标题： 一种隐式GNN求解器用于类似泊松的问题 显示英文标题

标题： An Implicit GNN Solver for Poisson-like problems

摘要： 本文提出了$\Psi$-GNN，一种用于解决具有混合边界条件的普遍泊松PDE问题的新颖图神经网络（GNN）方法。 通过利用隐层理论，$\Psi$-GNN建模了一个“无限”深的网络，从而避免了经验性调整所需的消息传递层数以获得解。 其原始架构明确考虑了边界条件，这是物理应用的关键前提，并能够适应任何最初提供的解。 $\Psi$-GNN使用“物理感知”的损失进行训练，其训练过程设计稳定，对初始化不敏感。 此外，该方法的一致性得到了理论证明，其灵活性和泛化效率通过实验得到验证：相同的已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，$\Psi$-GNN是第一个基于GNN的物理感知方法，能够在提供收敛保证的同时处理各种非结构化域、边界条件和初始解。 显示英文摘要

摘要： 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.