Region of Attraction Estimate Learning and Verification for Nonlinear Systems using Neural-Network-based Lyapunov Functions
Adel Bechihi, Aristotelis Kapnopoulos
TL;DR
The paper tackles certifiable stability and RoA estimation for nonlinear systems by learning a neural-network-based Lyapunov function with a composite form $V_\theta(x)$ that includes a neural component and a quadratic term. It introduces a homogeneous RoA-focused loss over a grid to maximize the discrete inner RoA $\hat{\mathcal{R}}_\mathcal{G}$ and uses SMT verification with the solver $dReal$ to certify Lyapunov conditions over a continuous domain. The approach reduces conservatism compared to traditional Lyapunov methods while providing formal guarantees for a verified RoA, demonstrated on two nonlinear benchmarks. This framework bridges data-driven approximation with formal stability certification, enabling scalable safety analysis for learning-based control in safety-critical applications.
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.