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

Yan Jingyue; Su Huan

doi:10.21656/1000-0887.460125

Volume 47 Issue 7

Jul. 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(7): 912-923

Yan Jingyue, Su Huan. Residual-Based Fourier Neural Networks for Solving Elliptic Differential Equations on Unbounded Domains and Convergence Analysis[J]. Applied Mathematics and Mechanics, 2026, 47(7): 912-923. doi: 10.21656/1000-0887.460125

Citation:

PDF( 3041 KB)

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

doi: 10.21656/1000-0887.460125

Yan Jingyue^,,
Su Huan

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

Received Date: 2025-06-08
Rev Recd Date: 2025-09-02
Publish Date: 2026-07-01

Abstract

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

FullText(HTML)

References(16)

References

[1]	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
[2]	Cai S, Mao Z, Wang Z. Physics-informed neural networks (PINNs) for fluid mechanics: a review[J]. Acta Mechanica Sinica, 2021, 37 (12): 1727-1738. doi: 10.1007/s10409-021-01148-1
[3]	Mao Z, Jagtap A D, Karniadakis G E. Physics-informed neural networks for high-speed flows[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112789. doi: 10.1016/j.cma.2019.112789
[4]	Jin X, Cai S, Li H. NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations[J]. Journal of Computational Physics, 2021, 426: 109951. doi: 10.1016/j.jcp.2020.109951
[5]	Arzani A, Wang J X, D'souza R M. Uncovering near-wall blood flow from sparse data with physics-informed neural networks[J]. Physics of Fluids, 2021, 33 (7): 071905. doi: 10.1063/5.0055600
[6]	Shin Y, Darbon J, Karniadakis G E. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs[J]. Communications in Computational Physics, 2020, 28 (5): 2042-2074. doi: 10.4208/cicp.OA-2020-0193
[7]	Kharazmi E, Zhang Z, Karniadakis G E. Hp-VPINNs: variational physics-informed neural networks with domain decomposition[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 374: 113547. doi: 10.1016/j.cma.2020.113547
[8]	Jagtap A D, Karniadakis G E. 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, 2020, 28 (5).
[9]	闵建, 傅卓佳, 郭远. 课程-迁移学习物理信息神经网络用于曲面长时间对流扩散行为模拟[J]. 应用数学和力学, 2024, 45 (9): 1212-1223. doi: 10.21656/1000-0887.440320 Min Jian, Fu Zhuojia, Guo Yuan. Curriculum-transfer-learning-based physics-informed neural networks for simulating long-term-evolution convection-diffusion behaviors on curved surfaces[J]. Applied Mathematics and Mechanics, 2024, 45 (9): 1212-1223. (in Chinese) doi: 10.21656/1000-0887.440320
[10]	Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators[J]. Neural networks, 1989, 2 (5): 359-366. doi: 10.1016/0893-6080(89)90020-8
[11]	Liu Z, Wang Y, Vaidya S. KAN: Kolmogorov-arnold networks[PP/OL]. V5. (2025-02-09). https://arxiv.org/abs/2404.19756.
[12]	Kolmogorov A N. On the representation of continuous functions of several variables as super-positions of continuous functions of a smaller number of variables[J]. Doklady Akademii Nauk SSSR, 1956, 108 (2): 369-373. (in Russian)
[13]	Liu Shuang. Fourier neural network for machine learning[C]// 2013 International Conference on Machine Learning and Cybernetics. Tianjin, 2014: 285-290.
[14]	Ngom M, Marin O. Fourier neural networks as function approximators and differential equation solvers[J]. Statistical Analysis and Data Mining: the ASA Data Science Journal, 2021, 14 (6): 647-661. doi: 10.1002/sam.11531
[15]	Rahaman N, Baratin A, Arpit D. On the spectral bias of neural networks[C]//Proceedings of the 36 th International Conference on Machine Learning, 2019: 5301-5310.
[16]	Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order[M]. Berlin, Heidelberg: Springer, 1977.