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 recomputation. 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 the experiments.

中文翻译：

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一种由偏微分方程描述的物理场模拟物理驱动的 GraphSAGE 方法


物理信息神经网络 （PINN） 已成功解决基于偏微分方程 （PDE） 的各种计算物理问题。然而，在解决与奇点和振荡等不规则性相关的问题时，经过训练的解决方案通常精度较低。此外，大多数当前工作仅提供预定输入参数的训练解。如果输入参数发生任何变化，则需要迁移学习或重新训练，而传统的数值技术也需要重新计算。在这项工作中，提出了一种基于 Galerkin 方法和分段多项式节点基函数的物理驱动的 GraphSAGE 方法 （PD-GraphSAGE），以解决由不规则偏微分方程控制的计算问题并开发参数化偏微分方程代理模型。这种方法采用物理域的图形表示，从而减少了由于局部细化而对评估点的需求。设计了一种与距离相关的边缘特征和特征映射策略，分别帮助奇点和振荡情况的训练和收敛。所提出的方法的优点通过几个案例得到了证明。此外，成功建立了由高斯随机场源参数化的热传导问题的鲁棒偏微分方程代理模型，不仅提供了准确的解，而且比实验中的有限元方法快了数倍。