To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics-informed neural networks and concept of neural operators. This approach generalizes and enhances each method, learning the parametric solution for mechanical problems without relying on data from other resources (e.g. other numerical solvers). We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics-based training yields higher accuracy compared to purely data-driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high-resolution solutions, avoiding the need to use basic interpolation techniques. First is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid point. Next, Fourier-based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. Comparisons with other well-known operator learning algorithms, further emphasize the advantages of the newly proposed method.

翻译：为获得固体力学中控制物理方程的快速求解方法，我们提出一种将有限元法核心思想与物理信息神经网络及神经算子概念相融合的技术。该方法对各类技术进行泛化与增强，能够在不依赖外部数据源（如其他数值求解器）的情况下学习力学问题的参数化解。我们提出直接利用有限元软件中现有的离散弱形式代数化构建损失函数，从而证明该方法即使在存在尖锐不连续性的情况下仍能获得有效解。本研究以微力学为例，其中获取特定非均质微结构在变形场与应力场方面的知识对后续设计应用至关重要。研究的主要参数是非均质固体系统中的杨氏模量分布。研究发现，对于未见过的微结构，基于物理的训练相比纯数据驱动方法具有更高精度。此外，我们提出两种直接改进高分辨率求解过程的方案，避免使用基础插值技术：其一是基于自编码器方法提升高分辨率网格点的计算效率；其二是利用傅里叶基参数化处理微力学中的复杂二维与三维问题，后者旨在通过傅里叶系数高效表征复杂微结构。与其他知名算子学习算法的对比进一步凸显了新方法的优势。