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:本文展示了 $\ Psi$- GNN, 这是在混合边界条件下解决无处不在的 Poisson PDE 问题的一种新颖的图形神经网络(GNN) 。 通过利用隐形层理论,$\ Psi$- GNN 模型“ 无限” 深度网络, 从而避免对所需的信息传递层数量进行实验性调整, 以达成解决方案。 其原始架构明确考虑到边界条件, 即物理应用的关键先决条件, 并能够适应任何最初提供的解决办法。 $\ Psi$- GNNN 是使用“ 物理知情” 损失来培训的, 培训过程通过设计而稳定, 并且对其初始化不敏感。 此外, 方法的一致性在理论上得到了证明, 其灵活性和一般化效率在实验上得到了证明: 同样的学习模型可以准确处理不同尺寸的无结构的模具, 以及不同的边界条件。 据我们所知, $\ Psi$- GNNN 是第一个基于物理知情的GNNN 方法, 能够处理各种非结构领域、 边界条件和初步解决方案。