  Numerical approximation methods for the Koopman operator have advanced
considerably in the last few years. In particular, data-driven approaches such
as dynamic mode decomposition (DMD) and its generalization, the extended-DMD
(EDMD), are becoming increasingly popular in practical applications. The EDMD
improves upon the classical DMD by the inclusion of a flexible choice of
dictionary of observables that spans a finite dimensional subspace on which the
Koopman operator can be approximated. This enhances the accuracy of the
solution reconstruction and broadens the applicability of the Koopman
formalism. Although the convergence of the EDMD has been established, applying
the method in practice requires a careful choice of the observables to improve
convergence with just a finite number of terms. This is especially difficult
for high dimensional and highly nonlinear systems. In this paper, we employ
ideas from machine learning to improve upon the EDMD method. We develop an
iterative approximation algorithm which couples the EDMD with a trainable
dictionary represented by an artificial neural network. Using the Duffing
oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our
algorithm can effectively and efficiently adapt the trainable dictionary to the
problem at hand to achieve good reconstruction accuracy without the need to
choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we
require fewer dictionary terms than EDMD with fixed dictionaries. This
alleviates an important shortcoming of the EDMD algorithm and enhances the
applicability of the Koopman framework to practical problems.
