Data-based Modeling of Complex Dynamical Systems on Rich Approximation Spaces

Stochastic and deterministic dynamical systems are widely used to model complex real-world processes. From a statistical point of view, such a system can be described by the semigroup of linear Koopman operators, or its infinitesimal generator. Numerous different methods to model these linear operators based on simulation data have been developed in past years. However, many of those methods require a fair amount of prior knowledge about the system of interest. In this talk, I will present some of my recent work to overcome these limitations. I will present data-driven models for the Koopman semigroup using the tensor train format, and also on reproducing kernel Hilbert spaces. I will also show how these models can be used to calculate eigenvalues and eigenfunctions of the operators in question, and how the underlying dynamical system can be further analyzed and manipulated based on them. I will conclude by sketching future research directions along these lines.