On dynamic mode decomposition: Theory and applications

By Jonathan H. Tu, Clarence W. Rowley, Dirk Martin Luchtenburg, Steven L. Brunton, J. Nathan Kutz. 2013

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Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistical and Nonlinear Physics

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Citations in the corpus

Deep Generative Markov State Models (2018, methodology): “The analogous approach for deterministic dynamical systems such as complex fluid flows is called Koopman analysis, where time propagation is approximated by a linear model in a suitable function space transformation of the flow variables [16, 26, 29, 4].”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “ Moore-Penrose inverse D+ 1 = P 1 P T P , the size of this matrix is usually prohibitively large (P2Rn s n s) and thus the DMD decomposition based on the eigenvalues of P (referred to as exact DMD in Tu et al. 2014) is rarely of practical interest. Many variants of the DMD have been developed in the”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, neutral): “94 If we adopt the training and testing terminologies from ML, both POD-and DMD-based models require the training snapshot data to build the ROM (i.e., data-driven modeling).”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, background): “2 Relationship to dynamic mode decomposition The SINDY regression is related to the dynamic mode decomposition (DMD), which originated in the fluids community to extract spatiotemporal coherent structures from large fluid data sets [41, 46, 52, 24].”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, positive): “Further, this architecture has allowed for the incorporation of recent innovations from compressive sensing providing insight into optimally measuring a system [21, 48, 7].”
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control (2019, methodology): “Algorithms such as DMD [1, 2], EDMD [3, 4], SINDy [5], and their various kernel- [3, 6, 7], tensor- [8, 9, 10], or neural network-based [11, 12, 13] extensions and generalizations have been successfully applied to a plethora of different problems, including molecular and fluid dynamics, meteorology,”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, positive): “Much of the success of DMD rests on it foundations in linear algebra, as the procedure is easy to implement [28], the results are highly interpretable, and the method lends itself naturally to extensions [17], such as compressed sensing [27, 29–31], disambiguating the effect of actuation [13], multi”
Data-driven operator inference for nonintrusive projection-based model reduction (2016, methodology): “This means that we compute multiple inferred operators and then interpolate between them, whereas DMD is typically applied to full models with parameter-independent operators only [35].”
VAMPnets for deep learning of molecular kinetics (2017, background): “The expectation values E account for stochasticity in the dynamics, such as in MD, and thus they can be omitted for deterministic dynamical systems [24, 26, 27].”
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control (2016, background): “trolled setting). The use of time-delayed measurements is classical in system identication theory (see, e.g., [12]) but has also appeared in the context of Koopman operator approximation (see, e.g., [1, 23]). Assume therefore that we are given a collection of data X~ = [ 1 ;:::; K ]; Y~ = [+ 1 ;:::;”
Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks (2018, methodology): “hese approaches due to advances in computing power, algorithmic innovations and the ample availability of data. A major beneciary of this convergence are data-driven dimensionality reduction methods [7, 2, 3, 5, 6, 4, 8], model identication procedures [11, 10, 12, 13, 14, 15, 16, 9] and forecasting ”
On the numerical approximation of the Perron-Frobenius and Koopman operator (2015, background): “For a detailed description, we refer to [55].”
Data-Driven Model Reduction and Transfer Operator Approximation (2017, methodology): “Since its introduction, several variants and extensions have been proposed, see [6, 7, 35, 36, 37].”