Non-overlapping domain decomposition methods are natural for solving interface problems arising from various disciplines, however, the numerical simulation requires technical analysis and is often available only with the use of high-quality grids, thereby impeding their use in more complicated situations. To remove the burden of mesh generation and to effectively tackle with the interface jump conditions, a novel mesh-free scheme, i.e., Dirichlet-Neumann learning algorithm, is proposed in this work to solve the benchmark elliptic interface problem with high-contrast coefficients as well as irregular interfaces. By resorting to the variational principle, we carry out a rigorous error analysis to evaluate the discrepancy caused by the boundary penalty treatment for each decomposed subproblem, which paves the way for realizing the Dirichlet-Neumann algorithm using neural network extension operators. The effectiveness and robustness of our proposed methods are demonstrated experimentally through a series of elliptic interface problems, achieving better performance over other alternatives especially in the presence of erroneous flux prediction at interface.

翻译：非重叠区域分解方法是解决跨学科界面问题的自然途径，然而数值模拟需要专业分析且通常依赖高质量网格，这制约了其在复杂场景中的应用。为消除网格生成负担并有效处理界面跳跃条件，本文提出了一种新颖的无网格方案——狄利克雷-诺伊曼学习算法，用于求解具有高对比度系数及不规则界面的基准椭圆界面问题。基于变分原理，我们开展了严格的误差分析以评估各分解子问题中边界罚函数处理带来的偏差，这为利用神经网络延拓算子实现狄利克雷-诺伊曼算法奠定了基础。通过一系列椭圆界面问题的实验验证，所提方法在有效性与鲁棒性上优于现有替代方案，尤其在处理界面处存在通量预测误差的情况下表现出更优性能。