Supervised Learning of Lyapunov Functions Using Laplace Averages of Approximate Koopman Eigenfunctions

Modern data-driven techniques have rapidly progressed beyond modelling and systems identification, with a growing interest in learning high-level dynamical properties of a system, such as safe-set invariance, reachability, input-to-state stability etc. In this letter, we propose a novel supervised Deep Learning technique for constructing Lyapunov certificates, by leveraging Koopman Operator theory-based numerical tools (Extended Dynamic Mode Decomposition and Generalized Laplace Analysis) to robustly and efficiently generate explicit ground truth data for training. This is in stark contrast to existing Deep Learning methods where the loss functions plainly penalize Lyapunov condition violation in the absence of labelled data for direct regression. Furthermore, our approach leads to a linear parameterization of Lyapunov candidate functions in terms of stable eigenfunctions of the Koopman operator, making them more interpretable compared to standard DNN-based architecture. We demonstrate and validate our approach numerically using 2-dimensional and 10-dimensional examples.