基于LM算法的多尺度神经网络求解椭圆界面问题
Solving Elliptic Interface Problems Using a Multi-Scale Neural Network Based on the LM Algorithm
DOI: 10.12677/aam.2025.1410431, PDF,   
作者: 粟钊阳：长沙理工大学数学与统计学院，湖南 长沙；长沙理工大学工程数学建模与分析湖南省重点实验室，湖南 长沙
关键词: 多尺度神经网络；LM算法；椭圆界面问题；Multi-Scale Neural Network； LM Algorithm； Elliptic Interface Problem
摘要: 近几年以来，利用深度学习方法解决偏微分方程问题引起了广泛关注和研究。本文介绍了一种基于改进的Levenberg-Marquardt (LM)优化方法的多尺度神经网络，该方法在求解椭圆界面问题的精度方面显示出不俗的潜力。文章通过单个神经网络框架解决界面问题，并选用不同的激活函数进行对比，通过几个具有规则和不规则界面的数值实验，以验证神经网络效果及相关理论。
Abstract: In recent years, the application of deep learning methods to solve partial differential equations has garnered significant attention and research. This paper introduces a multi-scale neural network based on an improved Levenberg-Marquardt (LM) optimization method, which demonstrates remarkable potential in achieving high accuracy for solving elliptic interface problems. The paper addresses boundary problems within a single neural network framework, comparing different activation functions. Numerical experiments involving both regular and irregular boundaries validate the neural network’s performance and related theoretical foundations.
文章引用：粟钊阳. 基于LM算法的多尺度神经网络求解椭圆界面问题[J]. 应用数学进展, 2025, 14(10): 189-198. https://doi.org/10.12677/aam.2025.1410431

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