accuracy of the algorithm is carried out via comparison principles of viscosity solutions. The effectiveness of the method is verified with various examples, including 10-dimensional linear quadratic regulator problems (LQRs).