Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations

Zecheng Zhang,Wing Tat Leung,Yalchin Efendiev,Eric T Chung

doi:10.3390/math8050720

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.

Highlights

We propose the use of deep learning methodology and avoid explicit clustering as in earlier works
To coarsen the uncertainty space, we propose a deep learning algorithm which will learn the clusters for each local neighborhood
We propose a deep learning clustering technique within generalized multiscale finite element method (GMsFEM) to solve flows in heterogeneous media

Summary

Introduction

Examples include problems in porous media, material sciences, biological sciences, and so on. In porous media applications, engineers can obtain fine-scale data about pore geometries or subsurface properties at very fine resolutions. These data are obtained in some spatial locations and generalized to the entire reservoir domain. One uses geostatistical or other statistical tools to populate the media properties in space. The resulting porous media properties are stochastic and one needs to deal with many porous media realizations, where each realization is multiscale and varies at very fine scales

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