Scalable Extended Dynamic Mode Decomposition using Random Kernel Approximation

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.

16 pages, 6 figures

References in corpus (14)

• Very Deep Convolutional Networks for Large-Scale Image Recognition
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• On Dynamic Mode Decomposition: Theory and Applications
• Sparsity-promoting dynamic mode decomposition
• Applied Koopmanism
• De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis
• Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators
• Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition
• Dynamic Mode Decomposition for Large and Streaming Datasets
• Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics
• Randomized Dynamic Mode Decomposition
• Toward compressed DMD: spectral analysis of fluid flows using sub-Nyquist-rate PIV data
• Randomized Low-Rank Dynamic Mode Decomposition for Motion Detection
• Compressive sampling and dynamic mode decomposition

Cited by in corpus (7)

• Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces
• Deep Learning Enhanced Dynamic Mode Decomposition
• Closed-Loop Koopman Operator Approximation
• Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction
• Data-Driven Batch Localization and SLAM Using Koopman Linearization
• Linear quadratic control of nonlinear systems with Koopman operator learning and the Nyström method
• A General Framework for Linking Free and Forced Fluctuations via Koopmanism