This paper presents Ψ-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems on general unstructured meshes with mixed boundary conditions. By leveraging the Implicit Layer Theory, Ψ-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 pre-requisite for physical applications, and is able to adapt to any initially provided solution. Ψ-GNN is trained using a physics-informed loss, and the training process is stable by design. 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, Ψ-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.

中文翻译：

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用于泊松问题的隐式 GNN 求解器


本文提出了 Ψ-GNN，这是一种新颖的图神经网络 （GNN） 方法，用于解决具有混合边界条件的一般非结构化网格上普遍存在的泊松偏微分方程问题。通过利用隐式层理论，Ψ-GNN 模拟了一个“无限”深度网络，从而避免了对所需消息传递层数量的经验调整以获得解决方案。其原始架构明确考虑了边界条件，这是物理应用的关键先决条件，并且能够适应任何最初提供的解决方案。Ψ-GNN 是使用物理知情损失进行训练的，训练过程在设计上是稳定的。此外，该方法的一致性在理论上得到了证明，其灵活性和泛化效率也得到了实验证明：相同的学习模型可以准确处理各种大小的非结构化网格，以及不同的边界条件。据我们所知，Ψ-GNN 是第一种基于物理学的基于 GNN 的方法，它可以处理各种非结构化域、边界条件和初始解，同时还提供收敛保证。