![](/images/graphics-bg.png)
Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems
Joint Authors
Nathan Kutz, J.
Proctor, J. L.
Brunton, S. L.
Source
Issue
Vol. 2018, Issue 2018 (31 Dec. 2018), pp.1-16, 16 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2018-12-02
Country of Publication
Egypt
No. of Pages
16
Main Subjects
Abstract EN
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.
American Psychological Association (APA)
Nathan Kutz, J.& Proctor, J. L.& Brunton, S. L.. 2018. Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems. Complexity،Vol. 2018, no. 2018, pp.1-16.
https://search.emarefa.net/detail/BIM-1134920
Modern Language Association (MLA)
Nathan Kutz, J.…[et al.]. Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems. Complexity No. 2018 (2018), pp.1-16.
https://search.emarefa.net/detail/BIM-1134920
American Medical Association (AMA)
Nathan Kutz, J.& Proctor, J. L.& Brunton, S. L.. Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems. Complexity. 2018. Vol. 2018, no. 2018, pp.1-16.
https://search.emarefa.net/detail/BIM-1134920
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1134920