  In this paper, we investigate and design multiscale simulations for
stochastic multiscale PDEs. As for the space, we consider a coarse grid and a
known multiscale method, the Generalized Multiscale Finite Element Method
(GMsFEM). In order to obtain a small dimensional representation of the solution
in each coarse block, the uncertainty space needs to be partitioned
(coarsened). This coarsening collects realizations that provide similar
multiscale features as outlined in GMsFEM (or other method of choice). This
step is known to be computationally demanding as it requires many local solves
and clustering based on them. In this paper, we take a different approach and
learn coarsening the uncertainty space. Our methods use deep learning
techniques in identifying clusters(coarsening) in the uncertainty space. We use
convolutional neural networks combined with some techniques in adversary neural
networks. We define appropriate loss functions in the proposed neural networks,
where the loss function is composed of several parts that includes terms
related to clusters and reconstruction of basis functions. We present numerical
results for channelized permeability fields in the examples of flows in porous
media.
