广西师范大学学报（自然科学版） ›› 2026, Vol. 44 ›› Issue (6): 134-145.doi: 10.16088/j.issn.1001-6600.2026010602

• 数学 • 上一篇    下一篇

基于傅里叶特征映射的物理信息残差网络求解高阶非线性偏微分方程

何荣泽1, 王庆祝1, 夏道勋2, 周晓军1*   

  1. 1.贵州师范大学 数学科学学院, 贵州 贵阳 550025;
    2.贵州师范大学 人工智能贵州省科技创新领军人才工作站, 贵州 贵阳 550025
  • 收稿日期:2026-01-06 修回日期:2026-03-18 出版日期:2026-11-05 发布日期:2026-11-10
  • 通讯作者: 周晓军(1982—),男,贵州遵义人,贵州师范大学教授,博士。E-mail: xjzhou@gznu.edu.cn
  • 基金资助:
    国家自然科学基金(62566013)

Physics-informed residual networks based on Fourier feature mapping for solving high-order nonlinear partial differential equations

He Rongze1, Wang Qingzhu1, Xia Daoxun2, Zhou Xiaojun1*   

  1. 1. School of Mathematical Sciences, Guizhou Normal University, Guiyang Guizhou 550025, China;
    2. Artificial Intelligence Leading Talent Workstation for Science and Technology Innovation in Guizhou Province, Guizhou Normal University, Guiyang Guizhou 550025, China
  • Received:2026-01-06 Revised:2026-03-18 Online:2026-11-05 Published:2026-11-10

摘要： 针对物理信息神经网络(PINN)在求解高阶非线性偏微分方程时面临频谱偏差和梯度病态的挑战,本文提出一种结合傅里叶特征映射与残差连接的改进模型FF-ResPINN。ResPINN利用傅里叶特征映射将输入坐标投影至高维特征空间,以调节神经正切核频谱带宽,增强网络对高频振荡解的捕捉能力。然后,在骨干网络中引入残差连接,建立梯度的直接传播通路,有效缓解计算高阶导数引发的梯度消失问题。数值实验选取四阶Boussinesq方程和五阶Sawada-Kotera方程进行验证,实验结果表明,相较于标准PINN模型,FF-ResPINN能明显提高求解精度和收敛稳定性,相对L2误差可达O(10-5),验证了该方法在求解高阶非线性偏微分方程的有效性。

关键词: 物理信息神经网络, 傅里叶特征映射, 残差网络, 高阶非线性偏微分方程, 神经正切核

Abstract: An improved model named FF-ResPINN, combining Fourier feature mapping with residual connections, is proposed to address the challenges of spectral bias and gradient pathology encountered by Physics-Informed Neural Networks (PINN) when solving high-order nonlinear partial differential equations. Firstly, input coordinates are projected into a high-dimensional feature space by Fourier feature mapping. The spectral bandwidth of the Neural Tangent Kernel is regulated, and the capability of the network to capture high-frequency oscillatory solutionsis enhanced. Secondly, residual connections are introduced into the network to establish direct propagation paths for gradients. Consequently, the vanishing gradient problem triggered by high-order derivative calculations is effectively mitigated. The fourth-order Boussinesq equation and the fifth-order Sawada-Kotera equationare selected for numerical verification. Experimental results show that, compared with the standard PINN model, the solution accuracy and convergence stability are obviously improved by FF-ResPINN. The relative L2 error reaches O(10-5). The effectiveness of the proposed method in solving high-order nonlinear partial differential equations is verified.

Key words: physics-informed neural networks, Fourier feature mapping, residual network, high-order nonlinear partial differential equations, neural tangent kernel

中图分类号:  O214.82

[1] Yan J L, Zhang Z Y, Zhao T J, et al. High-order energy-preserving schemes for the improved Boussinesq equation[J]. Numerical Methods for Partial Differential Equations, 2018, 34(4): 1145-1165. DOI: 10. 1002/num.22249.
[2] Su C M, Yao W Q. A Deuflhard-type exponential integrator Fourier pseudo-spectral method for the“good” Boussinesq equation[J]. Journal of Scientific Computing, 2020, 83: 4. DOI: 10.1007/s10915-020-01192-2.
[3] Su C M, Muslu G M. An exponential integrator sine pseudospectral method for the generalized improved Boussinesq equation[J]. BIT Numerical Mathematics, 2021, 61(4): 1397-1419. DOI: 10.1007/s10543-021-00865-0.
[4] Gupta A K, Ray S S. Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method[J]. Applied Mathematical Modelling, 2015, 39(17): 5121-5130. DOI: 10.1016/j.apm.2015.04.003.
[5]Alharthi N S. Dynamical and computational analysis of fractional Korteweg-de Vries-Burgers and Sawada-Kotera equations in terms of Caputo fractional derivative[J]. Fractal and Fractional, 2025, 9(7): 411. DOI: 10.3390/fractalfract9070411.
[6]Ali K K, Mehanna M S, Abdel-Aty A H, et al. New soliton solutions of dual mode Sawada Kotera equation using a new form of modified Kudryashov method and the finite difference method[J]. Journal of Ocean Engineering and Science, 2024, 9(3): 207-215. DOI: 10.1016/j.joes.2022.04.033.
[7] Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378: 686-707. DOI: 10.1016/j.jcp.2018.10.045.
[8] Jagtap A D, Em Karniadakis G. Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations[J]. Communications in Computational Physics, 2025, 28(5): 2002-2041. DOI: 10.4208/cicp.oa-2020-0164.
[9] Jagtap A D, Kharazmi E, Karniadakis G E. Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 365: 113028. DOI: 10.1016/j.cma.2020.113028.
[10]Yu J, Lu L, Meng X H, et al. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 393: 114823. DOI: 10.1016/j.cma.2022.114823.
[11]高飞, 郭晓斌, 袁冬芳, 等. 改进PINNs方法求解边界层对流占优扩散方程[J]. 广西师范大学学报(自然科学版), 2023, 41(6): 33-50. DOI: 10.16088/j.issn.1001-6600.2023032203.
[12]王宇铎, 陈嘉鑫, 李彪. 基于扩展混合训练物理信息神经网络的非线性薛定谔方程求解和参数发现[J]. 物理学报, 2025, 74(16): 1-8. DOI:10.7498/aps.74.20250422.
[13]韩烁, 江林峰, 杨建斌. 基于注意力机制PINNs方法求解圣维南方程[J]. 广西师范大学学报(自然科学版), 2025, 43(4): 58-68. DOI: 10.16088/j.issn.1001-6600.2024061802.
[14]张照阳, 王青旺. 信息表征-损失优化改进的物理信息神经网络求解偏微分方程[J]. 物理学报, 2025, 74(18): 384-398. DOI: 10.7498/aps.74.20250707.
[15]吴泽康, 王晓丽, 韩文静, 等. 物理信息神经网络求解五阶emKdV方程的正反问题[J]. 数学物理学报, 2024, 44(2): 484-499.
[16]谢翔, 江林峰, 杨凤莲. 基于自适应权重两阶段PINNs方法求解偏微分方程[J]. 广西师范大学学报(自然科学版), 2026, 44(2): 175-189. DOI: 10.16088/j.issn.1001-6600.2025050801.
[17]张亚斌. 基于PINN框架的若干非线性波动方程数据驱动解的研究[D]. 北京: 华北电力大学, 2023.
[18]Shafee A, Alkhezi Y, Jacob K. Data-driven modeling of Boussinesq-Burgers equations: comparing physics-informed neural networks (PINNs) with exact solutions[J]. European Journal of Pure and Applied Mathematics, 2025, 18(4): 6803. DOI: 10.29020/nybg.ejpam.v18i4.6803.
[19]Lin S N, Chen Y. A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions[J]. Journal of Computational Physics, 2022, 457: 111053. DOI: 10.1016/j.jcp.2022.111053.
[20]张挺, 鄢锐, 章思茜, 等. 基于傅里叶特征化物理信息神经网络的输流管道振动特性模拟[J]. 土木工程学报, 2025, 58(9): 86-95. DOI: 10.15951/j.tmgcxb.24010049.
[21]牛艺晓, 李佳芳, 杨春, 等. 基于ResNet-PINN求解中子方程算法研究[J]. 核动力工程, 2025, 46(2): 76-80. DOI: 10.13832/j.jnpe.2024.080035.
[22]Wang S F, Wang H W, Perdikaris P. On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 384: 113938. DOI: 10.1016/j.cma.2021.113938.
[23]Wang S F, Li B W, Chen Y H, et al. PirateNets: physics-informed deep learning with residual adaptive networks[J]. Journal of Machine Learning Research, 2024, 25(1): 19707-19757. DOI: 10. 5555/3722577.3722979.
[24]Ali Akbar M, Akinyemi L, Yao S W, et al. Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method[J]. Results in Physics, 2021, 25: 104228. DOI: 10.1016/j.rinp.2021.104228.
[25]Liu C F, Dai Z D. Exact soliton solutions for the fifth-order Sawada-Kotera equation[J]. Applied Mathematics and Computation, 2008, 206(1): 272-275. DOI: 10.1016/j.amc.2008.08.028.
[26]Tancik M, Srinivasan P P, Mildenhall B, et al. Fourier features let networks learn high frequency functions in low dimensional domains[C]//Proceedings of the 34th International Conference on Neural Information Processing Systems.New York, NY: ACM, 2020: 7537-7547.
[27]He K M, Zhang X Y, Ren S Q, et al. Deep residual learning for image recognition[C]//2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Los Alamitos, CA: IEEE Computer Society, 2016: 770-778. DOI: 10.1109/CVPR.2016.90.
[28]Cipolla R, Gal Y, Kendall A. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics[C]//2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. Los Alamitos, CA: IEEE Computer Society, 2018: 7482-7491. DOI: 10.1109/CVPR.2018.00781.
[1] 谢翔, 江林峰, 杨凤莲. 基于自适应权重两阶段PINNs方法求解偏微分方程[J]. 广西师范大学学报（自然科学版）, 2026, 44(2): 175-189.
[2] 肖宇庭, 吕晓琪, 谷宇, 刘传强. 基于拆分残差网络的糖尿病视网膜病变分类[J]. 广西师范大学学报（自然科学版）, 2024, 42(1): 91-101.
[3] 高飞, 郭晓斌, 袁冬芳, 曹富军. 改进PINNs方法求解边界层对流占优扩散方程[J]. 广西师范大学学报（自然科学版）, 2023, 41(6): 33-50.
[4] 蒋懿波, 刘会家, 吴田. 基于改进残差网络的输电线路雷击过电压识别研究[J]. 广西师范大学学报（自然科学版）, 2023, 41(4): 74-83.
[5] 万黎明, 张小乾, 刘知贵, 宋林, 周莹, 李理. 基于高效通道注意力的UNet肺结节CT图像分割[J]. 广西师范大学学报（自然科学版）, 2022, 40(3): 66-75.
[6] 邓文轩, 杨航, 靳婷. 基于注意力机制的图像分类降维方法[J]. 广西师范大学学报（自然科学版）, 2021, 39(2): 32-40.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 广西师范大学学报（自然科学版）编辑部
地址：广西桂林市三里店育才路15号　邮编：541004
电话：0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发