POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems.

We propose POLAR, a \textbf{pol}ynomial \textbf{ar}ithmetic framework that leverages polynomial overapproximations with interval remainders for bounded-time reachability analysis of neural network-controlled systems (NNCSs). Compared with existing arithmetic approaches that use standard Taylor models, our framework uses a novel approach to iteratively overapproximate the neuron output ranges layer-by-layer with a combination of Bernstein polynomial interpolation for continuous activation functions and Taylor model arithmetic for the other operations. This approach can overcome the main drawback in the standard Taylor model arithmetic, i.e. its inability to handle functions that cannot be well approximated by Taylor polynomials, and significantly improve the accuracy and efficiency of reachable states computation for NNCSs. To further tighten the overapproximation, our method keeps the Taylor model remainders symbolic under the linear mappings when estimating the output range of a neural network. We show that POLAR can be seamlessly integrated with existing Taylor model flowpipe construction techniques, and demonstrate that POLAR significantly outperforms the current state-of-the-art techniques on a suite of benchmarks.