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The Koopman operator is a linear, infinite-dimensional operator that governsthe dynamics of system observables; Extended Dynamic Mode Decomposition (EDMD)is a data-driven method for approximating the Koopman operator using functions(features) of the system state snapshots. This paper investigates an approachto EDMD in which the features used provide random approximations to aparticular kernel function. The objective of this is computational economy forlarge data sets: EDMD is generally ill-suited for problems with large statedimension, and its dual kernel formulation (KDMD) is well-suited for suchproblems only if the number of data snapshots is relatively small. We discusstwo specific methods for generating features: random Fourier features, and theNystrom method. The first method is a data-independent method fortranslation-invariant kernels only and involves random sampling in featurespace; the second method is a data-dependent empirical method that may be usedfor any kernel and involves random sampling of data. We first discuss how theseideas may be applied in an EDMD context, as well as a means for adaptivelyadding random Fourier features. We demonstrate these methods on two exampleproblems and conclude with an analysis of the relative benefits and drawbacksof each method.

Scalable Extended Dynamic Mode Decomposition using Random Kernel Approximation