Koopman operator theory has emerged as a powerful modeling approach for complex dynamical systems arising in physics, chemistry, materials science, and engineering. The basic idea is to leverage existing simulation data to learn a linear model that allows to predict expectation values of observable functions at future times. Though the resulting algorithm, known as Extended Dynamic Mode Decomposition (EDMD), is conceptually quite simple, its underlying mathematical structure (the Koopman operator semigroup) is very rich, and can be used for different purposes including control, coarse graining, or the identification of metastable states in complex molecules and materials.
In this talk, I will present recent results concerning the approximation error for the Koopman operator based on finite data. I will first focus on stationary systems, and present error bounds for EDMD with a finite-dimensional dictionary, as well as extensions to reproducing kernel Hilbert spaces (RKHS). Subsequently, I will discuss recent extensions of the theory to non-stationary systems.