Prediction of Discretization of GMsFEM Using Deep Learning

Min Wang,Eric T Chung,Yalchin Efendiev,Siu Wun Cheung,Wing Tat Leung,Yating Wang

doi:10.3390/math7050412

Abstract

In this paper, we propose a deep-learning-based approach to a class of multiscale problems. The generalized multiscale finite element method (GMsFEM) has been proven successful as a model reduction technique of flow problems in heterogeneous and high-contrast porous media. The key ingredients of GMsFEM include mutlsicale basis functions and coarse-scale parameters, which are obtained from solving local problems in each coarse neighborhood. Given a fixed medium, these quantities are precomputed by solving local problems in an offline stage, and result in a reduced-order model. However, these quantities have to be re-computed in case of varying media (various permeability fields). The objective of our work is to use deep learning techniques to mimic the nonlinear relation between the permeability field and the GMsFEM discretizations, and use neural networks to perform fast computation of GMsFEM ingredients repeatedly for a class of media. We provide numerical experiments to investigate the predictive power of neural networks and the usefulness of the resultant multiscale model in solving channelized porous media flow problems.

Highlights

Multiscale features widely exist in many engineering problems
The idea of local model reduction in these methods is based on idenfications of local multiscale basis functions supported in coarse regions on the fine grid, and replacement of the macroscopic equations by a coarse-scale system using a limited number of local multiscale basis functions
We present some numerical results for predicting the generalized multiscale finite element methods (GMsFEM) ingredients and solutions using our proposed method

Summary

Introduction

Multiscale features widely exist in many engineering problems. For instance, in porous media flow, the media properties typically vary over many scales and contain high contrast. Are designed for solving multiscale problems using local model reduction techniques. In these methods, the computational domain is partitioned into a coarse grid T H , which does not necessarily resolve all multiscale features. We further perform a refinement of T H to obtain a fine grid T h , which essentially resolves all multiscale features. The idea of local model reduction in these methods is based on idenfications of local multiscale basis functions supported in coarse regions on the fine grid, and replacement of the macroscopic equations by a coarse-scale system using a limited number of local multiscale basis functions. As in many model reduction techniques, the computations of multiscale basis functions, which constitute a small dimensional subspace, can be performed in an offline stage

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Conclusion