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

Matthew O. Williams, Yannis Kevrekidis, Clarence Worth Rowley

Research output: Contribution to journalArticlepeer-review

1048 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1307-1346
Number of pages40
JournalJournal of Nonlinear Science
Issue number6
StatePublished - Dec 1 2015

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


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


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