🤖 AI Summary
This work addresses stability analysis of nonlinear systems by proposing a Lyapunov function construction method that integrates physical priors with neural networks. Methodologically, it formulates the Zubov equation as a partial differential equation (PDE) constraint within a physics-informed neural network (PINN) framework and establishes theoretical guarantees for uniform approximation of the true region of attraction (ROA). It further introduces SMT-verifiable sufficient stability conditions—overcoming the scalability and conservatism limitations inherent in traditional sum-of-squares (SOS) and semidefinite programming (SDP) approaches. The method unifies PINN-based PDE solving, Zubov-type modeling, formal verification via SMT solvers (e.g., dReal, Barcelogic), and rigorous error convergence analysis. Experiments demonstrate that the proposed approach significantly improves ROA estimation tightness on multiscale and high-dimensional nonlinear systems, while achieving over an order-of-magnitude speedup in verification efficiency compared to SOS-SDP.
📝 Abstract
We provide a systematic investigation of using physics-informed neural networks to compute Lyapunov functions. We encode Lyapunov conditions as a partial differential equation (PDE) and use this for training neural network Lyapunov functions. We analyze the analytical properties of the solutions to the Lyapunov and Zubov PDEs. In particular, we show that employing the Zubov equation in training neural Lyapunov functions can lead to approximate regions of attraction close to the true domain of attraction. We also examine approximation errors and the convergence of neural approximations to the unique solution of Zubov's equation. We then provide sufficient conditions for the learned neural Lyapunov functions that can be readily verified by satisfiability modulo theories (SMT) solvers, enabling formal verification of both local stability analysis and region-of-attraction estimates in the large. Through a number of nonlinear examples, ranging from low to high dimensions, we demonstrate that the proposed framework can outperform traditional sums-of-squares (SOS) Lyapunov functions obtained using semidefinite programming (SDP).