边界感知傅里叶神经算子
Boundary-Aware Fourier Neural Operator
谭兴美 1佘骏1
作者信息
- 1. 重庆大学数学与统计学院，重庆 401331
- 折叠
摘要
偏微分方程是描述连续介质与场量演化规律的核心数学工具，广泛应用于各个领域，其高效求解对于科学计算与工程应用都具有重要意义。近年来，神经算子方法通过学习函数到函数的映射关系，为参数化偏微分方程的快速求解提供了新的思路，其中，傅里叶神经算子（FNO）通过在频域对卷积核进行参数化表示，并利用快速傅里叶变换实现全局信息传播，在参数化问题快速求解中展现出了独特优势。但FNO 的离散实现隐含周期兼容假设，当问题具有非周期边界条件时，模型在边界邻域容易产生误差集聚。在此基础上，本文提出一种边界感知傅里叶神经算子（BA-FNO）框架，通过在标准 FNO 主通路外引入独立的边界分支网络，对边界区域特征进行显式建模，并通过可插拔的特征融合机制实现全局谱信息与局部边界信息的协同表达。在二维达西方程上的数值实验表明，BA-FNO能有效降低全域与边界带误差，同时还保持了FNO的跨分辨率泛化特性，从而为非周期边界条件下偏微分方程的求解问题提供了一个更为稳健的求解方案。
Abstract
Partial differential equations (PDEs) are fundamental mathematical tools for describing the evolution of continuous media and field quantities, and they are widely used across many scientific and engineering disciplines. Efficient numerical solution of PDEs is therefore of great importance for both scientific computing and practical applications. In recent years, neural operator methods have emerged as a promising approach for solving parametric PDEs by learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) parameterizes convolution kernels in the frequency domain and leverages the Fast Fourier Transform (FFT) to achieve efficient global information propagation, demonstrating significant advantages in the rapid prediction of parametric PDE solutions. However, the discrete implementation of FNO implicitly assumes periodic compatibility. When the problem involves non-periodic boundary conditions, prediction errors tend to accumulate near the boundaries.To address this issue, a Boundary-Aware Fourier Neural Operator (BA-FNO) framework is proposed. The proposed model introduces an independent boundary branch in addition to the standard FNO backbone to explicitly capture boundary features. A plug-in feature fusion mechanism is further designed to integrate global spectral information with local boundary representations. Numerical experiments on the two-dimensional Darcy equation demonstrate that BA-FNO effectively reduces both global and boundary-band errors while preserving the cross-resolution generalization capability of the original FNO. These results indicate that the proposed framework provides a more robust solution strategy for PDE problems with non-periodic boundary conditions.关键词
计算数学/参数化偏微分方程/神经算子/傅里叶神经算子Key words
Computational mathematics/Parametric partial differential equations/Neural operator/Fourier neural operator引用本文复制引用
谭兴美,佘骏.边界感知傅里叶神经算子[EB/OL].(2026-03-26)[2026-03-28].http://www.paper.edu.cn/releasepaper/content/202603-257.学科分类
数学/计算技术、计算机技术
评论