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

Systems and Control;Machine Learning;Robotics;Systems and Control