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

2026-06-22

AI Summary

This study addresses the challenge of efficiently solving the Hamilton–Jacobi–Bellman equation for optimal control of high-dimensional nonlinear control-affine systems. The authors propose a novel approach that leverages the Pontryagin maximum principle to generate training data comprising the value function, its gradient, and Hessian. By integrating hyperbolic cross sparse polynomial expansions with weighted least squares regression, they construct a high-fidelity approximation model. A key innovation lies in explicitly incorporating Hessian information into the supervised learning framework, complemented by a partial Hessian strategy that balances computational efficiency and approximation accuracy. Experimental results demonstrate that, in high-dimensional settings, the proposed method reduces the required number of training samples by nearly an order of magnitude compared to approaches using only function values, while significantly improving both value function approximation accuracy and closed-loop control performance.

Abstract

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.

Problem

• Hamilton-Jacobi-Bellman PDEs
• optimal control
• value function approximation
• sample complexity
• high-dimensional problems

Innovation

• Hessian augmentation
• Hamilton-Jacobi-Bellman PDEs
• data-driven optimal control
• sparse polynomial regression
• Pontryagin Maximum Principle