The spatiotemporal coupling in delay-coordinates dynamic mode decomposition

TL;DR

This paper investigates the spectral structure of delay-coordinates DMD, revealing spatiotemporal coupling, and introduces a new mode selection method that improves reduced-order modeling of dynamical systems.

Contribution

It analyzes the spatiotemporal coupling in delay-coordinates DMD and proposes a novel component selection method for better reduced-order representations.

Findings

The spectral structure of delay-coordinates DMD exhibits spatiotemporal coupling.

The proposed mode selection method outperforms amplitude-based methods.

Demonstrated effectiveness on noisy signals and various dynamical systems.

Abstract

Dynamic mode decomposition (DMD) is a leading tool for equation-free analysis of high-dimensional dynamical systems from observations. In this work, we focus on a combination of delay-coordinates embedding and DMD, i.e., delay-coordinates DMD, which accommodates the analysis of a broad family of observations. An important utility of DMD is the compact and reduced-order spectral representation of observations in terms of the DMD eigenvalues and modes, where the temporal information is separated from the spatial information. From a spatiotemporal viewpoint, we show that when DMD is applied to delay-coordinates embedding, temporal information is intertwined with spatial information, inducing a particular spectral structure on the DMD components. We formulate and analyze this structure, which we term the spatiotemporal coupling in delay-coordinates DMD. Based on this spatiotemporal coupling, we propose a new method for DMD components selection. When using delay-coordinates DMD that comprises redundant modes, this selection is an essential step for obtaining a compact and reduced-order representation of the observations. We demonstrate our method on noisy simulated signals and various dynamical systems and show superior component selection compared to a commonly-used method that relies on the amplitudes of the modes.