Ritz神经网络求解微分方程的数值比较及敏感性分析

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Ritz神经网络求解微分方程的数值比较及敏感性分析
Numerical Comparison and Sensitivity Analysis of Differential Equations Solved by Ritz Neural Network

DOI: 10.12677/AAM.2024.135226, PDF, 下载: 19 浏览: 38 科研立项经费支持
作者: 史正梅^*, 刘云美, 范小林：贵州师范大学，数学科学学院，贵州贵阳
关键词: Ritz神经网络；复合楝形求积；复合Simpson求积；三点Gauss求积；蒙特卡罗求积；Ritz Neural Network； Compound Trapezoid Quadrature； Compound Simpson Quadrature； Three-Point Gauss Quadrature； Monte Carlo Quadrature

摘要: Ritz神经网络已经被广泛用来求解微分方程，其中关键一步是近似损失函数中所涉及的定积分，因此求积方法的选取对神经网络逼近解尤为重要。本文通过一些例子进行数值比较分析。首先，我们引入Dirichlet和Neumann边界的边值问题模型。其次，构造神经网络进行模型训练。另外，详细介绍复合楝形求积、复合Simpson求积、三点Gauss求积以及蒙特卡罗求积法。然后，对算例进行数值比较分析，结果表明三点Gauss求积方法更好。最后对神经网络参数敏感性做进一步研究，神经网络的精度随着训练集的增大而提高，当神经元数量达到4 时，再增加神经元数量并不能明显提高精度，而增加隐藏层数量和更换激活函数并没有太大的影响。

Abstract: Ritz neural networks have been widely used to solve differential equations, in which the key step is to approximate the definite integral involved in the loss function, so the selection of quadrative methods is particularly important for neural networks to approximate the solution. In this paper, some examples are used for numerical comparison. First, we introduce the boundary value problem model of Dirichlet and Neumann boundaries. Secondly, the neural network is constructed to train the model. In addition, the methods of compound trapezoidal quadrature, compound Simpson quadrature, three-point Gauss quadrature and Monte Carlo quadrature are introduced in detail. Then, the numerical comparison and analysis of the examples show that the three-point Gauss quadrature method is better. Finally, the sensitivity of neural network parameters was further studied, and the accuracy of neural network was improved with the increase of training set. When the number of neurons reached 4, increasing the number of neurons could not significantly improve the accuracy, while increasing the number of hidden layers and changing the activation function had no significant impact.

文章引用：史正梅, 刘云美, 范小林. Ritz神经网络求解微分方程的数值比较及敏感性分析[J]. 应用数学进展, 2024, 13(5): 2380-2391. https://doi.org/10.12677/AAM.2024.135226

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