Guiding Reachability Learning with Pontryagin's Maximum Principle

Hamilton–Jacobi (HJ) reachability provides a framework for analyzing the safety of autonomous systems under worst-case disturbances. Specifically, its value function identifies states from which failure is unavoidable, while its spatial gradient determines how a controller should act to preserve safety. Computing this function with traditional grid-based solvers is prohibitively expensive in high dimensions. Neural methods such as DeepReach improve scalability, but training on the HJ partial differential equation residual alone provides limited direct supervision for the gradient, while existing methods that generate additional value labels can be costly. We propose a sample-efficient method that supplements residual training with value and gradient labels generated using Pontryagin’s Maximum Principle (PMP). At intervals aligned with the training curriculum, we initialize state–costate pairs from the network’s current level sets and integrate the PMP characteristic equations over a short horizon to generate supervision beyond the curriculum frontier. At the same time, since the network provides the initial conditions, generating these labels avoids solving the expensive two-point boundary-value problems associated with PMP, thereby enabling efficient training. In experiments with a 13-dimensional quadrotor and an eight-dimensional racing vehicle, the method reduces training time and improves the accuracy of the learned safety value function in challenging settings. We further deploy the learned value function as a safety filter on a full-scale electric vehicle driven near the friction limit and against an adversarial human driver instructed to leave the track.