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

TL;DR

POLAR introduces a novel polynomial arithmetic framework combining Bernstein Bézier form and symbolic remainders to efficiently verify neural-network controlled systems, overcoming limitations of existing methods with improved accuracy and performance.

Contribution

It integrates Bernstein Bézier form and symbolic remainders into Taylor Model arithmetic, enabling verification of NNCSs with non-differentiable activations and reducing error accumulation.

Findings

POLAR outperforms state-of-the-art tools in efficiency.

POLAR provides tighter overapproximations of reachable sets.

The framework effectively handles non-differentiable activation functions.

Abstract

We present POLAR, a polynomial arithmetic-based framework for efficient bounded-time reachability analysis of neural-network controlled systems (NNCSs). Existing approaches that leverage the standard Taylor Model (TM) arithmetic for approximating the neural-network controller cannot deal with non-differentiable activation functions and suffer from rapid explosion of the remainder when propagating the TMs. POLAR overcomes these shortcomings by integrating TM arithmetic with \textbf{Bernstein B{\'e}zier Form} and \textbf{symbolic remainder}. The former enables TM propagation across non-differentiable activation functions and local refinement of TMs, and the latter reduces error accumulation in the TM remainder for linear mappings in the network. Experimental results show that POLAR significantly outperforms the current state-of-the-art tools in terms of both efficiency and tightness of the reachable set overapproximation. The source code can be found in https://github.com/ChaoHuang2018/POLAR_Tool