Learning Metastable Dynamics: Application to Molecular Dynamics

Metastablility is a phenomenon which often inhibits the efficient simulation of dynamical systems, or the generation of samples from high-dimensional probability measures. In particular, it is frequently encountered in computer simulations of biological macromolecules using molecular dynamics. It is well-known that metastable transitions and their time scales are encoded in the dominant spectrum of certain transition operators, also called Koopman operators. The study of Koopman operators, and their data-driven approximation by algorithms like the Extended Dynamic Mode Decomposition (EDMD), have gained significant traction in recent years.
In this talk, I will report on recent progress concerning the data-driven analysis of metastable systems using Koopman operators. First, I will introduce approximation methods based on reproducing kernel Hilbert spaces (RKHS), which allow the use of rich approximation spaces, and explain how the resulting large-scale linear problems can be solved efficiently using random Fourier features (RFF). Second, I will explain how similar ideas can be applied to learn models for the infinitesimal generator, which allows for a more detailed system analysis, including interpolation across statistical ensembles, or the definition of reduced (coarse grained) models.