🤖 AI Summary
Existing Koopman-based methods struggle to simultaneously learn continuous-time dynamics and guarantee stability for unknown nonlinear systems under low-frequency sampling.
Method: This paper proposes a novel framework that jointly learns the Koopman generator and a Lyapunov function, tightly integrating high-accuracy Koopman generator estimation with physics-informed neural networks (PINNs) and embedding Lyapunov stability constraints. Attractor domain verification is performed formally using an SMT solver.
Contribution/Results: The approach significantly expands the verifiable region of the estimated region of attraction while reducing conservatism. It provides mathematically certified stability guarantees for dynamical systems under sparse observations—overcoming a key limitation of current Koopman methods, which lack rigorous stability certification under low-sampling-rate conditions.
📝 Abstract
Koopman operator theory has gained significant attention in recent years for identifying discrete-time nonlinear systems by embedding them into an infinite-dimensional linear vector space. However, providing stability guarantees while learning the continuous-time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman-based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high-accuracy Koopman generator learning for capturing transient system transitions and physics-informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.