Formal Verification of Stochastic Systems with ReLU Neural Network Controllers

TL;DR

This paper presents a formal verification framework for stochastic systems with ReLU neural network controllers, estimating safety probabilities and refining system abstractions to ensure safety within a confidence level.

Contribution

It introduces a novel SMC-based approach for safety verification of stochastic CPS with neural network controllers, including a refinement heuristic for improved bounds.

Findings

Effective safety probability bounds for stochastic systems

Successful application to robot navigation example

Comparison shows improved accuracy over existing methods

Abstract

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.