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Numerical approximation methods for the Koopman operator have advancedconsiderably in the last few years. In particular, data-driven approaches suchas dynamic mode decomposition (DMD) and its generalization, the extended-DMD(EDMD), are becoming increasingly popular in practical applications. The EDMDimproves upon the classical DMD by the inclusion of a flexible choice ofdictionary of observables that spans a finite dimensional subspace on which theKoopman operator can be approximated. This enhances the accuracy of thesolution reconstruction and broadens the applicability of the Koopmanformalism. Although the convergence of the EDMD has been established, applyingthe method in practice requires a careful choice of the observables to improveconvergence with just a finite number of terms. This is especially difficultfor high dimensional and highly nonlinear systems. In this paper, we employideas from machine learning to improve upon the EDMD method. We develop aniterative approximation algorithm which couples the EDMD with a trainabledictionary represented by an artificial neural network. Using the Duffingoscillator and the Kuramoto Sivashinsky PDE as examples, we show that ouralgorithm can effectively and efficiently adapt the trainable dictionary to theproblem at hand to achieve good reconstruction accuracy without the need tochoose a fixed dictionary a priori. Furthermore, to obtain a given accuracy werequire fewer dictionary terms than EDMD with fixed dictionaries. Thisalleviates an important shortcoming of the EDMD algorithm and enhances theapplicability of the Koopman framework to practical problems.

Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator