计算机科学

计算机科学 ›› 2025, Vol. 52 ›› Issue (9): 144-151.doi: 10.11896/jsjkx.240700122

• 高性能计算 • 上一篇    下一篇

基于局部增强傅里叶神经算子的偏微分方程求解方法

罗驰1, 陆凌云2, 刘飞1   

  1. 1 华南理工大学软件学院 广州 510006
    2 南京电子工程研究所 南京 210007
  • 收稿日期:2024-07-19 修回日期:2024-10-22 出版日期:2025-09-15 发布日期:2025-09-11
  • 通讯作者: 刘飞(feiliu@scut.edu.cn)
  • 作者简介:(202121046555@scut.edu.cn)
  • 基金资助:
    国家自然科学基金(62273153);广东省基础与应用基础研究基金(2024A1515010900)

Partial Differential Equation Solving Method Based on Locally Enhanced Fourier NeuralOperators

LUO Chi1, LU Lingyun2, LIU Fei1   

  1. 1 School of Software Engineering,South China University of Technology,Guangzhou 510006,China
    2 Nanjing Research Institute of Electronics Engineering,Nanjing 210007,China
  • Received:2024-07-19 Revised:2024-10-22 Online:2025-09-15 Published:2025-09-11
  • About author:LUO Chi,born in 1997,postgraduate.His main research interest is deep learning.
    LIU Fei,born in 1976,Ph.D,professor,Ph.D supervisor,is a member of CCF(No.B9231M).His main research interests include modeling and simulation,and artificial intelligence.
  • Supported by:
    National Natural Science Foundation of China(62273153) and Guangdong Basic and Applied Basic Research Foundation(2024A1515010900).

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摘要： 偏微分方程(PDE)是描述现实系统的重要数学工具,对其进行求解可以预测和分析系统的行为。PDE的解析解通常难以获取,一般通过数值法进行近似解算,但数值法求解参数化PDE时效率较低。近年来,利用深度学习求解PDE的方法在应对上述问题时展现出了优势,特别是傅里叶神经算子FNO(Fourier Neural Operator)已在此类问题中展现出显著成效。然而,FNO仅通过频域上的卷积来提取全局信息,难以捕获PDE的多尺度信息。针对此挑战,提出一种基于局部增强的FNO模型,在傅里叶层引入并行多尺寸卷积模块,通过不同尺寸的卷积提高模型捕获局部多尺度信息的能力。同时,在线性层后引入一种多分支特征融合模块,通过将数据提升到不同通道进行学习来提高模型整合多通道信息的能力。实验结果表明,该模型在Burgers方程的求解中误差降低了30.9%,在Darcy Flow方程的求解中误差降低了18.5%,在Navier-Stokes方程的求解中误差降低了5.5%。

关键词: 深度学习, 偏微分方程, 傅里叶神经算子, 多尺寸卷积, 多分支特征融合, 多尺度PDE

Abstract: Partial differential equations(PDEs) are crucial mathematical tools for describing real-world systems,and solving them is key for predicting and analyzing system behavior.Analytical solutions for PDEs are often difficult to obtain,and numerical methods are typically used for approximate solutions.However,numerical solutions for parameterized PDEs can be inefficient.In recent years,the use of deep learning for solving PDEs has shown its advantages in addressing these issues,and particularly the Fourier Neural Operator(FNO) has proven effective.However,FNO only captures global information through convolution in the frequency domain and struggles with multi-scale information of PDEs.To address this challenge,a locally-enhanced FNO model is proposed,incorporating a parallel multi-size convolution module in the Fourier layer to enhance the model’s capability to capture local multi-scale information.Behind the linear layer,a multi-branch feature fusion module is introduced,enhancing the model’s ability to integrate multi-channel information by elevating the data across different channels.Experimental results demonstrate that the model reduces errors by 30.9% in solving Burgers’ equation,18.5% in Darcy Flow equations,and 5.5% in Navier-Stokes equations.

Key words: Deep learning, Partial differential equations, Fourier neural operator, Multi-size convolution, Multi-branch feature fusion, Multiscale PDE

中图分类号: 

  • TP391
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