Data-driven Random-feature Approximation of Koopman Generators of Nonlinear Continuous-time Systems

Learning continuous-time dynamical systems from data is an active area of research. The Koopman formalism is often used here, because it is a flexible approach that is informative about the system's fundamental properties. The formalism represents dynamical systems through linear operators acting on a suitable function space. A practical difficulty arising in the numerical approximation of Koopman operators is the choice of a ``dictionary'', meaning an effective finite-dimensional representation of the function space. In this paper, we focus on learning dynamical systems in continuous-time, and so we consider numerical approximations of the Koopman generator instead of the Koopman operator. Our main contribution is a data-driven random-feature method to obtain a parsimonious and effective function representation for Galerkin approximation methods such as gEDMD. The representation we construct is data-informed, which allows us to improve upon data-agnostic methods. We empirically demonstrate the suitability of our method for forecasting tasks and system analysis. The numerical experiments show that our approach is accurate, and faster than iterative optimization. Further, we prove convergence of our approximation of the vector field and the resulting trajectories. In addition to the numerical results, we also make theoretical contributions, including a general convergence criterion for the semigroup generated by the gEDMD approximation and a counterexample showing that gEDMD does not generally preserve exponential boundedness. With our contribution, we advance the frontier of efficiently learning continuous-time dynamics from data, towards interpretable and mathematically robust modelling based on the Koopman framework.