Learning operators mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, 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 in both the input functions and the output solutions across the interface, IONet divides the entire domain into several separate subdomains according to the interface and uses multiple branch nets and trunk nets. Each branch net extracts latent representations of input functions at a fixed number of sensors on a specific subdomain, and each trunk net is responsible for output solutions on one subdomain. Additionally, tailored physics-informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the training dataset requirement and makes IONet effective without any paired input-output observations inside the computational domain. Extensive numerical studies demonstrate that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy 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，其中将不同的系数、源项和边界条件视为输入特征。为捕捉界面两侧输入函数和输出解的不连续性，IONet根据界面将整个计算域划分为若干独立子域，并采用多个分支网络与主干网络。每个分支网络在特定子域的固定传感器位置提取输入函数的隐式表征，每个主干网络负责一个子域上的输出解。此外，本文设计了定制化的IONet物理信息损失函数以保证物理一致性，这显著降低了训练数据需求，使得IONet在计算域内无需任何配对输入-输出观测数据即可有效工作。大量数值研究表明，IONet在精度与泛化能力方面均优于现有最先进的深度算子网络。