Journal of Nonlinear Science

, Volume 25, Issue 6, pp 1307–1346 | Cite as

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

  • Matthew O. Williams
  • Ioannis G. Kevrekidis
  • Clarence W. Rowley


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.


Data mining Koopman spectral analysis Set oriented methods Spectral methods Reduced order models 

Mathematics Subject Classification

Primary: 65P99 37M25 Secondary: 47B33 



The authors would like to thank Igor Mezić, Jonathan Tu, Maziar Hemati, and Scott Dawson for interesting and useful discussions on dynamic mode decomposition and the Koopman operator. M.O.W. gratefully acknowledges support from NSF DMS-1204783. I.G.K acknowledges support from AFOSR FA95550-12-1-0332 and NSF CMMI-1310173. C.W.R acknowledges support from AFOSR FA9550-12-1-0075.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Program in Applied and Computational Mathematics (PACM)Princeton UniversityPrincetonUSA
  2. 2.Chemical and Biological Engineering Department & PACMPrinceton UniversityPrincetonUSA
  3. 3.Mechanical and Aerospace Engineering DepartmentPrinceton UniversityPrincetonUSA

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