  Data-driven approximations of the Koopman operator are promising for
predicting the time evolution of systems characterized by complex dynamics.
Among these methods, the approach known as extended dynamic mode decomposition
with dictionary learning (EDMD-DL) has garnered significant attention. Here we
present a modification of EDMD-DL that concurrently determines both the
dictionary of observables and the corresponding approximation of the Koopman
operator. This innovation leverages automatic differentiation to facilitate
gradient descent computations through the pseudoinverse. We also address the
performance of several alternative methodologies. We assess a 'pure' Koopman
approach, which involves the direct time-integration of a linear,
high-dimensional system governing the dynamics within the space of observables.
Additionally, we explore a modified approach where the system alternates
between spaces of states and observables at each time step -- this approach no
longer satisfies the linearity of the true Koopman operator representation. For
further comparisons, we also apply a state space approach (neural ODEs). We
consider systems encompassing two and three-dimensional ordinary differential
equation systems featuring steady, oscillatory, and chaotic attractors, as well
as partial differential equations exhibiting increasingly complex and intricate
behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the
state space approach offers superior performance compared to the 'pure' Koopman
approach where the entire time evolution occurs in the space of observables.
When the temporal evolution of the Koopman approach alternates between states
and observables at each time step, however, its predictions become comparable
to those of the state space approach.
