Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations
Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations
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DOI：
10.3390/math8050720
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发表时间：
2020-04
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影响因子：
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通讯作者：
Zecheng Zhang;Eric T. Chung;Y. Efendiev;W. Leung
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文献类型：
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作者：
Zecheng Zhang;Eric T. Chung;Y. Efendiev;W. Leung
本文研究和设计了随机多尺度偏微分方程组的多尺度模拟。对于空间，我们考虑了一种粗网格和一种已知的多尺度方法--广义多尺度有限元方法。为了在每个粗块中获得解的小维度表示，需要对不确定性空间进行划分(粗化)。这种粗化收集的实现提供了与GMsFEM(或其他选择方法)中概述的类似的多尺度特征。众所周知，这一步骤计算要求很高，因为它需要许多局部求解和基于它们的聚类。在这项工作中，我们采取了一种不同的方法，学习了粗化不确定性空间。我们的方法使用深度学习技术来识别不确定空间中的簇(粗化)。在对抗性神经网络中，我们结合了卷积神经网络的一些技术。我们在所提出的神经网络中定义了适当的损失函数，其中损失函数由若干部分组成，其中包括与聚类和基函数重构相关的项。以渗流为例，给出了沟道化渗透场的数值结果。
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.