  Koopman operators model nonlinear dynamics as a linear dynamic system acting
on a nonlinear function as the state. This nonstandard state is often called a
Koopman observable and is usually approximated numerically by a superposition
of functions drawn from a \textit{dictionary}. A widely used algorithm, is
\textit{Extended Dynamic Mode Decomposition}, where the dictionary functions
are drawn from a fixed, homogeneous class of functions. Recently, deep learning
combined with EDMD has been used to learn novel dictionary functions in an
algorithm called deep dynamic mode decomposition (deepDMD). The learned
representation both (1) accurately models and (2) scales well with the
dimension of the original nonlinear system. In this paper we analyze the
learned dictionaries from deepDMD and explore the theoretical basis for their
strong performance. We discover a novel class of dictionary functions to
approximate Koopman observables. Error analysis of these dictionary functions
show they satisfy a property of subspace approximation, which we define as
uniform finite approximate closure. We discover that structured mixing of
heterogeneous dictionary functions drawn from different classes of nonlinear
functions achieve the same accuracy and dimensional scaling as deepDMD. This
mixed dictionary does so with an order of magnitude reduction in parameters,
while maintaining geometric interpretability. Our results provide a hypothesis
to explain the success of deep neural networks in learning numerical
approximations to Koopman operators.
