  Recently, a general data driven numerical framework has been developed for
learning and modeling of unknown dynamical systems using fully- or
partially-observed data. The method utilizes deep neural networks (DNNs) to
construct a model for the flow map of the unknown system. Once an accurate DNN
approximation of the flow map is constructed, it can be recursively executed to
serve as an effective predictive model of the unknown system. In this paper, we
apply this framework to chaotic systems, in particular the well-known Lorenz 63
and 96 systems, and critically examine the predictive performance of the
approach. A distinct feature of chaotic systems is that even the smallest
perturbations will lead to large (albeit bounded) deviations in the solution
trajectories. This makes long-term predictions of the method, or any data
driven methods, questionable, as the local model accuracy will eventually
degrade and lead to large pointwise errors. Here we employ several other
qualitative and quantitative measures to determine whether the chaotic dynamics
have been learned. These include phase plots, histograms, autocorrelation,
correlation dimension, approximate entropy, and Lyapunov exponent. Using these
measures, we demonstrate that the flow map based DNN learning method is capable
of accurately modeling chaotic systems, even when only a subset of the state
variables are available to the DNNs. For example, for the Lorenz 96 system with
40 state variables, when data of only 3 variables are available, the method is
able to learn an effective DNN model for the 3 variables and produce accurately
the chaotic behavior of the system.
