Kernel methods for Koopman mode analysis and prediction: Ergodic and skew-product systems

We discuss a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic and skew-product dynamical systems. This framework is based on a representation of the Koopman group of unitary operators governing dynamical evolution in a smooth orthonormal basis acquired from time-ordered data through the diffusion maps algorithm. Using this
representation, we compute Koopman eigenfunctions through a regularized advection-diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. We also use this basis to build nonparametric forecast models for arbitrary probability densities and observables. In the first part of the talk, we focus on the ergodic case and discuss a connection between Koopman operators and diffusion operators obtained from Takens delay-coordinate mapped data that helps improve the efficiency and noise robustness of numerically computed Koopman eigenfunctions. Then, we discuss extensions of this framework to skew-product systems; our primary motivation being the identification and prediction of coherent spatiotemporal patterns in time-dependent fluid flows. We illustrate how Koopman operators for skew-product systems lead to a natural definition of spatiotemporal coherent patterns through their eigenfunctions, as well as model-free prediction schemes for these patterns. This work is in collaboration with Tyrus Berry (George Mason), Shuddho Das (NYU), Matina Gkioulidou (Johns Hopkins APL), John Harlim (Penn State), and Joanna Slawinska (Rutgers).