Latent Modes of Nonlinear Flows

Abstract

Extracting the latent underlying structures of complex nonlinear local and nonlocal flows is essential for their analysis and modeling. In this Element the authors attempt to provide a consistent framework through Koopman theory and its related popular discrete approximation - dynamic mode decomposition (DMD). They investigate the conditions to perform appropriate linearization, dimensionality reduction and representation of flows in a highly general setting. The essential elements of this framework are Koopman eigenfunctions (KEFs) for which existence conditions are formulated. This is done by viewing the dynamic as a curve in state-space. These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics. They examine the limitations of DMD through the analysis of Koopman theory and propose a new mode decomposition technique based on the typical time profile of the dynamics.

Functional Dimensionality of Koopman Eigenfunction Space

“…Treating Koopman Eigenfunction space as an infinite dimensional vector space yields various techniques to represent the system as a linear one with truncated dimensionality [22,6,14,24,23]. Naturally, these methods occasionally result in an overly redundant spectral decomposition [19,26], which is often inaccurate [9,10]. Thus, the challenge of extracting meaningful information about the dynamics from samples remains open [1].…”