Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled data, and physics-informed DeepONets encounter training challenges. In this paper, we introduce a novel physics-informed tailored finite point operator network (PI-TFPONet) method to solve parametric interface problems without the need for labeled data. Our method fully leverages the prior physical information of the problem, eliminating the need to include the PDE residual in the loss function, thereby avoiding training challenges. The PI-TFPONet is specifically designed to address certain properties of the problem, allowing us to naturally obtain an approximate solution that closely matches the exact solution. Our method is theoretically proven to converge if the local mesh size is sufficiently small and the training loss is minimized. Notably, our approach is uniformly convergent for singularly perturbed interface problems. Extensive numerical studies show that our unsupervised PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised deep operator networks in terms of accuracy and versatility.

翻译：近年来，利用神经网络学习参数化偏微分方程（PDEs）的算子受到了广泛关注。然而，深度算子网络（DeepONets）等标准方法需要大量标注数据，而基于物理信息的DeepONets则面临训练挑战。本文提出了一种新颖的基于物理信息的定制有限点算子网络（PI-TFPONet）方法，用于求解参数化界面问题，且无需标注数据。我们的方法充分利用了问题的先验物理信息，无需在损失函数中包含PDE残差，从而避免了训练难题。PI-TFPONet专门设计用于处理问题的某些特性，使我们能够自然地获得与精确解高度吻合的近似解。理论上证明，若局部网格尺寸足够小且训练损失最小化，我们的方法能够收敛。值得注意的是，对于奇异摄动界面问题，我们的方法具有一致收敛性。大量数值研究表明，在精度和泛化能力方面，我们这种无监督的PI-TFPONet与现有最先进的监督式深度算子网络相当甚至更优。