  We present efficient deep learning techniques for approximating flow and
transport equations for both single phase and two-phase flow problems. The
proposed methods take advantages of the sparsity structures in the underlying
discrete systems and can be served as efficient alternatives to the system
solvers at the full order. In particular, for the flow problem, we design a
network with convolutional and locally connected layers to perform model
reductions. Moreover, we employ a custom loss function to impose local mass
conservation constraints. This helps to preserve the physical property of
velocity solution which we are interested in learning. For the saturation
problem, we propose a residual type of network to approximate the dynamics. Our
main contribution here is the design of custom sparsely connected layers which
take into account the inherent sparse interaction between the input and output.
After training, the approximated feed-forward map can be applied iteratively to
predict solutions in the long range. Our trained networks, especially in
two-phase flow where the maps are nonlinear, show their great potential in
accurately approximating the underlying physical system and improvement in
computational efficiency. Some numerical experiments are performed and
discussed to demonstrate the performance of our proposed techniques.
