基于PINN方法的KdV类方程新孤子解的研究

邱天威; 魏光美; 宋禹欣; 王振

doi:10.21656/1000-0887.450122

基于PINN方法的KdV类方程新孤子解的研究

doi: 10.21656/1000-0887.450122

北京航空航天大学数学科学学院，北京 100191

基金项目:

国家自然科学基金(面上项目)(52171251)

详细信息

作者简介:
邱天威(2002—)，男，硕士生(E-mail: qiutw2002@163.com);魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn);王振(1981—)，男，教授，博士，博士生导师(E-mail: wangzmath@163.com).

通讯作者:
魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn).

中图分类号: O241
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出版历程
- 收稿日期: 2024-04-30
- 修回日期: 2024-06-04

Novel Soliton Solutions to KdV-Type Equations Based on Physics-Informed Neural Networks

School of Mathematical Sciences, Beihang University, Beijing 100191, P.R.China

Funds:

The National Science Foundation of China(52171251)

摘要

摘要: 该文采用物理信息神经网络(physicsinformed neural network，PINN）方法结合广义Miura变换，深入研究了三个KdV类方程，获得了一系列新的孤子解.具体而言，研究成果包括：基于改进的PINN方法，获得了mKdV方程的扭结钟形解的解析形式；通过Miura变换，发现了KdV方程的新单孤子解；结合广义Miura变换与PINN方法，预测出非线性较强的KdV类方程的暗孤子解.通过将PINN方法的数值结果与理论分析结果进行对比可以得知，基于广义Miura变换的PINN方法是发现偏微分方程新数值解的有效途径，同时对理论研究具有重要的启示意义.
- PINN方法 /
- Miura变换 /
- KdV类方程 /
- 孤立子 /
- 可积系统
Abstract: Physics-informed neural networks (PINNs) were applied in combination with generalized Miura transformations to investigate 3 KdV-type equations. Several novel soliton solutions, including the kink-bell solution of the mKdV equation, were derived analytically with the improved PINN method; a single-soliton-like solution of the KdV equation, was achieved through the Miura transformation; and a dark-soliton solution of a strongly nonlinear KdV equation, was obtained by means of both the generalized Miura transformation and the PINN methods. Comparison of the numerical results obtained under the PINN framework with the exact solutions from theoretical analysis shows that, the proposed algorithm effectively uncovers new numerical solutions of partial differential equations and offers valuable insights for theoretical research.
- physics-informed neural networks /
- Miura transformation /
- KdV-type equation /
- soliton /
- integrable system

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参考文献(19)

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基于PINN方法的KdV类方程新孤子解的研究

doi: 10.21656/1000-0887.450122

作者简介:
邱天威(2002—)，男，硕士生(E-mail: qiutw2002@163.com);魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn);王振(1981—)，男，教授，博士，博士生导师(E-mail: wangzmath@163.com).

通讯作者:
魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn).

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Novel Soliton Solutions to KdV-Type Equations Based on Physics-Informed Neural Networks

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留言板

基于PINN方法的KdV类方程新孤子解的研究

doi: 10.21656/1000-0887.450122

作者简介: 邱天威(2002—)，男，硕士生(E-mail: qiutw2002@163.com);魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn);王振(1981—)，男，教授，博士，博士生导师(E-mail: wangzmath@163.com).

通讯作者: 魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn).

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出版历程

Novel Soliton Solutions to KdV-Type Equations Based on Physics-Informed Neural Networks

计量

出版历程

目录

作者简介:
邱天威(2002—)，男，硕士生(E-mail: qiutw2002@163.com);魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn);王振(1981—)，男，教授，博士，博士生导师(E-mail: wangzmath@163.com).

通讯作者:
魏光美(1967—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: gmwei@buaa.edu.cn).