  In this paper, we propose a deep learning based reduced order modeling method
for stochastic underground flow problems in highly heterogeneous media. We aim
to utilize supervised learning to build a reduced surrogate model from the
stochastic parameter space that characterizes the possible highly heterogeneous
media to the solution space of a stochastic flow problem to have fast online
simulations. Dominant POD modes obtained from a well-designed spectral problem
in a global snapshot space are used to represent the solution of the flow
problem. Due to the small dimension of the solution, the complexity of the
neural network is significantly reduced. We adopt the generalized multiscale
finite element method (GMsFEM), in which a set of local multiscale basis
functions that can capture the heterogeneity of the media and source
information are constructed to efficiently generate globally defined snapshot
space. Rigorous theoretical analyses are provided and extensive numerical
experiments for linear and nonlinear stochastic flows are provided to verify
the superior performance of the proposed method.
