Recently, a class of machine learning methods called physics-informed neural networks (PINNs) has been proposed and gained prevalence in solving various scientific computing problems. This approach enables the solution of partial differential equations (PDEs) via embedding physical laws into the loss function. Many inverse problems can be tackled by simply combining the data from real life scenarios with existing PINN algorithms. In this paper, we present a multi-task learning method using uncertainty weighting to improve the training efficiency and accuracy of PINNs for inverse problems in linear elasticity and hyperelasticity. Furthermore, we demonstrate an application of PINNs to a practical inverse problem in structural analysis: prediction of external loads of diverse engineering structures based on limited displacement monitoring points. To this end, we first determine a simplified loading scenario at the offline stage. By setting unknown boundary conditions as learnable parameters, PINNs can predict the external loads with the support of measured data. When it comes to the online stage in real engineering projects, transfer learning is employed to fine-tune the pre-trained model from offline stage. Our results show that, even with noisy gappy data, satisfactory results can still be obtained from the PINN model due to the dual regularization of physics laws and prior knowledge, which exhibits better robustness compared to traditional analysis methods. Our approach is capable of bridging the gap between various structures with geometric scaling and under different loading scenarios, and the convergence of training is also greatly accelerated through not only the layer freezing but also the multi-task weight inheritance from pre-trained models, thus making it possible to be applied as surrogate models in actual engineering projects.

翻译：近年来，一类称为物理信息神经网络（PINNs）的机器学习方法被提出，并广泛应用于求解各类科学计算问题。该方法通过将物理定律嵌入损失函数，实现偏微分方程（PDEs）的求解。诸多反问题可简单通过结合实际场景数据与现有PINN算法得以解决。本文提出一种基于不确定性加权的多任务学习方法，以提升PINN在线弹性及超弹性材料反问题中的训练效率与精度。进一步，我们展示了PINN在结构分析实际反问题中的应用：基于有限位移监测点预测不同工程结构的外荷载。为此，首先在离线阶段确定简化荷载工况，通过将未知边界条件设为可学习参数，PINN可利用实测数据预测外荷载。在实际工程项目的在线阶段，采用迁移学习对离线阶段预训练模型进行微调。结果表明，即便存在含噪声的稀疏数据，由于物理定律与先验知识的双重正则化作用，PINN模型仍可获得满意结果，较传统分析方法展现出更优鲁棒性。本方法能够弥合不同几何缩放结构及多种荷载工况间的差异，同时通过层冻结及预训练模型的多任务权重继承显著加速训练收敛，从而使其具备作为实际工程代理模型的可行性。