Abstract
This article presents a review on two methods based on dynamic mode decomposition and its multiple applications, focusing on higher order dynamic mode decomposition (which provides a purely temporal Fourier-like decomposition) and spatiotemporal Koopman decomposition (which gives a spatiotemporal Fourier-like decomposition). These methods are purely data-driven, using either numerical or experimental data, and permit reconstructing the given data and identifying the temporal growth rates and frequencies involved in the dynamics and the spatial growth rates and wavenumbers in the case of the spatiotemporal Koopman decomposition. Thus, they may be used to either identify and extrapolate the dynamics from transient behavior to permanent dynamics or construct efficient, purely data-driven reduced order models.
Highlights
Uncovering the quantitative essence of complex signals, either numerical or experimental, coming from nonlinear systems is an interesting topic in data science and fluid dynamics
With the above in mind, the main object of this paper is to present the higher order DMD (HODMD) and spatiotemporal Koopman decomposition (STKD) methods, illustrating them in simple toy models and applying them to various problems of scientific and technological interest
As anticipated, retaining only those terms appearing in the dynamic mode decomposition (DMD) expansion (1) that exhibit zero growth rate, the final attractor may be obtained from transient data
Summary
Uncovering the quantitative essence of complex signals, either numerical or experimental, coming from nonlinear systems is an interesting topic in data science and fluid dynamics. Similar spatiotemporal expansions have been already derived (via a not purely data-driven method) for the Navier-Stokes (NS) equations in [31] by first projecting the NS equations into the temporal modes and applying spatial DMD to these modes using appropriate basis functions. DMD, HODMD, and STKD provide approximations of type (9) and (10) in a purely data-driven fashion for arbitrary fully nonlinear dynamics, which are great advantages, specially in the case of the analysis of experimental data Note in this context that expansions (9) and (10) can be seen as semianalytical expressions that allow for the fast online computation of the associated spatiotemporal data, which leads to a purely data-driven reduced order model (ROM).
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