*Article* **Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations**

**Zecheng Zhang 1, Eric T. Chung 2, Yalchin Efendiev 1,3,\* and Wing Tat Leung 4**


Received: 21 March 2020; Accepted: 24 April 2020; Published: 4 May 2020

**Abstract:** 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 coarsenining 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 work, 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.

**Keywords:** generalized multiscale finite element method; multiscale model reduction; clustering; deep learning
