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Safety certification of data-driven control techniques remains a major openproblem. This work investigates backward reachability as a framework forproviding collision avoidance guarantees for systems controlled by neuralnetwork (NN) policies. Because NNs are typically not invertible, existingmethods conservatively assume a domain over which to relax the NN, which causesloose over-approximations of the set of states that could lead the system intothe obstacle (i.e., backprojection (BP) sets). To address this issue, weintroduce DRIP, an algorithm with a refinement loop on the relaxation domain,which substantially tightens the BP set bounds. Furthermore, we introduce aformulation that enables directly obtaining closed-form representations ofpolytopes to bound the BP sets tighter than prior work, which required solvinglinear programs and using hyper-rectangles. Furthermore, this work extends theNN relaxation algorithm to handle polytope domains, which further tightens thebounds on BP sets. DRIP is demonstrated in numerical experiments on controlsystems, including a ground robot controlled by a learned NN obstacle avoidancepolicy.

DRIP: Domain Refinement Iteration with Polytopes for Backward Reachability Analysis of Neural Feedback Loops