  Dynamical models underpin our ability to understand and predict the behavior
of natural systems. Whether dynamical models are developed from
first-principles derivations or from observational data, they are predicated on
our choice of state variables. The choice of state variables is driven by
convenience and intuition, and in the data-driven case the observed variables
are often chosen to be the state variables. The dimensionality of these
variables (and consequently the dynamical models) can be arbitrarily large,
obscuring the underlying behavior of the system. In truth, these variables are
often highly redundant and the system is driven by a much smaller set of latent
intrinsic variables. In this study, we combine the mathematical theory of
manifolds with the representational capacity of neural networks to develop a
method that learns a system's intrinsic state variables directly from time
series data, and also learns predictive models for their dynamics. What
distinguishes our method is its ability to reduce data to the intrinsic
dimensionality of the nonlinear manifold they live on. This ability is enabled
by the concepts of charts and atlases from the theory of manifolds, whereby a
manifold is represented by a collection of patches that are sewn together -- a
necessary representation to attain intrinsic dimensionality. We demonstrate
this approach on several high-dimensional systems with low-dimensional
behavior. The resulting framework provides the ability to develop dynamical
models of the lowest possible dimension, capturing the essence of a system.
