Region of Attraction Estimate Learning and Verification for Nonlinear Systems using Neural-Network-based Lyapunov Functions

Abstract: Estimating the Region of Attraction (RoA) for nonlinear dynamical systems is a fundamental problem in control theory, with direct implications for stability analysis and safe controller design. Traditional approaches rely on analytically derived Lyapunov functions, which are often conservative and challenging to construct for high-dimensional or highly nonlinear systems. In this work, we propose a data-driven framework for learning and verifying RoA estimates for nonlinear systems using neural-network-based Lyapunov functions. Our method employs a composite Lyapunov function that combines a quadratic term with a neural-network-based component, providing both structure and flexibility. We introduce a novel homogeneous loss function for training, which removes the imbalance typically caused by the two non-homogeneous Lyapunov conditions. Together, these two aspects enable efficient training of the Lyapunov candidate. To guarantee the correctness of the learned Lyapunov function, we employ a Satisfiability Modulo Theories (SMT) solver to formally verify the stability results. Lastly, we perform a deeper analysis near the origin to overcome numerical artifacts, ensuring strict asymptotic stability. We demonstrate the effectiveness of our approach on benchmark nonlinear systems, showing that it significantly reduces conservatism compared to traditional Lyapunov methods while maintaining verifiability. This framework bridges the gap between function approximation and stability certification, paving the way for scalable safety analysis in learning-based control and safety-critical applications.