  Hamilton-Jacobi (HJ) reachability analysis provides a formal method for
guaranteeing safety in constrained control problems. It synthesizes a value
function to represent a long-term safe set called feasible region. Early
synthesis methods based on state space discretization cannot scale to
high-dimensional problems, while recent methods that use neural networks to
approximate value functions result in unverifiable feasible regions. To achieve
both scalability and verifiability, we propose a framework for synthesizing
verified neural value functions for HJ reachability analysis. Our framework
consists of three stages: pre-training, adversarial training, and
verification-guided training. We design three techniques to address three
challenges to improve scalability respectively: boundary-guided backtracking
(BGB) to improve counterexample search efficiency, entering state
regularization (ESR) to enlarge feasible region, and activation pattern
alignment (APA) to accelerate neural network verification. We also provide a
neural safety certificate synthesis and verification benchmark called
Cersyve-9, which includes nine commonly used safe control tasks and supplements
existing neural network verification benchmarks. Our framework successfully
synthesizes verified neural value functions on all tasks, and our proposed
three techniques exhibit superior scalability and efficiency compared with
existing methods.
