Applied Koopman theory for partial differential equations and data-driven modeling of spatio-temporal systems
type: Item
instance of: scholarly article
title: Applied Koopman theory for partial differential equations and data-driven modeling of spatio-temporal systems
zbMATH Open document ID: 1409.37029
DOI: 10.1155/2018/6010634
author: J. N. Kutz
author: J. L. Proctor
author: S. L. Brunton
published in: Complexity
publication date: 2019-02-19T00:00:00Z
review text: Summary: We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers' equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 37M10
Mathematics Subject Classification ID: 68Q32
Mathematics Subject Classification ID: 35Q56
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 37L65
zbMATH DE Number: 7025171
zbMATH Keywords: Koopman theory
zbMATH Keywords: cubic-quintic Ginzburg-Landau equation
zbMATH Keywords: observables
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/6010634
OpenAlex ID: W2902132588
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description: scientific article; zbMATH DE number 7025171