{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T01:44:22Z","timestamp":1774143862665,"version":"3.50.1"},"reference-count":47,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2018,12,2]],"date-time":"2018-12-02T00:00:00Z","timestamp":1543708800000},"content-version":"vor","delay-in-days":335,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["HR0011-16-C-0016"],"award-info":[{"award-number":["HR0011-16-C-0016"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-17-1-0329"],"award-info":[{"award-number":["FA9550-17-1-0329"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We consider the application of Koopman theory to nonlinear partial differential equations and data\u2010driven spatio\u2010temporal 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\u2010dimensional 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\u2010temporal 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\u2019 equation, the nonlinear Schr\u00f6dinger equation, the cubic\u2010quintic Ginzburg\u2010Landau equation, and a reaction\u2010diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.<\/jats:p>","DOI":"10.1155\/2018\/6010634","type":"journal-article","created":{"date-parts":[[2018,12,2]],"date-time":"2018-12-02T23:33:45Z","timestamp":1543793625000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":52,"title":["Applied Koopman Theory for Partial Differential Equations and Data\u2010Driven Modeling of Spatio\u2010Temporal Systems"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6004-2275","authenticated-orcid":false,"given":"J.","family":"Nathan Kutz","sequence":"first","affiliation":[]},{"given":"J. L.","family":"Proctor","sequence":"additional","affiliation":[]},{"given":"S. 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