  A common problem in time series analysis is to predict dynamics with only
scalar or partial observations of the underlying dynamical system. For data on
a smooth compact manifold, Takens theorem proves a time delayed embedding of
the partial state is diffeomorphic to the attractor, although for chaotic and
highly nonlinear systems learning these delay coordinate mappings is
challenging. We utilize deep artificial neural networks (ANNs) to learn
discrete discrete time maps and continuous time flows of the partial state.
Given training data for the full state, we also learn a reconstruction map.
Thus, predictions of a time series can be made from the current state and
several previous observations with embedding parameters determined from time
series analysis. The state space for time evolution is of comparable dimension
to reduced order manifold models. These are advantages over recurrent neural
network models, which require a high dimensional internal state or additional
memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to
predict chaotic behavior from a scalar observation on a manifold of dimension
three via the Lorenz system. We also consider multivariate observations on the
Kuramoto-Sivashinsky equation, where the observation dimension required for
accurately reproducing dynamics increases with the manifold dimension via the
spatial extent of the system.
