A Survey on the Methods and Results of Data-Driven Koopman Analysis in the Visualization of Dynamical Systems

Koopman mode decomposition is a flow analysis technique developed by Igor Mezić in 2004, based upon the Koopman operator first proposed by Bernard Koopman in 1931. Via Koopman decomposition any non-chaotic well-sampled dynamic system – linear, non-linear, laminar or turbulent – is broken down into single-frequency repetitive components (modes). This paper presents a survey consolidating published information regarding data-driven Koopman analysis techniques. It is intended to aid researchers exploring the suitability of data-driven Koopman analysis in anticipation of developing their own modeling. A basic mathematical explanation of Koopman analysis is given with emphasis toward the data-driven Dynamic Mode Decomposition (DMD) solution, which converges to the Koopman operator given a highly-sampled dataset. The four primary uses of Koopman analysis: flow analysis, power grid analysis, building thermal analysis, and biomedical analysis are discussed, along with other public. Finally, weaknesses and problems inherent within Koopman analysis/DMD will be enumerated, alongside potential solutions. Koopman analysis is a computationally complex, yet often suitable method for determining periodic motion in any highly-sampled dataset. When compared to a similar analysis method, Proper Orthogonal Decomposition, Koopman analysis often provides additional detail regarding the structure of less significant modes present, albeit at the cost of increased computational complexity.

1 Introduction

The Koopman operator is an infinite-dimensional linear operator capable of expressing the time-evolution of a nonlinear dynamic system. It was first defined by mathematician Bernard Koopman in 1931 [1] in his construction of Koopman-von Neumann classical mechanics. Due to the high computational requirements for analyzing complicated fluid flow using the Koopman operator, Koopman analysis was not substantially revisited until 2004-2005 by Igor Mezic, who developed the Koopman Mode Decomposition technique for analysis of the spectral properties of this operator, and computing the ‘modes’ of the observed system [2], [3]. These modes represent the projection of a field of observables onto an eigenfunction, and their eigenvalues may be used to decompose any non-completely-chaotic system into its almost-periodic components. Today, Koopman analysis is frequently used to simplify the modeling of high-dimensional or complex dynamical physical systems [3]. A data-driven technique, Dynamic Mode Decomposition (DMD), was introduced by Peter Schmid in 2009-2010 [4], [5] and was later shown to connect directly to the Koopman operator, as DMD converges to the Koopman operator via increasing the number of sample points [6], [7]. Since then, Koopman analysis has seen ever-increasing use in the modeling of complex dynamic physical systems.