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基于卷积神经网络模型数值求解双曲型偏微分方程的研究

高普阳 赵子桐 杨扬

高普阳, 赵子桐, 杨扬. 基于卷积神经网络模型数值求解双曲型偏微分方程的研究[J]. 应用数学和力学, 2021, 42(9): 932-947. doi: 10.21656/1000-0887.420050
引用本文: 高普阳, 赵子桐, 杨扬. 基于卷积神经网络模型数值求解双曲型偏微分方程的研究[J]. 应用数学和力学, 2021, 42(9): 932-947. doi: 10.21656/1000-0887.420050
GAO Puyang, ZHAO Zitong, YANG Yang. Study on Numerical Solutions to Hyperbolic Partial Differential Equations Based on the Convolutional Neural Network Model[J]. Applied Mathematics and Mechanics, 2021, 42(9): 932-947. doi: 10.21656/1000-0887.420050
Citation: GAO Puyang, ZHAO Zitong, YANG Yang. Study on Numerical Solutions to Hyperbolic Partial Differential Equations Based on the Convolutional Neural Network Model[J]. Applied Mathematics and Mechanics, 2021, 42(9): 932-947. doi: 10.21656/1000-0887.420050

基于卷积神经网络模型数值求解双曲型偏微分方程的研究

doi: 10.21656/1000-0887.420050
基金项目: 

国家自然科学基金(11901051

陕西省自然科学基础研究计划青年项目(2020JQ-338;2019JQ-625)

11971075)

详细信息
    作者简介:

    高普阳(1991—),男,讲师,博士(通讯作者. E-mail: gaopuyang@chd.edu.cn).

    通讯作者:

    高普阳(1991—),男,讲师,博士(通讯作者. E-mail: gaopuyang@chd.edu.cn).

  • 中图分类号: O29

Study on Numerical Solutions to Hyperbolic Partial Differential Equations Based on the Convolutional Neural Network Model

Funds: 

The National Natural Science Foundation of China(11901051

11971075)

  • 摘要: 人工神经网络近年来得到了快速发展,将此方法应用于数值求解偏微分方程是学者们关注的热点问题.相比于传统方法其具有应用范围广泛(即同一种模型可用于求解多种类型方程)、网格剖分条件要求低等优势,并且能够利用训练好的模型直接计算区域中任意点的数值.该文基于卷积神经网络模型,对传统有限体积法格式中的权重系数进行优化,以得到在粗粒度网格下具有较高精度的新数值格式,从而更适用于复杂问题的求解.该网络模型可以准确、有效地求解Burgers方程和level set方程,数值结果稳定,且具有较高数值精度.
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    [14]毕卉. 基于Godunov格式的求解Burgers方程的有限差分法[D]. 硕士学位论文. 哈尔滨: 哈尔滨工业大学, 2006.(BI Hui. Finite difference method for solving Burgers equation based on Godunov scheme[D]. Master Thesis. Harbin: Harbin Institute of Technology, 2006.(in Chinese))
    [15]KINGMA D P, BA J L. Adam: a method for stochastic optimization[R/OL]. 2014.[2021-03-23]. https://perso.ensta-paris.fr/~pcarpent/StochOpt/PDF/Articles/Kingma_arXiv_2017.pdf.
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    [17]包立平, 洪文珍. 一维弱噪声随机Burgers方程的奇摄动解[J]. 应用数学和力学, 2018,39(1): 113-122.(BAO Liping, HONG Wenzhen. Singular perturbation solutions to 1D stochastic burgers equations under weak noises[J].Applied Mathematics and Mechanics,2018,39(1): 113-122.(in Chinese))
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出版历程
  • 收稿日期:  2021-02-23
  • 修回日期:  2021-04-21
  • 网络出版日期:  2021-09-29

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