Conventional finite element methods are known to be tedious in adaptive refinements due to their conformal regularity requirements. Further, the enrichment functions for adaptive refinements are often not readily available in general applications. This work introduces a novel neural network-enriched Partition of Unity (NN-PU) approach for solving boundary value problems via artificial neural networks with a potential energy-based loss function minimization. The flexibility and adaptivity of the NN function space are utilized to capture complex solution patterns that the conventional Galerkin methods fail to capture. The NN enrichment is constructed by combining pre-trained feature-encoded NN blocks with an additional untrained NN block. The pre-trained NN blocks learn specific local features during the offline stage, enabling efficient enrichment of the approximation space during the online stage through the Ritz-type energy minimization. The NN enrichment is introduced under the Partition of Unity (PU) framework, ensuring convergence of the proposed method. The proposed NN-PU approximation and feature-encoded transfer learning forms an adaptive approximation framework, termed the neural-refinement (n-refinement), for solving boundary value problems. Demonstrated by solving various elasticity problems, the proposed method offers accurate solutions while notably reducing the computational cost compared to the conventional adaptive refinement in the mesh-based methods.

翻译：传统有限元方法因保形正则性要求，在自适应细化过程中常显繁琐。此外，一般应用场景中自适应细化的增强函数往往不易获取。本文提出一种新型神经网络增强单位分解法（NN-PU），通过基于势能损失函数最小化的人工神经网络求解边界值问题。该方法利用NN函数空间的灵活性与自适应性，捕捉传统伽辽金方法难以表征的复杂解模式。NN增强通过组合预训练特征编码NN模块与额外未训练NN模块构建：预训练模块在离线阶段学习特定局部特征，使在线阶段可通过里兹型能量最小化高效增强近似空间；NN增强在单位分解（PU）框架下引入，保证了方法的收敛性。所提出的NN-PU近似与特征编码迁移学习形成自适应近似框架（称为神经细化n-refinement）以求解边界值问题。通过求解多种弹性力学问题验证，该方法在保持高精度解的同时，相较于传统基于网格方法的自适应细化显著降低了计算成本。