  Recent work has focused on data-driven learning of the evolution of unknown
systems via deep neural networks (DNNs), with the goal of conducting long term
prediction of the dynamics of the unknown system. In many real-world
applications, data from time-dependent systems are often collected on a time
scale that is coarser than desired, due to various restrictions during the data
acquisition process. Consequently, the observed dynamics can be severely
under-sampled and do not reflect the true dynamics of the underlying system.
This paper presents a computational technique to learn the fine-scale dynamics
from such coarsely observed data. The method employs inner recurrence of a DNN
to recover the fine-scale evolution operator of the underlying system. In
addition to mathematical justification, several challenging numerical examples,
including unknown systems of both ordinary and partial differential equations,
are presented to demonstrate the effectiveness of the proposed method.
