一种基于神经网络的多尺度可杂交间断Galerkin方法，用于求解多孔介质中的偏微分方程
A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media
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中文总结 AI 辅助
该研究提出NN-MsHDG方法，用神经网络预测局部算子替代细尺度求解，中等对比度下可降低5至16倍在线计算成本，同时指出高对比度下需改进粗空间及算子表示。
中文摘要 AI 辅助
我们开发了一种神经网络加速的多尺度可杂交间断Galerkin（MsHDG）方法，用于求解具有异质系数的椭圆型问题。该方法保留了标准MsHDG的局部到全局结构：粗块上的细尺度HDG问题定义离散Dirichlet-to-Neumann算子，这些算子通过标准MsHDG的全局骨架方程组装而成。为降低构造这些局部算子的成本，我们在参考块上定义的系数场上训练神经网络，并用预测的算子替代重复的细尺度局部求解。数值实验评估了所得NN-MsHDG方法的精度和在线效率。对于中等对比度的二维二元渗透率场，该神经网络方法以百分之几的相对建模误差重现了标准MsHDG的解，同时根据粗迹线维度的不同，将总在线计算成本降低了约5至16倍。在高对比度区域，所选的多项式粗迹线空间已使标准MsHDG方法产生显著的离散误差，表明需要更有效的粗空间，如适配系数的谱迹线空间。此外，学习到的局部算子会引入建模误差，且随着迹线空间的丰富，误差会变得严重。这些结果证明了神经代理在加速多尺度HDG计算方面的潜力，同时也凸显了改进算子表示和更好理解高对比度问题中误差放大的必要性。
英文摘要
We develop a neural-network-accelerated multiscale hybridizable discontinuous Galerkin method for elliptic problems with heterogeneous coefficients. The method preserves the standard MsHDG local-to-global structure: fine-scale HDG problems on coarse blocks define discrete Dirichlet-to-Neumann operators, which are assembled through the standard MsHDG global skeleton equations. To reduce the cost of constructing these local operators, we train a neural network on coefficient fields defined on a reference block and use the predicted operators in place of repeated fine-scale local solves. The numerical experiments assess both the accuracy and online efficiency of the resulting NN-MsHDG method. For two-dimensional binary permeability fields with moderate contrast, the neural method reproduces the standard MsHDG solution with relative modeling errors of a few percent while reducing the total online computational cost by factors of approximately 5 to 16, depending on the coarse trace dimension. In the high-contrast regime, the chosen polynomial coarse trace spaces already yield substantial discretization errors in the standard MsHDG method, indicating the need for more effective coarse spaces, such as coefficient-adapted spectral trace spaces. In addition, the learned local operators introduce modeling errors that become severe as the trace space is enriched. These results demonstrate the potential of neural surrogates for accelerating multiscale HDG computations while also highlighting the need for improved operator representations and a better understanding of error amplification in high-contrast problems.
发表机构
- ECPI University（ECPI大学）
- Old Dominion University（老道明大学）
机构由 AI 辅助整理，请以论文原文为准。
