Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems

By J. Nathan Kutz, Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor. 2016

1,558 citations recorded by the source corpus.

Subfields: Astronomy and Astrophysics, Molecular Biology, Statistical and Nonlinear Physics, Time-Series & Forecasting

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Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, methodology): “Several additional works [167, 43, 259] used sparse regression and dynamic mode decomposition to discover the governing equa-”
Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, methodology): “here wdenotes the vorticity, uthe x-component of the velocity eld, and v the y-component. To generate a training dataset for this problem we follow the exact same instructions as the ones provided in [45,1]. Specically, we simulate the Navier-Stokes equations describing the two-dimensional uid ow pa”
Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems (2018, methodology): “he uid ow past a cylinder (see gure4) at Reynolds number 100 using direct numerical simulations of the two dimensional Navier-Stokes equations. In particular, following the problem setup presented in [23] and [24], we simulate the Navier-Stokes equations describing the two-dimensional uid ow past a ”
Hidden physics models: Machine learning of nonlinear partial differential equations (2017, methodology): “As for the data, following the exact same instructions as the ones provided in [62] and [2], we simulate the Navier–Stokes equations describing the two-dimensional fluid flow past a circular cylinder at Reynolds number 100 using the Immersed Boundary Projection Method [63,64].”
Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control (2021, background): “…exist, including equation-free modeling [1], normal form identiﬁcation [2–4], nonlinear Laplacian spectral analysis [5], Koop-man analysis [6, 7] and dynamic mode decomposition (DMD) [8–10], symbolic regression [11–15], sparse regression [16, 17], Gaussian processes [18], and deep learning [19–24].”
Resurrecting Recurrent Neural Networks for Long Sequences (2023, methodology): “We show how Koopman operator theory (Koopman & Neumann, 1932), combined with recent advances in dynamic mode decomposition (Schmid, 2010; Kutz et al., 2016; Williams et al., 2015), can provide a solid theoretical foundation for understanding the class of functions that can be approximated by linear ”
Data-driven discovery of coordinates and governing equations (2019, methodology): “Methods such as dynamic mode decomposition (DMD) [4, 5, 10, 74], extended DMD [75], kernel DMD [76], and time-delay DMD [77, 78] build linear models for dynamics, but these methods rely on a proper set of coordinates for linearization.”
Machine Learning for Fluid Mechanics (2019, background): “The dynamic mode decomposition (DMD) (Schmid 2010; Kutz et al. 2016) is a modern approach, to extract spatiotemporal coherent structures from time-series data of ﬂuid ﬂows, resulting in a low-dimensional linear model for the evolution of these dominant coherent structures.”
Deep learning in fluid dynamics (2017, methodology): “…has capitalized on machine learning efforts with dimensionality-reduction techniques such as proper orthogonal decomposition or dynamic mode decomposition, which compute interpretable low-rank modes and subspaces that characterize spatio-temporal ﬂow data (Holmes et al. 1998; Kutz et al. 2016).”
On Numerical Approximations of the Koopman Operator (2020, background): “It has a number of extensions (many of which are summarized in [23]), for example, Exact DMD [24]; Bayesian/subspace DMD [25]; Optimized DMD [26,27]; Recursive DMD [28]; Variational DMD [29]; DMD with control [30,31]; sparsity promoting DMD [32]; DMD for noisy systems [33–35].”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Much of the modern Koopman operator theory has been recently developed [74]–[77], and it has been shown that, under certain conditions, DMD approximates the Koopman operator [20]–[23]; sensor fusion is also possible in the Koopman framework [78].”
Constrained sparse Galerkin regression (2016, methodology): “s are becoming increasingly popular and encompass a large variety of dierent techniques such as the eigensystem realisation algorithm (ERA) (Juang and Pappa, 1985), dynamic mode decomposition (DMD) (Kutz et al., 2016; Rowley et al., 2009; Schmid, 2010), Koopman theory (Mezic, 2005, 2013) and variant”
Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition (2020, background): “…interactions inherent in fluid flows, modal decomposition approaches aid in the design and implementation of active flow control since the instabilities targeted often consist of discrete frequency, spatio-temporal coherent flow structures (modes) (Kutz et al. 2016; Taira et al. 2017, 2019).”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “The core functionality of DMD algorithms is computing the eigendecomposition of matrix A = Y1Y † 0 [13], [21], where Y † 0 is the Moore– Penrose pseudoinverse of Y0.”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, methodology): “t capture the relevant dynamics for a given objective. Increasingly, ﬁrst principles models are giving way to data-driven approaches, for example in turbulence, epidemiology, neuroscience, and ﬁnance [1]. Although these methods offer tremendous promise, there has been slow progress in distilling phy”