  The objective of this paper is to design novel multi-layer neural network
architectures for multiscale simulations of flows taking into account the
observed data and physical modeling concepts. Our approaches use deep learning
concepts combined with local multiscale model reduction methodologies to
predict flow dynamics. Using reduced-order model concepts is important for
constructing robust deep learning architectures since the reduced-order models
provide fewer degrees of freedom. Flow dynamics can be thought of as
multi-layer networks. More precisely, the solution (e.g., pressures and
saturations) at the time instant $n+1$ depends on the solution at the time
instant $n$ and input parameters, such as permeability fields, forcing terms,
and initial conditions. One can regard the solution as a multi-layer network,
where each layer, in general, is a nonlinear forward map and the number of
layers relates to the internal time steps. We will rely on rigorous model
reduction concepts to define unknowns and connections for each layer. In each
layer, our reduced-order models will provide a forward map, which will be
modified ("trained") using available data. It is critical to use reduced-order
models for this purpose, which will identify the regions of influence and the
appropriate number of variables. Because of the lack of available data, the
training will be supplemented with computational data as needed and the
interpolation between data-rich and data-deficient models. We will also use
deep learning algorithms to train the elements of the reduced model discrete
system. We will present main ingredients of our approach and numerical results.
Numerical results show that using deep learning and multiscale models, we can
improve the forward models, which are conditioned to the available data.
