  We present a data-driven numerical approach for modeling unknown dynamical
systems with missing/hidden parameters. The method is based on training a deep
neural network (DNN) model for the unknown system using its trajectory data. A
key feature is that the unknown dynamical system contains system parameters
that are completely hidden, in the sense that no information about the
parameters is available through either the measurement trajectory data or our
prior knowledge of the system. We demonstrate that by training a DNN using the
trajectory data with sufficient time history, the resulting DNN model can
accurately model the unknown dynamical system. For new initial conditions
associated with new, and unknown, system parameters, the DNN model can produce
accurate system predictions over longer time.
