Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs
For researchers in optimal control, this method improves sample efficiency and accuracy of value function approximation, though it is an incremental extension of existing regression techniques.
A data-driven method approximates value functions for deterministic optimal control problems using gradients and Hessians from the Pontryagin Maximum Principle, achieving up to an order-of-magnitude reduction in training samples compared to value-only regression.
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.