  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 dictionary. In a widely used algorithm, Extended
Dynamic Mode Decomposition, the dictionary functions are drawn from a fixed
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 explore State-Inclusive
Logistic Lifting (SILL) 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. Our results provide a hypothesis to explain the success of
deep neural networks in learning numerical approximations to Koopman operators.
Part 2 of this paper will extend this explanation by demonstrating the subspace
invariant of heterogeneous dictionaries and presenting a head-to-head numerical
comparison of deepDMD and low-parameter heterogeneous dictionary learning.
