We present a domain decomposition strategy for developing structure-preserving finite element discretizations from data when exact governing equations are unknown. On subdomains, trainable Whitney form elements are used to identify structure-preserving models from data, providing a Dirichlet-to-Neumann map which may be used to globally construct a mortar method. The reduced-order local elements may be trained offline to reproduce high-fidelity Dirichlet data in cases where first principles model derivation is either intractable, unknown, or computationally prohibitive. In such cases, particular care must be taken to preserve structure on both local and mortar levels without knowledge of the governing equations, as well as to ensure well-posedness and stability of the resulting monolithic data-driven system. This strategy provides a flexible means of both scaling to large systems and treating complex geometries, and is particularly attractive for multiscale problems with complex microstructure geometry. While consistency is traditionally obtained in finite element methods via quasi-optimality results and the Bramble-Hilbert lemma as the local element diameter $h\rightarrow0$, our analysis establishes notions of accuracy and stability for finite h with accuracy coming from matching data. Numerical experiments and analysis establish properties for $H(\operatorname{div})$ problems in small data limits ($\mathcal{O}(1)$ reference solutions).

翻译：本文提出了一种域分解策略，用于在精确控制方程未知的情况下，从数据中开发结构保持的有限元离散格式。在子域上，采用可训练的Whitney形式单元从数据中识别结构保持模型，提供狄利克雷-诺伊曼映射，该映射可用于全局构造砂浆法。在第一性原理模型推导难以处理、未知或计算成本过高的情况下，可通过离线训练降阶局部单元来复现高保真度的狄利克雷数据。在此类情形中，需特别注意在未知控制方程的情况下，保持局部层面和砂浆层面的结构，并确保最终整体数据驱动系统的适定性和稳定性。该策略为大规模系统扩展和复杂几何处理提供了灵活手段，特别适用于具有复杂微观结构几何的多尺度问题。传统有限元方法通过准最优性结果和Bramble-Hilbert引理在局部单元直径$h\rightarrow0$时获得一致性，而我们的分析建立了有限h情况下的精度与稳定性概念，其精度来源于对数据的匹配。数值实验与分析确立了$H(\operatorname{div})$问题在小数据极限（$\mathcal{O}(1)$参考解）下的性质。