  Despite many of the most common chaotic dynamical systems being continuous in
time, it is through discrete time mappings that much of the understanding of
chaos is formed. Henri Poincar\'e first made this connection by tracking
consecutive iterations of the continuous flow with a lower-dimensional,
transverse subspace. The mapping that iterates the dynamics through consecutive
intersections of the flow with the subspace is now referred to as a Poincar\'e
map, and it is the primary method available for interpreting and classifying
chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit
form for such a mapping remains outstanding. This work proposes a method for
obtaining explicit Poincar\'e mappings by using deep learning to construct an
invertible coordinate transformation into a conjugate representation where the
dynamics are governed by a relatively simple chaotic mapping. The invertible
change of variable is based on an autoencoder, which allows for dimensionality
reduction, and has the advantage of classifying chaotic systems using the
equivalence relation of topological conjugacies. Indeed, the enforcement of
topological conjugacies is the critical neural network regularization for
learning the coordinate and dynamics pairing. We provide expository
applications of the method to low-dimensional systems such as the R\"ossler and
Lorenz systems, while also demonstrating the utility of the method on
infinite-dimensional systems, such as the Kuramoto--Sivashinsky equation.
