Scalable Extended Dynamic Mode Decomposition using Random Kernel Approximation

TL;DR

This paper presents a scalable approach to Extended Dynamic Mode Decomposition (EDMD) by using random kernel approximations, enabling efficient analysis of large datasets and high-dimensional systems.

Contribution

It introduces random Fourier features and Nystrom methods for kernel approximation in EDMD, improving computational efficiency for large-scale problems.

Findings

Random Fourier features enable efficient kernel approximation.

Nystrom method provides a data-dependent kernel approximation.

The methods are demonstrated on example problems with analysis of benefits and drawbacks.

Abstract

The Koopman operator is a linear, infinite-dimensional operator that governs the 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 approach to EDMD in which the features used provide random approximations to a particular kernel function. The objective of this is computational economy for large data sets: EDMD is generally ill-suited for problems with large state dimension, and its dual kernel formulation (KDMD) is well-suited for such problems only if the number of data snapshots is relatively small. We discuss two specific methods for generating features: random Fourier features, and the Nystrom method. The first method is a data-independent method for translation-invariant kernels only and involves random sampling in feature space; the second method is a data-dependent empirical method that may be used for any kernel and involves random sampling of data. We first discuss how these ideas may be applied in an EDMD context, as well as a means for adaptively adding random Fourier features. We demonstrate these methods on two example problems and conclude with an analysis of the relative benefits and drawbacks of each method.