The interface problem is highly challenging due to its non-smoothness, discontinuity, and interface complexity. In this paper, a new and simple Deep Interface Alternation Method (DIAM) is developed to solve elliptic interface problems to avoid dealing with interfaces. It combines the ideas of domain decomposition methods and deep learning methods. Specifically, we first transform the interface problem with discontinuous derivatives into multiple continuous subproblems based on the Dirichlet–Dirichlet algorithm of domain decomposition. Then, we establish different fully connected neural networks for each subproblem to approximate parallelly the continuous solutions in the subdomain. The interface information is especially exchanged among the different loss functions of each subdomain neural network while minimizing the loss functions of each subdomain separately to obtain solutions to the entire interface problem. Numerical experiments were conducted on two-dimensional and three-dimensional elliptical interface problems with different coefficient contrasts and interface complexity. The results indicate that the Deep Interface Alternation Method has effectiveness and accuracy.

中文翻译：

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基于域分解的深度界面交替法（DIAM）求解椭圆界面问题


界面问题由于其非光滑性、不连续性和界面复杂性而极具挑战性。在本文中，开发了一种新的简单的深度界面交替方法（DIAM）来解决椭圆界面问题，以避免处理界面。它结合了领域分解方法和深度学习方法的思想。具体来说，我们首先基于域分解的Dirichlet-Dirichlet算法将具有不连续导数的界面问题转化为多个连续子问题。然后，我们为每个子问题建立不同的全连接神经网络，以并行逼近子域中的连续解。特别是在每个子域神经网络的不同损失函数之间交换界面信息，同时分别最小化每个子域的损失函数，以获得整个界面问题的解。对不同系数对比和界面复杂度的二维和三维椭圆界面问题进行了数值实验。结果表明，深度界面交替方法具有有效性和准确性。