A Solution Method for Partial Differential Equations by Fusing Fourier Operators and U-Net
Year: 2025
Type: article
Abstract: In scientific and engineering computations, the efficient solution of partial differential equations (PDEs) is of great significance. This paper proposes an innovative method based on the combination of the U-Net neural network and the Fourier neural operator, aiming to improve the accuracy and efficiency of solving PDEs. The unique encoder-decoder structure and skip connections of the U-Net neural network can effectively extract and integrate spatial domain features, accurately depicting the spatial structure of PDEs. The Fourier neural operator, by means of the Fourier transform, deeply explores and processes features in the frequency domain, capturing the frequency characteristics of the solutions of PDEs. By organically combining the two, this method not only preserves spatial details but also makes full use of frequency domain information. It significantly reduces the model's dependence on large scale data and enhances its generalization ability. Experimental results show that compared with traditional methods, this method performs outstandingly in various complex partial differential equation(PDE) solving tasks, achieving higher accuracy. It provides a new and highly promising solution for the solution of PDEs and is expected to be widely applied in related fields.
Source: Mathematics
DOI: 10.3390/math13071033
Authors: Tao Zhang, Xinran Zhu
Institution: Yangtze University
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W4408776621