Applied Koopmanism.

By Marko Budisic, Ryan Mohr, Igor Mezi'c. 2012

963 citations recorded by the source corpus.

A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré's geometric picture that focuses on "dynamics of states." While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of "big data" measurements. This overview article presents an alternative framework for dynamical systems, based on the "dynamics of observables" picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity. For each concept, we provide a summary of theoretical concepts required to define and study them, numerical methods that have been developed for their analysis, and, when possible, applications that made use of them. The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice. Therefore, the paper highlights its strengths in applied and numerical contexts. Additionally, we point out areas where an additional research push is needed before the approach is adopted as an off-the-shelf framework for analysis and design.

Subfields: Electrical and Electronic Engineering, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, background): “, 2009), nonlinear Laplacian spectral analysis (Giannakis and Majda, 2012), modeling emergent behavior (Roberts, 2014), Koopman analysis (Mezić, 2005; Budǐsić et al., 2012; Mezić, 2013; Brunton et al., 2017), automated inference of dynamics (Schmidt et al.”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “The above discussion can also be applied straightforwardly to continuous-time dynamical systems [4], [5].”
Data-driven discovery of Koopman eigenfunctions for control (2017, background): “Many classical results have been extended to the Koopman formalism [13,43].”
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition (2014, methodology): “GLA and similar methods have been applied to extract meaningful spatio-temporal structures using sensor data from buildings and power systems [24–27].”
Data-driven non-Markovian closure models (2014, background): “18), an approach based on averaging along trajectories is furthermore consistent with other Koopman operator techniques developed recently for the spectral analysis of time series [104].”
On the numerical approximation of the Perron-Frobenius and Koopman operator (2015, methodology): “2 Koopman operator While the Perron–Frobenius operator describes the evolution of densities, the Koopman operator describes the evolution of observables [10].”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, neutral): “The rapid adoption of Koopman theory across a number of scientific and engineering fields [8, 30] is not surprising.”
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control (2016, background): “For a detailed survey on Koopman operator and its applications, see [3].”
De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets (2015, methodology): “For instance, DMD has been utilized in the fields of epidemiology [41], medicine [5], neuroscience [7], power systems [8], robotics [4], sustainable buildings [8], and video processing [22].”
Linearly-Recurrent Autoencoder Networks for Learning Dynamics (2017, neutral): “For example, level sets of Koopman eigenfunctions may be used to form partitions of the phase space into ergodic sets along with periodic and wandering chains of sets [2].”