  We present a general numerical approach for constructing governing equations
for unknown dynamical systems when only data on a subset of the state variables
are available. The unknown equations for these observed variables are thus a
reduced system of the complete set of state variables. Reduced systems possess
memory integrals, based on the well known Mori-Zwanzig (MZ) formulism. Our
numerical strategy to recover the reduced system starts by formulating a
discrete approximation of the memory integral in the MZ formulation. The
resulting unknown approximate MZ equations are of finite dimensional, in the
sense that a finite number of past history data are involved. We then present a
deep neural network structure that directly incorporates the history terms to
produce memory in the network. The approach is suitable for any practical
systems with finite memory length. We then use a set of numerical examples to
demonstrate the effectiveness of our method.
