Analyzing Nonlinear Dynamics via Data‐Driven Dynamic Mode Decomposition‐Like Methods

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.

Finding characteristically rich nonlinear solution space: a statistical mechanics approach

“…Based on the nonlinear state-space Equation (10) and Liouville's theorem (12), derive a secondary equation from which we can obtain basis modes that will span the characteristically rich spectrum of the nonlinear solution space given a rich space of ICs, BCs, or system inputs. Again, we are not seeking for a parametrically rich space.…”

“…Since we are not considering parameter variations within the system, the only way it can be subjected to other types of changes is through an IC or the system interface, that is, a BC or an external input. Without loss of generality and keeping in line with the earlier exposition, we will assume that the solutions are a result of applying the random impulse inputs with zero ICs as described by the stochastic nonlinear Equation (10) and the Liouville Equation (12). In the limit, as M becomes very large we will let the input vectors b i (i = 1, 2, … M) be continuously distributed and responsible for the distinct solutions, x i (t) ′ s. Statistically, the uncertainties introduced in (x, t) is due to the uncertainties present in how b i 's are weighed in against each other.…”

“…At the heart of the new approach are the dynamic eigen decomposition (DED) and modally equivalent perturbed system (MEPS), first introduced by the author. 5 Not to be confused with dynamic mode decomposition (DMD), 12 the DED is a radical departure from the traditional eigen decomposition in that it is executed on a frequency or time-dependent matrix function rather than on a constant matrix. A major advantage of the technique is that its eigenmodes, known as the dynamic eigenmodes (DE), are invariant to the parameter variations and hence the system response can be expressed by the same modes for all possible values and combinations of the parameters.…”

The Structure of Urban Flows

“…Using data-driven techniques to extract the main features driving the flow, and combining the acquired physical knowledge with new machine-learning strategies to create ROMs is a research topic of high interest that should be explored more in detail. This Appendix presents a brief summary of some relevant data-driven techniques based on modal decompositions exhibiting good performance when analyzing complex flows [151,152,154,[156][157][158] (see more details and other techniques in Refs. [147,153,159]).…”

Dynamic Characteristics of an Offshore Wind Turbine with Tripod Suction Buckets via Full-Scale Testing

“…Among renewable and sustainable energies, offshore wind power shows a variety of advantages including high energy density, low turbulence, and low wind shear [1]. Technological advances are also making wind energy competitive from the economic perspective [2][3][4][5][6][7][8][9]. In 2018, 409 new offshore wind turbines were commissioned in Europe, which provided an additional capacity of 2,649 MW, and the cumulative capacity of wind farms was 18,499 MW [10].…”