A kernel-based method for data-driven koopman spectral analysis

A data-driven, kernel-based method for approximating the leading Koopmaneigenvalues, eigenfunctions, and modes in problems with high-dimensional statespaces is presented.This approach uses a set of scalar observables (functions that map a state to a scalar value) that are defined implicitly by the feature map associated with a user-defined kernel function.This circumvents the computational issues that arise due to the number offunctions required to span a ``sufficiently rich'' subspace of all possible scalar observables in such applications.We illustrate this method on two examples: the first is the FitzHugh-Nagumo PDE, a prototypical one-dimensional reaction-diffusionsystem, and the second is a set of vorticity data computed from experimentally obtainedvelocity data from flow past a cylinder at Reynolds number 413.In both examples, we use the output of Dynamic Mode Decomposition, which has a similar computational cost, as the benchmark for our approach.