In this work, we introduce two innovative types of discontinuity-removing physics-informed neural networks (DR-PINNs) aimed at solving variable coefficient elliptic interface problems on curved surfaces, i.e., decoupling DR-PINN and coupling DR-PINN. Initially, by leveraging the level set function associated with the surface, we reframe the surface differential operators as conventional differential operators within the framework of Eulerian coordinates. The resultant solution is divided into two parts: continuous and discontinuous components, which are each approximated using separate neural network surrogates that can be trained independently or together. The decoupling DR-PINN utilizes a sequential training procedure for the two components. The first neural network focuses on the discontinuous part and pre-trains on solution jumps to facilitate the learning of the complementary PDE conditions by the subsequent network. This decoupling strategy relies heavily on the cusp-enforced level set function of the interface, which may lead to inefficiencies since the training process is separated into two distinct stages. To address these challenges, we propose a novel coupling approach of the DR-PINNs, in which both components simultaneously learn complementary conditions within a unified network, thereby removing the requirement for cusp-enforcing level set functions of the interface. Ultimately, we perform numerical experiments to validate the precision and effectiveness of both types of DR-PINNs. To the best of our knowledge, this is the first work on neural networks to solve surface interface problems.

中文翻译：

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两种新型的去断面 PINN，用于求解曲面上的变系数椭圆界面问题


在这项工作中，我们引入了两种创新类型的不连续性消除物理信息神经网络 （DR-PINN），旨在解决曲面上的变系数椭圆界面问题，即解耦 DR-PINN 和耦合 DR-PINN。最初，通过利用与表面相关的水平集函数，我们在欧拉坐标框架内将表面微分算子重新构建为常规微分算子。得到的解决方案分为两部分：连续分量和不连续分量，每部分都使用单独的神经网络代理进行近似处理，这些代理项可以独立训练，也可以一起训练。解耦 DR-PINN 对这两个组件使用顺序训练过程。第一个神经网络专注于不连续部分，并对解跳转进行预训练，以促进后续网络学习互补偏微分方程条件。这种解耦策略在很大程度上依赖于接口的 cusp 强制级别集功能，这可能会导致效率低下，因为训练过程分为两个不同的阶段。为了应对这些挑战，我们提出了一种新的 DR-PINN 耦合方法，其中两个组件同时学习统一网络内的互补条件，从而消除了对接口的尖点强制级别集功能的要求。最终，我们进行数值实验以验证两种类型的 DR-PINN 的精度和有效性。据我们所知，这是解决表面界面问题的神经网络的第一项工作。