  A central challenge in data-driven model discovery is the presence of hidden,
or latent, variables that are not directly measured but are dynamically
important. Takens' theorem provides conditions for when it is possible to
augment these partial measurements with time delayed information, resulting in
an attractor that is diffeomorphic to that of the original full-state system.
However, the coordinate transformation back to the original attractor is
typically unknown, and learning the dynamics in the embedding space has
remained an open challenge for decades. Here, we design a custom deep
autoencoder network to learn a coordinate transformation from the delay
embedded space into a new space where it is possible to represent the dynamics
in a sparse, closed form. We demonstrate this approach on the Lorenz,
R\"ossler, and Lotka-Volterra systems, learning dynamics from a single
measurement variable. As a challenging example, we learn a Lorenz analogue from
a single scalar variable extracted from a video of a chaotic waterwheel
experiment. The resulting modeling framework combines deep learning to uncover
effective coordinates and the sparse identification of nonlinear dynamics
(SINDy) for interpretable modeling. Thus, we show that it is possible to
simultaneously learn a closed-form model and the associated coordinate system
for partially observed dynamics.
