论文标题
使用奇异性分裂深丽兹方法解决奇异源的椭圆形问题
Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method
论文作者
论文摘要
在这项工作中，我们开发了一个有效的求解器，该求解器基于具有可变系数和奇异源的二阶椭圆方程的神经网络。这类问题涵盖了一般点源，线路来源和点线来源的组合，并且具有广泛的实际应用。所提出的方法是基于将真实溶液分解为一个单一部分，该部分使用拉普拉斯方程的基本解决方案在分析上以分析性的方式，以及一个正常零件，可满足适合修改的椭圆形PDE，并使用更顺滑的来源，然后使用深层Ritz方法为常规部分求解。建议提出一种路径遵循的策略，以选择执行Dirichlet边界条件的惩罚参数。提出了具有点源，线条源或其组合的两维空间和多维空间的广泛数值实验，以说明所提出的方法的效率，并提供了一些基于神经网络的现有方法的比较研究，这显然显示了其对特定问题类别的竞争力。此外，我们简要讨论该方法的误差分析。
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination of point-line sources, and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable modified elliptic PDE with a smoother source, and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for enforcing the Dirichlet boundary condition. Extensive numerical experiments in two- and multi-dimensional spaces with point sources, line sources or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches based on neural networks is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach.
