A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

By Matthew O. Williams, Ioannis G. Kevrekidis, Clarence W. Rowley. Journal of nonlinear science, 2014

2,293 citations recorded by the source corpus.

The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator. We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005). Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions.

Subfields: Artificial Intelligence, Materials Chemistry, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/10cab2f43c29fe12e5a0d0718eb6e1ff8c9d4777

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Citations in the corpus

Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “Note that the Koopman operator and modal decomposition based on it can be extended to random dynamical systems actuated by process noise [4], [14], [18].”
Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks (2018, background): “A major beneficiary of this convergence are data-driven dimensionality reduction methods [7, 2, 3, 5, 6, 4, 8], model identification procedures [11, 10, 12, 13, 14, 15, 16, 9] and forecasting techniques [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] that aim to provide precise short-te”
Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems (2017, methodology): “Various algorithms have extended these methods using dynamic and extended dynamic mode decomposition, both for autonomous and controlled systems [7], [8], [9].”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “Current data-driven methods to approximate the Koopman operator include extended DMD (EDMD) [70,71,29] and the variational approach of conformation dynamics (VAC) [49,50].”
De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets (2015, methodology): “, the eigenvalues, modes, and eigenfunctions of K) has been made possible by means of DMD-like methods [8, 38,42,55,56], in which one approximates the Koopman operator from data obtained from experiments or simulations, without explicit knowledge of the map f .”
On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator (2017, methodology): “Abstract Extended Dynamic Mode Decomposition (EDMD) [27] is an algorithm that approximates the action of the Koopman operator on an N -dimensional subspace of the space of observables by sampling at M points in the state space.”
Data-Driven Model Reduction and Transfer Operator Approximation (2017, methodology): “The Koopman modes are vectors that, together with the eigenvalues and eigenfunctions, allow us to reconstruct and to propagate the system’s state [13].”
VAMPnets for deep learning of molecular kinetics (2017, methodology): “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].”
Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition (2020, background): “…DMD provides a finite-dimensional discrete-time linear system that approximates the Koopman operator (an infinite-dimensional linear description for nonlinear dynamical systems) that evolves the dynamics one time step forward (Tu et al. 2014; Schmid 2010; Rowley et al. 2009; Williams et al. 2015).”
Chaos as an intermittently forced linear system (2016, methodology): “Linear models can be obtained in entire basins of attraction of fixed points or periodic orbits using Koopman theory with the correct choice of measurement functions [61, 101].”
SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics (2020, background): “, system identification) is a rapidly developing field [1], with a range of techniques including classic linear approaches [2, 3], dynamic mode decomposition (DMD) [4, 5] and Koopman theory [6–9], nonlinear autoregressive models [10, 11], neural networks [12–20], Gaussian process regression [21, 22]”
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control (2019, methodology): “EDMD [38, 4] was developed for the approximation of the Koopman or Perron–Frobenius operator from data.”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, methodology): “It is, however, possible to obtain linear representations of entire basins of attractions of certain fixed points using eigenfunction coordinates [8, 36], where it may be possible to invert to find the state.”
Linearly-Recurrent Autoencoder Networks for Learning Dynamics (2017, neutral): “The Extended Dynamic Mode Decomposition (EDMD) [28] is an algorithm for approximating the Koopman operator on a dictionary of observable functions using data.”