# Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations

arXiv: https://arxiv.org/abs/1907.05317 · HTML: https://mlsanity.com/paper/1907.05317/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2019.

## Built on by (most cited first, 30 of 40)

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- [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)
- [Approximating Discontinuous Nash Equilibrial Values of Two-Player General-Sum Differential Games](https://mlsanity.com/paper/2207.01773/) (2022)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation](https://mlsanity.com/paper/2401.05629/) (2024)
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- [Formally Verified Physics-Informed Neural Control Lyapunov Functions](https://mlsanity.com/paper/2409.20528/) (2024)
- [Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs](https://mlsanity.com/paper/2207.09299/) (2022)
- [Offline Supervised Learning V.S. Online Direct Policy Optimization: A Comparative Study and A Unified Training Paradigm for Neural Network-Based Optimal Feedback Control](https://mlsanity.com/paper/2211.15930/) (2022)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [Value Approximation for Two-Player General-Sum Differential Games with State Constraints](https://mlsanity.com/paper/2311.16520/) (2023)
- [Pontryagin Neural Operator for Solving Parametric General-Sum Differential Games](https://mlsanity.com/paper/2401.01502/) (2024)
- [End-to-End Training of High-Dimensional Optimal Control with Implicit Hamiltonians via Jacobian-Free Backpropagation](https://mlsanity.com/paper/2510.00359/) (2025)
- [An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps](https://mlsanity.com/paper/2203.16874/) (2022)
- [Neural Network Optimal Feedback Control with Guaranteed Local Stability](https://mlsanity.com/paper/2205.00394/) (2022)
- [Nonuniform random feature models using derivative information](https://mlsanity.com/paper/2410.02132/) (2024)
- [Continuous-Time Value Iteration for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2509.09135/) (2025)
- [Neural network optimal feedback control with enhanced closed loop stability](https://mlsanity.com/paper/2109.07466/) (2021)
- [Multi-level Optimal Control with Neural Surrogate Models](https://mlsanity.com/paper/2402.07763/) (2024)
- [Systematic construction of continuous-time neural networks for linear dynamical systems](https://mlsanity.com/paper/2403.16215/) (2024)
- [Domain Adaptive Safety Filters via Deep Operator Learning](https://mlsanity.com/paper/2410.14528/) (2024)
- [Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions](https://mlsanity.com/paper/2508.07559/) (2025)
- [Optimality-Informed Neural Networks for Solving Parametric Optimization Problems](https://mlsanity.com/paper/2512.20270/) (2025)
- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)

## Abstract (arXiv)

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semi-global solutions to HJB equations for general high-dimensional nonlinear systems and compute candidate optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known physics of the problem and using the partially-trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations corresponding to the attitude control of a six-dimensional nonlinear rigid body, and nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time feedback control of these systems.