ConDR: Conservative Reachability through Neural Hamilton-Jacobi Subsolutions

Kendra Givens, Taylor T. Johnson, Bryan C. Ward, Thomas Beckers

Hamilton-Jacobi (HJ) Reachability identifies the unsafe states of a dynamical system from which it can be forced into failure under worst-case disturbance. However, classical grid-based solvers scale exponentially with state dimension. DeepReach improves scalability by replacing the grid with a neural value function trained on the HJ-Isaacs (HJI) variational inequality (VI), but penalizes both sides of the residual equally. This allows the learned value to lie above the true one and unsafe states to be misclassified as safe. We propose Conservative DeepReach (ConDR) which aims to learn a subsolution of the HJI VI such that the learned unsafe set contains the true one. ConDR enforces the obstacle branch and terminal condition by construction, pushes the HJI residual to a safety margin $ε$ below zero with a one-sided loss, and prevents the degenerate solution of classifying everything as unsafe. We prove that the exact solution of the margin-shifted VI that ConDR trains toward is a subsolution of the original VI so its unsafe set contains the true one and its value lies within $ετ$ of the true value function. We compare against DeepReach and MADR in the established systems of…