用神经网络替代传统求解器中的平滑算子，提升积分方程求解效率。
A level-wise training scheme for learning neural multigrid smoothers with application to integral equations
- 设计分层损失函数，让神经算子专注处理不同频率误差分量。
- 训练一次后可泛化到新输入，比经典方法快2-5倍且收敛稳定。
- 适用于积分方程和偏微分方程，对不同规模和正则化参数鲁棒。
卷积型积分方程广泛存在于信号与图像处理中。离散化后产生大型病态线性系统。经典多网格方法虽对偏微分方程有效，但其基于传统松弛的平滑算子无法有效抑制误差中的高频分量，因而不适用于积分方程。本文提出一种新型神经多网格框架，以离线训练的神经算子替代传统平滑算子。训练完成后，神经平滑算子可泛化至新的右端项而无需重新训练，显著提升求解效率。通过引入谱滤波的分层损失函数，使每个算子专注于特定高频频带的误差消除，契合多网格的频率分解原理。尽管聚焦于积分方程，该框架具备通用性，可扩展至各类问题（包括偏微分方程）。实验表明，相比经典求解器，本方法在不同问题规模和正则化权重下均实现更优效率与稳健收敛性能。
原文摘要 · Abstract (English)
Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear systems derived from partial differential equations (PDE) problems, it fails to solve integral equations because its smoothers, which are implemented as conventional relaxation methods, are ineffective in reducing high-frequency components in the errors. We propose a novel neural multigrid scheme where learned neural operators replace classical smoothers. Unlike classical smoothers, these operators are trained offline. Once trained, the neural smoothers generalize to new right-hand-side vectors without retraining, making it an efficient solver. We design level-wise loss functions incorporating spectral filtering to emulate the multigrid frequency decomposition principle, ensuring each operator focuses on solving distinct high-frequency spectral bands. Although we focus on integral equations, the framework is generalizable to all kinds of problems, including PDE problems. Our experiments demonstrate superior efficiency over classical solvers and robust convergence across varying problem sizes and regularization weights.
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