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

TL;DR

This paper introduces a neural-network-based framework for estimating and verifying the Region of Attraction in nonlinear systems, combining data-driven learning with formal verification to improve accuracy and scalability over traditional methods.

Contribution

It presents a novel composite Lyapunov function with a homogeneous loss function and uses SMT solvers for formal verification, advancing scalable stability analysis for complex nonlinear systems.

Findings

Reduces conservatism in RoA estimates compared to traditional Lyapunov methods

Employs a neural network with a composite Lyapunov function for better flexibility

Successfully verifies stability using SMT solvers on benchmark systems

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.