This paper presents $\Psi$-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems with mixed boundary conditions. By leveraging the Implicit Layer Theory, $\Psi$-GNN models an ''infinitely'' deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical prerequisite for physical applications, and is able to adapt to any initially provided solution. $\Psi$-GNN is trained using a ''physics-informed'' loss, and the training process is stable by design, and insensitive to its initialization. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, $\Psi$-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.

翻译：本文提出了$\Psi$-GNN，一种新颖的图神经网络（GNN）方法，用于求解具有混合边界条件的泊松偏微分方程（PDE）问题。通过利用隐式层理论，$\Psi$-GNN建模了一个“无限”深网络，从而避免了为达到解而需要经验性调整消息传递层数的问题。其原始架构显式地考虑了边界条件（这是物理应用的关键前提），并能适应任何初始提供的解。$\Psi$-GNN采用“物理信息”损失函数进行训练，且训练过程设计上保持稳定，对初始化不敏感。此外，该方法的相容性得到了理论证明，其灵活性与泛化效率在实验中得到验证：同一学习模型能够准确处理不同尺寸的非结构化网格以及不同的边界条件。据我们所知，$\Psi$-GNN是首个能处理多种非结构化域、边界条件和初始解，同时提供收敛性保证的基于物理信息的GNN方法。