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.

本文介绍了一种基于图神经网络和暗示层理论的物理信息的新型混合边界条件下解决泊松方程问题的方法，通过使用‘物理信息’的损失函数对该方法进行训练，证明了其一致性，并在实验中证明了其灵活性和广泛适用性。