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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.

LEARNING REDUCED SYSTEMS VIA DEEP NEURAL NETWORKS WITH MEMORY