Learning dynamical systems from data: Gradient-based dictionary optimization

Dynamical systems can be used to describe the motion of atoms, fluids, and planets as well as biological and chemical processes to name just a few examples. Deriving mathematical models for such complex problems can be challenging. Even if we do have mathematical models, the resulting dynamical systems will often be high-dimensional and highly nonlinear, which makes their analysis difficult or sometimes impossible. The goal of data-driven modeling approaches is to learn the governing equations or transfer operators associated with the system from measurement data. Instead of analyzing individual trajectories of the system, transfer operators such as the Koopman operator and Perron-Frobenius operator describe the evolution of observables and probability densities [1, 2, 3, 4]. Data-driven methods allow us to study the global behavior of the system without requiring detailed mathematical models, see [5] for an overview of different applications. Of particular interest are the eigenvalues and eigenfunctions of transfer operators since they contain important information about timescales and slowly evolving spatiotemporal patterns of the systems.