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

Chao Huang, Jiameng Fan, Xin Chen, Wenchao Li, Qi Zhu

Keywords: Neural network controlled systems, safety, verification, Taylor model arithmetic

Abstract:

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 via a combination of Bernstein polynomial interpolation for continuous activation functions and Taylor model arithmetic for the other operations. This approach overcomes 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 propagating Taylor models across the neural-network controller.
We show that POLAR can be seamlessly integrated with existing Taylor model flowpipe construction techniques, and POLAR significantly outperforms the current state-of-the-art techniques on a suite of benchmarks.

One-sentence Summary: A new Taylor model arithmetic based approach for reachability analysis of neural network controlled systems, with the current best performance.