  In this work, we address the problem of formal safety verification for
stochastic cyber-physical systems (CPS) equipped with ReLU neural network (NN)
controllers. Our goal is to find the set of initial states from where, with a
predetermined confidence, the system will not reach an unsafe configuration
within a specified time horizon. Specifically, we consider discrete-time LTI
systems with Gaussian noise, which we abstract by a suitable graph. Then, we
formulate a Satisfiability Modulo Convex (SMC) problem to estimate upper bounds
on the transition probabilities between nodes in the graph. Using this
abstraction, we propose a method to compute tight bounds on the safety
probabilities of nodes in this graph, despite possible over-approximations of
the transition probabilities between these nodes. Additionally, using the
proposed SMC formula, we devise a heuristic method to refine the abstraction of
the system in order to further improve the estimated safety bounds. Finally, we
corroborate the efficacy of the proposed method with simulation results
considering a robot navigation example and comparison against a
state-of-the-art verification scheme.
