Statistical Certification of Neural Barrier Functions for invariance and Reach-Avoidance in Learning-Enabled Systems

Certifying safety and reach-avoidance for learning-enabled systems with unknown dynamics is challenging, as exact verification is intractable and finite data provide limited evidence. We propose a joint training-and-certification framework that combines multi-step statistical certification of neural barrier functions with reinforcement learning of control policies. Within this framework, PID-barrier feedback jointly trains the policy and barrier to improve safety and performance. After training, we certify probably approximately correct (PAC) multi-step safe invariant and reach-avoid sets using scenario approaches and the trained barrier function. Because the PAC statement bounds the portion of the certified set that may violate the property, rather than certifying individual states, the guarantees are reusable online as long as the system remains within the certified set and the dynamics and controller remain unchanged. The method attains $O(1/\alpha )$ sample complexity, improving on $O(1/{\alpha }^2)$ Monte Carlo bounds. Experiments on an inverted pendulum, Van der Pol oscillator, and 10-D Hopfield system demonstrated improved return-safety trade-offs and valid PAC certificates.