Learning operator mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this work, we propose an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms and boundary conditions are considered as input features. To capture the discontinuities of both input functions and output solutions across the interface, IONet divides the entire domain into several separate sub-domains according to the interface, and leverages multiple branch networks and truck networks. Each branch network extracts latent representations of input functions at a fixed number of sensors on a specific sub-domain, and each truck network is responsible for output solutions on one sub-domain. In addition, tailored physics-informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the requirement for training datasets and makes IONet effective without any paired input-output observations in the interior of the computational domain. Extensive numerical studies show that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy, efficiency, and versatility.

翻译：近年来，通过神经网络学习无限维巴拿赫空间之间的算子映射引起了广泛关注。本文提出了一种界面算子网络（IONet），用于求解参数化椭圆界面偏微分方程，其中不同系数、源项和边界条件均被作为输入特征考虑。为捕捉输入函数和输出解在界面上的不连续性，IONet根据界面将整个计算域划分为若干独立子域，并利用多个分支网络和主干网络。每个分支网络提取特定子域上固定传感器位置处输入函数的潜在表示，而每个主干网络则负责生成对应子域上的输出解。此外，我们提出了针对物理特性的IONet损失函数以确保物理一致性，这极大降低了训练数据的需求，使得IONet在无需计算域内部成对输入-输出观测数据的情况下仍能高效运行。大量数值研究表明，IONet在精度、效率和通用性方面均优于现有最先进的深度算子网络。