Analytical derivation of Lyapunov functions using Lyapunov Kolmogorov-Arnold Networks (L-KANs)

This paper presents a solution for the automatic derivation of analytic Lyapunov functions for dynamical systems, making use of Kolmogorov-Arnold networks (KANs) as a symbolic regression tool. We propose a modified unsupervised machine learning model, named Lyapunov-KAN (L-KAN), which incorporates model-driven Lyapunov function verification criteria. Given a dynamical system model and an equilibrium point, two algorithms are presented that learn, directly from data sampled according to the system dynamics, an L-KAN representing a Lyapunov function in a neighborhood ℬ of the equilibrium point. The found L-KAN models a symbolic Lyapunov function, whose analytical expression allows not only to formally certify inner approximations of the Region of Attraction (RoA) of the equilibrium, but also to fine-tune the function itself – extending its certified domain beyond the training region and, potentially, upgrading it to a global Lyapunov function with a formally verifiable certificate. The symbolic nature of L-KANs hence opens up unprecedented possibilities precluded to standard neural stability certificates, surpassing also the polynomial restrictions of sum-of-squares methods. The effectiveness of the proposed approach is demonstrated through various numerical examples, which highlight its practical usefulness.