SEROAISE: Advancing ROA Estimation for ReLU and PWA Dynamics through Estimating Certified Invariant Sets

This paper presents a novel framework for constructing the Region of Attraction (RoA) for dynamics derived either from Piecewise Affine (PWA) functions or from Neural Networks (NNs) with Rectified Linear Units (ReLU) activation function. This method, described as Sequential Estimation of RoA based on Invariant Set Estimation (SEROAISE), computes a Lyapunov-like PWA function over a certified PWA invariant set. While traditional approaches search for Lyapunov functions by enforcing Lyapunov conditions over pre-selected domains, this framework enforces Lyapunov-like conditions over a certified invariant subset obtained using the Iterative Invariant Set Estimator(IISE). Compared to the state-of-the-art, IISE provides systematically larger certified invariant sets. In order to find a larger invariant subset, the IISE utilizes a novel concept known as the Non-Uniform Growth of Invariant Set (NUGIS). A number of examples illustrating the efficacy of the proposed methods are provided, including dynamical systems derived from learning algorithms. The implementation is publicly available at: https://github.com/PouyaSamanipour/SEROAISE.git.