Physics-Informed Neural Networks (PINNs) have gained considerable interest in diverse engineering domains thanks to their capacity to integrate physical laws into deep learning models. Recently, geometry-aware PINN-based approaches that employ the strong form of underlying physical system equations have been developed with the aim of integrating geometric information into PINNs. Despite ongoing research, the assessment of PINNs in problems with various geometries remains an active area of investigation. In this work, we introduce a novel physics-informed framework named the Geometry-Aware Deep Energy Method (GADEM) for solving structural mechanics problems on different geometries. As the weak form of the physical system equation (or the energy-based approach) has demonstrated clear advantages compared to the strong form for solving solid mechanics problems, GADEM employs the weak form and aims to infer the solution on multiple shapes of geometries. Integrating a geometry-aware framework into an energy-based method results in an effective physics-informed deep learning model in terms of accuracy and computational cost. Different ways to represent the geometric information and to encode the geometric latent vectors are investigated in this work. We introduce a loss function of GADEM which is minimized based on the potential energy of all considered geometries. An adaptive learning method is also employed for the sampling of collocation points to enhance the performance of GADEM. We present some applications of GADEM to solve solid mechanics problems, including a loading simulation of a toy tire involving contact mechanics and large deformation hyperelasticity. The numerical results of this work demonstrate the remarkable capability of GADEM to infer the solution on various and new shapes of geometries using only one trained model.

翻译：物理信息神经网络（PINNs）因其将物理定律融入深度学习模型的能力，已在多个工程领域引起广泛关注。近期，基于PINN的几何感知方法通过采用物理系统方程的强形式，旨在将几何信息融入PINNs。尽管已有研究不断推进，但针对不同几何形状问题的PINNs评估仍是活跃研究方向。本文提出一种名为几何感知深度能量方法（GADEM）的新型物理信息框架，用于求解不同几何形状的结构力学问题。鉴于物理系统方程的弱形式（或基于能量的方法）在固体力学问题求解中相比强形式具有显著优势，GADEM采用弱形式，旨在推断多种几何形状上的解。将几何感知框架融入基于能量的方法，可在精度和计算成本方面形成有效的物理信息深度学习模型。本文研究了几何信息的不同表示方式及几何潜在向量的编码方法，提出了基于所有考虑几何势能最小化的GADEM损失函数，并采用自适应学习方法进行配点采样以提升GADEM性能。通过固体力学问题的应用实例（包括涉及接触力学与大变形超弹性的玩具轮胎加载模拟），数值结果表明GADEM仅需单个训练模型即可在不同几何形状与新形状上实现优异的解推断能力。