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Koopman operators globally linearize nonlinear dynamical systems and theirspectral information is a powerful tool for the analysis and decomposition ofnonlinear dynamical systems. However, Koopman operators areinfinite-dimensional, and computing their spectral information is aconsiderable challenge. We introduce measure-preserving extended dynamic modedecomposition ($\texttt{mpEDMD}$), the first truncation method whoseeigendecomposition converges to the spectral quantities of Koopman operatorsfor general measure-preserving dynamical systems. $\texttt{mpEDMD}$ is adata-driven algorithm based on an orthogonal Procrustes problem that enforcesmeasure-preserving truncations of Koopman operators using a general dictionaryof observables. It is flexible and easy to use with any pre-existing DMD-typemethod, and with different types of data. We prove convergence of$\texttt{mpEDMD}$ for projection-valued and scalar-valued spectral measures,spectra, and Koopman mode decompositions. For the case of delay embedding(Krylov subspaces), our results include the first convergence rates of theapproximation of spectral measures as the size of the dictionary increases. Wedemonstrate $\texttt{mpEDMD}$ on a range of challenging examples, its increasedrobustness to noise compared with other DMD-type methods, and its ability tocapture the energy conservation and cascade of experimental measurements of aturbulent boundary layer flow with Reynolds number $> 6\times 10^4$ andstate-space dimension $>10^5$.

The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems