The Multiscale Finite Element Method (MsFEM) decomposes the problem of solving partial differential equations with multiscale characteristics into two subproblems at two discrete resolution levels, i.e., the macroscopic one on a coarse mesh and the microscopic one on a fine mesh. The microscopic subproblems are used for constructing the Equivalent Stiffness Matrices (ESMs) of the coarse elements, and the calculation of them is the most time-consuming part in the MsFEM. Using a pure data-driven model that is independent of mechanical knowledge to directly predict ESMs, even with a pretty high-precision model, the outputs may still lack basic physical rationality. The core challenge lies in the strict assurance of the basic physical characteristics of the predicted ESMs, that is, the Rigid Displacement Properties (RDPs), which require the ESM to produce zero-strain energy under rigid body displacement. In terms of the mechanical essence, this requirement is closely correlated with the physical meaning of the eigenvectors and eigenvalues of the ESMs. Based on the above deep mechanics prior knowledge, a surrogate model based on Deep Learning (DL) and orthogonal decomposition techniques is developed. The inputs of the DL neural networks are the geometry parameters of the coarse element, while the outputs are eigenvectors and eigenvalues of the ESM. The dimensions of the outputs are reduced by directly specifying a specific number of zero eigenvalues and eigenvectors. The RDPs are embedded in the reconstruction calculation of the ESMs based on the outputs in a structured manner, assuring the physical reasonability of the predictions. Numerical examples demonstrate the performance of the proposed method.

中文翻译：

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Deep mechanics prior - 用于多尺度有限元方法

多尺度有限元法 （MsFEM） 将求解具有多尺度特性的偏微分方程的问题分解为两个离散分辨率水平的子问题，即粗网格上的宏观问题和细网格上的微观子问题。微观子问题用于构建粗化单元的等效刚度矩阵 （ESM），计算它们是 MsFEM 中最耗时的部分。使用独立于机械知识的纯数据驱动模型直接预测 ESM，即使使用相当高精度的模型，输出可能仍然缺乏基本的物理合理性。核心挑战在于严格保证预测的 ESM 的基本物理特性，即刚性位移特性 （RDP），这要求 ESM 在刚体位移下产生零应变能量。就力学本质而言，这一要求与 ESM 的特征向量和特征值的物理意义密切相关。基于上述深度力学先验知识，开发了一种基于深度学习 （DL） 和正交分解技术的代理模型。DL 神经网络的输入是粗略单元的几何参数，而输出是 ESM 的特征向量和特征值。通过直接指定特定数量的零特征值和特征向量来减小输出的维度。RDP 以结构化的方式嵌入到基于输出的 ESM 的重建计算中，确保预测的物理合理性。数值算例验证了所提方法的性能。