Replacing K-infinity Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems

In this paper, a systematic framework is presented for determining piecewise affine PWA barrier functions and their corresponding invariant sets for dynamical systems identified via Rectified Linear Unit (ReLU) neural networks or their equivalent PWA representations. A common approach to determining the invariant set is to use Nagumo's condition, or to utilize the barrier function with a class K-infinity function. It may be challenging to find a suitable class K-infinity function in some cases. We propose leaky ReLU as an efficient substitute for the complex nonlinear K-infinity function in our formulation. Moreover, we propose the Union of Invariant Sets (UIS) method, which combines information from multiple invariant sets in order to compute the largest possible PWA invariant set. The proposed framework is validated through multiple examples, showcasing its potential to enhance the analysis of invariant sets in ReLU-based dynamical systems. Our code is available at: https://github.com/PouyaSamanipour/UIS.git.