PINN models have demonstrated impressive capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well-established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite domain, which conflicts with the finite boundaries typical of most solid structures; and b) the solution space utilised by PINN is Euclidean, which is inadequate for addressing the complex geometries often present in solid structures. This work proposes a PINN architecture used for general solid mechanics problems, termed the Finite-PINN model. The proposed model aims to effectively address these two challenges while preserving as much of the original implementation of PINN as possible. The unique architecture of the Finite-PINN model addresses these challenges by separating the approximation of stress and displacement fields, and by transforming the solution space from the traditional Euclidean space to a Euclidean-topological joint space. Several case studies presented in this paper demonstrate that the Finite-PINN model provides satisfactory results for a variety of problem types, including both forward and inverse problems, in both 2D and 3D contexts. The developed Finite-PINN model offers a promising tool for addressing general solid mechanics problems, particularly those not yet well-explored in current research.

翻译：PINN模型在解决流体偏微分方程问题方面已展现出卓越能力，其在固体力学领域的潜力也正逐渐显现。本研究识别出使用PINN求解一般固体力学问题时面临的两个关键挑战。通过将PINN的局限性与其固体力学中常用的成熟数值方法（如有限元法）进行比较，这些挑战尤为明显：a) PINN模型在无限域上生成解，这与大多数固体结构典型的有限边界条件相冲突；b) PINN采用的解空间为欧几里得空间，难以处理固体结构中常见的复杂几何形状。本文提出一种用于一般固体力学问题的PINN架构，称为有限PINN模型。该模型旨在有效解决上述两个挑战，同时尽可能保留PINN的原始实现方式。有限PINN模型通过分离应力场与位移场的近似表达，并将解空间从传统欧几里得空间转换至欧几里得-拓扑联合空间，以独特架构应对这些挑战。本文展示的多个案例研究表明，有限PINN模型在二维与三维场景中，对包括正问题与反问题在内的各类问题均能提供令人满意的结果。所开发的有限PINN模型为处理一般固体力学问题，特别是当前研究中尚未深入探索的问题，提供了具有前景的工具。