Physics-informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and oscillations, trained solutions usually suffer low accuracy. In addition, most current works only offer the trained solution for predetermined input parameters. If any change occurs in input parameters, transfer learning or retraining is required, and traditional numerical techniques also need an independent simulation. In this work, a physics-driven GraphSAGE approach (PD-GraphSAGE) based on the Galerkin method and piecewise polynomial nodal basis functions is presented to solve computational problems governed by irregular PDEs and to develop parametric PDE surrogate models. This approach employs graph representations of physical domains, thereby reducing the demands for evaluated points due to local refinement. A distance-related edge feature and a feature mapping strategy are devised to help training and convergence for singularity and oscillation situations, respectively. The merits of the proposed method are demonstrated through a couple of cases. Moreover, the robust PDE surrogate model for heat conduction problems parameterized by the Gaussian random field source is successfully established, which not only provides the solution accurately but is several times faster than the finite element method in our experiments.

翻译：物理信息神经网络（PINNs）已成功解决基于偏微分方程（PDE）的各类计算物理问题。然而，在处理奇异性、振荡等不规则性问题时，训练所得解通常精度较低。此外，现有多数工作仅能针对预设输入参数提供训练解。若输入参数发生改变，则需进行迁移学习或重新训练，而传统数值方法也需要独立进行模拟仿真。本研究提出一种基于伽辽金法及分段多项式节点基函数的物理驱动GraphSAGE方法（PD-GraphSAGE），用于求解不规则PDE控制的计算问题并建立参数化PDE代理模型。该方法采用物理域图表示，通过局部细化减少所需评估点。针对奇异性和振荡情形，分别设计距离相关边特征与特征映射策略以促进训练与收敛。通过多个案例验证了所提方法的优势。此外，成功建立了由高斯随机场源参数化的热传导问题鲁棒PDE代理模型，该模型不仅能够精确求解，且在实验中比有限元方法快数倍。