Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method

Tianhao Hu, Bangti Jin, Zhi Zhou
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引用次数: 3

Abstract

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.
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用奇异分裂深里兹方法求解奇异源椭圆型问题
在这项工作中,我们开发了一种基于神经网络的二阶变系数奇异源椭圆方程的有效求解器。这类问题涵盖了一般的点源、线源和点线源的组合,具有广泛的实际应用。该方法基于将真解分解为利用拉普拉斯方程基本解解析已知的奇异部分和满足具有较光滑源的适当修正椭圆偏微分方程的正则部分,然后利用深里兹方法求解正则部分。提出了一种路径跟踪策略来选择执行狄利克雷边界条件的惩罚参数。在二维和多维空间中进行了大量的点源、线源或它们的组合的数值实验,以说明该方法的有效性,并与几种现有的基于神经网络的方法进行了比较研究,清楚地表明了该方法在特定类别问题上的竞争力。此外,我们还简要讨论了该方法的误差分析。
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