Interface problems pose significant challenges due to the discontinuity of their solutions, particularly when they involve singular perturbations or high-contrast coefficients, resulting in intricate singularities that complicate resolution. The increasing adoption of deep learning techniques for solving partial differential equations has spurred our exploration of these methods for addressing interface problems. In this study, we introduce Tailored Finite Point Operator Networks (TFPONets) as a novel approach for tackling parameterized interface problems. Leveraging DeepONets and integrating the Tailored Finite Point method (TFPM), TFPONets offer enhanced accuracy in reconstructing solutions without the need for intricate equation manipulation. Experimental analyses conducted in both one- and two-dimensional scenarios reveal that, in comparison to existing methods such as DeepONet and IONet, TFPONets demonstrate superior learning and generalization capabilities even with limited locations.

翻译：界面问题因其解的不连续性而带来显著挑战，尤其当涉及奇异摄动或高对比度系数时，会产生复杂的奇异性，从而使得求解过程变得困难。随着深度学习技术日益广泛地应用于偏微分方程的求解，我们开始探索利用这些方法来解决界面问题。在本研究中，我们提出了一种新颖的定制化有限点算子网络，用于处理参数化界面问题。该方法基于DeepONets，并整合了定制化有限点方法，从而在无需复杂方程操作的情况下，实现了更高的解重构精度。在一维和二维场景下的实验分析表明，与现有的DeepONet和IONet等方法相比，即使在采样点有限的情况下，定制化有限点算子网络也展现出更优的学习与泛化能力。