基于残差的Fourier神经网络求解广域椭圆型微分方程以及收敛性证明

颜景越; 苏欢

doi:10.21656/1000-0887.460125

基于残差的Fourier神经网络求解广域椭圆型微分方程以及收敛性证明

doi: 10.21656/1000-0887.460125

颜景越^,,
苏欢

哈尔滨工业大学（威海）理学院，山东威海 264209

基金项目:

山东省自然科学基金面上项目（ZR202102220411）

详细信息

作者简介:
颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）；苏欢（1981—），女，副教授，博士，博士生导师（E-mail: suhuantg@hitwh.edu.cn）.

通讯作者:
颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）.

中图分类号: O241|TP183
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- 文章访问数: 10
- HTML全文浏览量: 2
- PDF下载量: 3
- 被引次数: 0
出版历程
- 收稿日期: 2025-06-08
- 修回日期: 2025-09-02
- 网络出版日期: 2026-07-23

Residual-Based Fourier Neural Networks for Solving Elliptic Differential Equations on Unbounded Domains 

Yan Jingyue^,,
Su Huan

School of Science, Harbin Institute of Technology(Weihai), Weihai, Shandong 264209, P.R. China

摘要

摘要: 物理信息神经网络(physicsinformed neural network, PINN)在求解偏微分方程的正反问题中应用广泛.然而，传统PINN在求解广域微分方程时，存在精度不足、易陷入局部最优解的局限，且缺乏收敛性理论保证.针对上述问题，提出了一种基于残差的Fourier神经网络.该网络将残差Fourier层集成到传统PINN框架中，利用三角函数的周期性特征有效提升模型在求解广域问题时的精度.特别地，针对线性椭圆型微分方程，建立了基于残差的Fourier神经网络的收敛性理论分析.数值实验结果表明，基于残差的Fourier神经网络在求解正反问题时，相较于传统PINN，具有更高的求解精度和更快的收敛速度.
- 物理信息神经网络 /
- 残差 /
- Fourier神经网络 /
- 广域微分方程 /
- 线性椭圆型微分方程 /
- 收敛性分析
Abstract: Physics-informed neural network (PINN) are widely applied for solving both forward and inverse problems of differential equations. However, traditional PINNs exhibit limitations when solving differential equations over large domains, including insufficient accuracy, susceptibility to local optima, and a lack of theoretical guarantees of convergence. To address these issues, a residual-based Fourier neural network (Res-FNN) was proposed. This network integrates residual Fourier layers into the traditional PINN framework, leveraging the periodicity property of trigonometric functions to enhance the model’s accuracy in solving large-domain problems effectively. In particular, a theoretical convergence analysis was established for the Res-FNN applied to linear elliptic differential equations. Numerical experiments demonstrate that, the Res-FNN outperforms the traditional PINN in solving both forward and inverse problems, achieving higher accuracy and faster convergence speed.
- physics-informed neural network /
- residual /
- Fourier neural network /
- large-domain differential equation /
- linear elliptic differential equation /
- convergence analysis

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参考文献(16)

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基于残差的Fourier神经网络求解广域椭圆型微分方程以及收敛性证明

doi: 10.21656/1000-0887.460125

作者简介:
颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）；苏欢（1981—），女，副教授，博士，博士生导师（E-mail: suhuantg@hitwh.edu.cn）.

通讯作者:
颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）.

计量

Residual-Based Fourier Neural Networks for Solving Elliptic Differential Equations on Unbounded Domains 

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目录

留言板

基于残差的Fourier神经网络求解广域椭圆型微分方程以及收敛性证明

doi: 10.21656/1000-0887.460125

作者简介: 颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）；苏欢（1981—），女，副教授，博士，博士生导师（E-mail: suhuantg@hitwh.edu.cn）.

通讯作者: 颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）.

计量

出版历程

Residual-Based Fourier Neural Networks for Solving Elliptic Differential Equations on Unbounded Domains 

计量

出版历程

目录

作者简介:
颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）；苏欢（1981—），女，副教授，博士，博士生导师（E-mail: suhuantg@hitwh.edu.cn）.

通讯作者:
颜景越（2001—），男，硕士生（通信作者. E-mail: 23s030162@stu.hit.edu.cn）.