A Solution Method for Partial Differential Equations by Fusing Fourier Operators and U-Net

Tao Zhang () and Xinran Zhu
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Tao Zhang: School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
Xinran Zhu: School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

Mathematics, 2025, vol. 13, issue 7, 1-17

Abstract: In scientific and engineering calculation, the effective solution of partial differential equations (PDEs) has great significance. This paper presents an innovative method based on the combination of a U-Net neural network with Fourier neural operators, aiming to improve the accuracy and efficiency of solving partial differential equations. U-Net neural networks with a unique encoding–decoder structure and hopping connections can efficiently extract and integrate spatial-domain features and accurately describe the spatial structure of PDEs. The Fourier neural operator combines the advantages of neural networks and Fourier transform to learn the mapping relationship of high-dimensional data by training the linear transformation parameters in the detailed frequency domain of the data. Fourier transform is used to explore and process the frequency-domain features and capture the frequency characteristics of the solution of partial differential equations. By combining the two, this method not only retains spatial details but also makes full use of frequency-domain information, which significantly reduces the dependence of the model on large-scale data and improves the generalization ability of the model. The experimental results show that compared with the traditional method, this method performs well in various complex partial differential equation (PDE) solving tasks and achieves high accuracy.

Keywords: neural operator; partial differential equation; U-Net; Fourier transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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