  We present a general numerical approach for learning unknown dynamical
systems using deep neural networks (DNNs). Our method is built upon recent
studies that identified the residue network (ResNet) as an effective neural
network structure. In this paper, we present a generalized ResNet framework and
broadly define residue as the discrepancy between observation data and
prediction made by another model, which can be an existing coarse model or
reduced-order model. In this case, the generalized ResNet serves as a model
correction to the existing model and recovers the unresolved dynamics. When an
existing coarse model is not available, we present numerical strategies for
fast creation of coarse models, to be used in conjunction with the generalized
ResNet. These coarse models are constructed using the same data set and thus do
not require additional resources. The generalized ResNet is capable of learning
the underlying unknown equations and producing predictions with accuracy higher
than the standard ResNet structure. This is demonstrated via several numerical
examples, including long-term prediction of a chaotic system.
