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

Yan Jingyue; Su Huan

doi:10.21656/1000-0887.460125

Volume 47 Issue 7

Jul. 2026

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

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

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

Available Online: 2026-07-23

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

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References(16)

References

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