We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of solutions across neighboring subdomains, we adopt a generalized Robin-type interface condition, assigning unique Robin parameters to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch on the Robin interface condition, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. It represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. A key strength of our method lies in its ability to learn a Robin parameter for each subdomain, thereby enhancing information exchange with its neighboring subdomains. We observe that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach.

翻译：我们提出了一种基于人工神经网络的无网格Schwarz型非重叠区域分解方法，用于求解涉及偏微分方程(partial differential equations, PDEs)的正问题和反问题。为确保相邻子域间解的一致性，我们采用广义Robin型界面条件，为每个子域分配唯一的Robin参数。这些子域特定的Robin参数通过最小化Robin界面条件上的不匹配度进行学习，从而在训练过程中促进高效的信息交换。该方法适用于拉普拉斯方程和亥姆霍兹方程。它通过独立的神经网络模型表示局部解，该模型经过训练以最小化控制PDE的损失函数，同时通过增广拉格朗日形式严格施加边界和界面条件。该方法的一个关键优势在于能够为每个子域学习一个Robin参数，从而增强与其相邻子域的信息交换。我们观察到，学习到的Robin参数能够适应解的局部行为、区域划分方式以及子域相对于整体域的位置。在正问题和反问题上的大量实验，包括存在交叉点的一向和二向分解，证明了我们提出方法的通用性和性能。