# Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

arXiv: https://arxiv.org/abs/2606.23827 · HTML: https://mlsanity.com/paper/2606.23827/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [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)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [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)
- [Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations](https://mlsanity.com/paper/1507.04769/) (2015)
- [Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs](https://mlsanity.com/paper/1702.04400/) (2017)
- [Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation](https://mlsanity.com/paper/2002.08625/) (2020)
- [Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations](https://mlsanity.com/paper/1711.03314/) (2017)
- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-dependent Riccati Equations](https://mlsanity.com/paper/2103.04091/) (2021)
- [Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks](https://mlsanity.com/paper/2406.10758/) (2024)
- [Data-driven Tensor Train Gradient Cross Approximation for Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/2205.05109/) (2022)
- [Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems](https://mlsanity.com/paper/2305.06122/) (2023)
- [Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control](https://mlsanity.com/paper/1408.5354/) (2014)
- [Numerical realization of the Mortensen observer via a Hessian-augmented polynomial approximation of the value function](https://mlsanity.com/paper/2310.20321/) (2023)
- [Computing Solutions to the Polynomial-Polynomial Regulator Problem](https://mlsanity.com/paper/2410.22291/) (2024)
- [Separable Approximations of Optimal Value Functions and Their Representation by Neural Networks](https://mlsanity.com/paper/2502.08559/) (2025)
- [Recovery of the optimal control value function in reproducing kernel Hilbert spaces from verification conditions](https://mlsanity.com/paper/2512.07477/) (2025)

## Abstract (arXiv)

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.