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As neural networks (NNs) become more prevalent in safety-criticalapplications such as control of vehicles, there is a growing need to certifythat systems with NN components are safe. This paper presents a set of backwardreachability approaches for safety certification of neural feedback loops(NFLs), i.e., closed-loop systems with NN control policies. While backwardreachability strategies have been developed for systems without NN components,the nonlinearities in NN activation functions and general noninvertibility ofNN weight matrices make backward reachability for NFLs a challenging problem.To avoid the difficulties associated with propagating sets backward throughNNs, we introduce a framework that leverages standard forward NN analysis toolsto efficiently find over-approximations to backprojection (BP) sets, i.e., setsof states for which an NN policy will lead a system to a given target set. Wepresent frameworks for calculating BP over approximations for both linear andnonlinear systems with control policies represented by feedforward NNs andpropose computationally efficient strategies. We use numerical results from avariety of models to showcase the proposed algorithms, including ademonstration of safety certification for a 6D system.

Backward Reachability Analysis of Neural Feedback Loops: Techniques for Linear and Nonlinear Systems