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:
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
计量
- 文章访问数: 45
- HTML全文浏览量: 13
- PDF下载量: 11
- 被引次数: 0
出版历程
- 收稿日期: 2025-06-08
- 修回日期: 2025-09-02
- 刊出日期: 2026-07-01

Residual-Based Fourier Neural Networks for Solving Elliptic Differential Equations on Unbounded Domains and Convergence Analysis

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

HTML全文

图 1 PINN训练误差随区间长度b变化曲线

Figure 1. The PINN training error variation with interval length b

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图 2 区间[0, 6]中的PINN训练结果与解析解比较

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 2. Comparison of PINN training results with analytical solutions in the interval [0, 6]

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图 3 Res-FNN结构示意图

Figure 3. The Res-FNN structure diagram

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图 4 Res-FNN随区间长度b训练误差变化

Figure 4. The Res-FNN training error variation with interval length b

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图 5 两个不同神经网络L₂训练误差随区间长度b的变化对比

Figure 5. Comparison of L₂ training errors with interval length b for 2 different neural networks

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图 6 三种方法损失函数值的下降曲线

Figure 6. The decline curves of loss function values for 3 methods

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图 7 Res-FNN、FNN和PINN求解Poisson方程的结果

Figure 7. The results of solving Poisson equation with the Res-FNN, the FNN and the PINN

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图 8 三种方法损失函数值的下降曲线

Figure 8. The decline curves of loss function values for 3 methods

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图 9 Res-FNN、FNN和PINN求解Helmholtz方程的结果

Figure 9. The results of solving the Helmholtz equation with Res-FNN, FNN and PINN

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图 10 Res-FNN求解Poisson方程反问题的结果

Figure 10. The results of solving the inverse problem of the Poisson equation with the Res-FNN

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图 11 Res-FNN求解Helmholtz方程反问题的结果

Figure 11. The results of solving the inverse problem of the Helmholtz equation with the Res-FNN

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表 1 不同神经网络模型训练结果的L_∞误差和L₂误差及其对应训练次数

Table 1. The training result errors of L_∞ and L₂ for different neural network models, as well as the corresponding numbers of training times

model	error of L_∞	error of L₂	number of training times (Adam+L-BFGS)
Res-FNN	5.757×10^-5	1.091×10^-5	5 000+1 500
FNN	6.184×10^-5	1.950×10^-5	5 000+1 500
PINN	1.627×10^-4	3.475×10^-5	5 000+1 500

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表 2 不同神经网络模型训练结果的L_∞误差和L₂误差及其对应训练次数

Table 2. The training result errors of L_∞ and L₂ for different neural network models, as well as the corresponding numbers of training times

model	error of L_∞	error of L₂	number of training times (Adam+L-BFGS)
Res-FNN	4.110×10^-3	1.140×10^-3	10 000+25 000
FNN	5.234×10^-3	1.204×10^-3	10 000+25 000
PINN	5.434×10^-3	1.221×10^-3	10 000+25 000

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表 3 Res-FNN求解Poisson方程和Helmholtz方程在干净数据和1%噪声数据下反问题结果

Table 3. Res-FNN solutions to the inverse problems of the Poisson equation and the Helmholtz equation on clean data and 1% noisy datas

correct PDE	identified PDE (clean data)	identified PDE (1% noise)
Δu=sin(πx)sin(πy)	Δu=1.000 05sin(πx)sin(πy)	Δu=1.000 54sin(πx)sin(πy)
Δu+λu－f(x,y)	Δu+2.003 34u－f(x,y)	Δu+2.009 93u－f(x,y)

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表 4 Res-FNN求解Poisson方程和Helmholtz方程在不同噪声强度下的反问题结果

Table 4. The inverse problem results of solving the Poisson equation and the Helmholtz equation with the Res-FNN under different noise intensities

noise/%	relative error of the Poisson equation/%	relative error of the Helmholtz equation/%
0	0.005	0.167
1	0.054	0.497
5	0.105	2.976
10	0.664	4.216

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