  We present a numerical framework for recovering unknown non-autonomous
dynamical systems with time-dependent inputs. To circumvent the difficulty
presented by the non-autonomous nature of the system, our method transforms the
solution state into piecewise integration of the system over a discrete set of
time instances. The time-dependent inputs are then locally parameterized by
using a proper model, for example, polynomial regression, in the pieces
determined by the time instances. This transforms the original system into a
piecewise parametric system that is locally time invariant. We then design a
deep neural network structure to learn the local models. Once the network model
is constructed, it can be iteratively used over time to conduct global system
prediction. We provide theoretical analysis of our algorithm and present a
number of numerical examples to demonstrate the effectiveness of the method.
