With recent advancements in computer hardware and software platforms, there has been a surge of interest in solving partial differential equations with deep learning-based methods, and the integration with domain decomposition strategies has attracted considerable attention owing to its enhanced representation and parallelization capacities of the network solution. While there are already several works that substitute the subproblem solver with neural networks for overlapping Schwarz methods, the non-overlapping counterpart has not been extensively explored because of the inaccurate flux estimation at interface that would propagate errors to neighbouring subdomains and eventually hinder the convergence of outer iterations. In this study, a novel learning approach for solving elliptic boundary value problems, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to enable reliable flux transmission across subdomain interfaces, thereby allowing us to construct effective learning algorithms for realizing non-overlapping domain decomposition methods (DDMs) in the presence of erroneous interface conditions. Numerical experiments on a variety of elliptic problems, including regular and irregular interfaces, low and high dimensions, two and four subdomains, and smooth and high-contrast coefficients are carried out to validate the effectiveness of our proposed algorithms.

翻译：随着计算机硬件和软件平台的近期进展，基于深度学习的方法求解偏微分方程引起了广泛兴趣，而领域分解策略的整合因其在网络解中增强的表征能力和并行化能力而备受关注。尽管已有若干工作使用神经网络替代子问题求解器用于重叠型Schwarz方法，但由于界面处不精确的通量估计会将误差传播至相邻子域并最终阻碍外部迭代的收敛，非重叠型领域分解方法尚未得到充分探索。本研究提出一种求解椭圆边值问题的新型学习方法，即使用神经网络延拓算子的补偿型深度Ritz方法，该方法能够实现跨子域界面的可靠通量传递，从而在存在不准确界面条件的情况下构建有效的非重叠领域分解方法（DDMs）学习算法。通过对多种椭圆问题（包括规则与不规则界面、低维与高维、两个与四个子域、光滑与高对比系数）进行数值实验，验证了所提算法的有效性。