  In this paper, we combine deep learning concepts and some proper orthogonal
decomposition (POD) model reduction methods for predicting flow in
heterogeneous porous media. Nonlinear flow dynamics is studied, where the
dynamics is regarded as a multi-layer network. The solution at the current time
step is regarded as a multi-layer network of the solution at the initial time
and input parameters. As for input, we consider various sources, which include
source terms (well rates), permeability fields, and initial conditions. We
consider the flow dynamics, where the solution is known at some locations and
the data is integrated to the flow dynamics by modifying the reduced-order
model. This approach allows modifying the reduced-order formulation of the
problem. Because of the small problem size, limited observed data can be
handled. We consider enriching the observed data using the computational data
in deep learning networks. The basis functions of the global reduced order
model are selected such that the degrees of freedom represent the solution at
observation points. This way, we can avoid learning basis functions, which can
also be done using neural networks. We present numerical results, where we
consider channelized permeability fields, where the network is constructed for
various channel configurations. Our numerical results show that one can achieve
a good approximation using forward feed maps based on multi-layer networks.
