A level-wise training scheme for learning neural multigrid smoothers with application to integral equations

Lingfeng Li; Yin King Chu; Raymond Chan; Justin Wan

cs.LGMar 1, 2026

A level-wise training scheme for learning neural multigrid smoothers with application to integral equations

Authors: Lingfeng Li, Yin King Chu, Raymond Chan, Justin Wan

Organizations: Hetao Institute of Mathematics and Interdisciplinary Sciences, Shenzhen, Guangdong, China. · Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. · Department of Data Science, Department of Operations and Risk Management,Lingnan University, Tuen Mun, Hong Kong, China. · David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada.

Abstract

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.

Figures & tables

Figure 1: Plot of $1-\mu_{\phi}$ for $\phi\in[-\pi,\pi]$ when $\omega=1/2$ .

Figure 2: Eigenvectors corresponding to different eigenvalues of the linear system discretized from the Poisson equation (first row) and the Integral equation (second row).

Panel of Figure 3 — Figure 3: Error vectors when solving different linear systems using the Jacobi method.

Figure 4: Graph illustration of a V-cycle multigrid scheme with $L=3$

Panel of Figure 5 — Figure 5: Explanation of the frequency domain decomposition (L=3).

Panel of Figure 6 — Figure 6: Plot of $\mathbf{m}_{l}$ for $l=1,2,3$ .

Figure 7: The graph illustration of the neural multigrid cycle for one-dimensional problems when $L=3$ .

Panel of Figure 8 — Figure 8: Plot of $\hat{M}_{l}$ for $l=1,2,3$ .

		$\alpha=10^{-4}$	$\alpha=10^{-5}$	$\alpha=10^{-6}$
proposed method	$n=256$	13.7(0.17s) $\pm 1.8$	13.7(0.18s) $\pm 2.5$	14.6(0.19s) $\pm 0.8$
	$n=512$	13.3(0.21s) $\pm 0.6$	13.1(0.20s) $\pm 0.9$	14.2(0.24s) $\pm 0.6$
	$n=1024$	14.0(0.28s) $\pm 1.2$	13.4(0.27s) $\pm 1.7$	14.4(0.30s) $\pm 1.0$
multigrid method	$n=256$	9556(7s) $\pm 139$	27423(16s) $\pm 542$	35375(21s) $\pm 699$
	$n=512$	9570(16s) $\pm 87$	27484(45s) $\pm 330$	35455(58s) $\pm 425$
	$n=1024$	9589(56s) $\pm 73$	27600(162s) $\pm 286$	35604(210s) $\pm 369$

Table 1: The average number of iterations and evaluation time required for solving the one-dimensional integral equation.

Figure 9: Magnitude spectrums of unsolved errors at the fine grid after applying neural smoothers at each level.

		$\alpha=10^{-4}$	$\alpha=10^{-5}$	$\alpha=10^{-6}$
proposed method	$n=256$	13.2(0.15s) $\pm 1.2$	13.8(0.16s) $\pm 0.9$	14.4(0.19s) $\pm 1.3$
	$n=512$	13.4(0.21s) $\pm 1.3$	14.0(0.23s) $\pm 1.1$	14.5(0.25s) $\pm 0.8$
	$n=1024$	13.4(0.27s) $\pm 1.1$	13.4(0.26s) $\pm 1.0$	14.7(0.32s) $\pm 1.3$
multigrid method	$n=256$	5130(4s) $\pm$ 173	20325(13s) $\pm$ 1034	31707(21s) $\pm$ 1613
	$n=512$	5184(9s) $\pm$ 121	20603(36s) $\pm$ 728	32140(56s) $\pm$ 1136
	$n=1024$	5207(30s) $\pm$ 70	20742(118s) $\pm$ 420	32355(184s) $\pm$ 655

Table 2: The average number of iterations and evaluation time required for solving the one-dimensional integral equation.

	$n=256$	$n=512$	$n=1024$
proposed method	14.0(0.17s) $\pm 0.6$	13.0(0.21s) $\pm 1.8$	13.7(0.28s) $\pm 1.8$
multigrid method	8(0.00s) $\pm 0$	8(0.01s) $\pm 0$	8(0.04s) $\pm 0$

Table 3: The average number of iterations (cycles) and evaluation time required for solving the one-dimensional PDE

		$\alpha=10^{-3}$	$\alpha=10^{-4}$	$\alpha=10^{-5}$
proposed method	$n=256$	$11.3$ (4s) $\pm 2.2$	$13.8$ (5s) $\pm 3.0$	$14.2$ (5s) $\pm 3.7$
	$n=512$	$12.7$ (12s) $\pm 4.5$	$14.3$ (13s) $\pm 2.4$	$15$ (14s) $\pm 4.5$
	$n=1024$	13.6(55s) $\pm 4.3$	14.2(59s) $\pm 3.18$	14.6(61s) $\pm 2.2$
multigrid method	$n=256$	$1140$ (17s) $\pm 5$	$9510$ (2min) $\pm 59$	$71263$ (18min) $\pm 703$
	$n=512$	$1145$ (57s) $\pm 4$	$9555$ (8min) $\pm 34$	$77288$ (1hr) $\pm 397$
	$n=1024$	$1146$ (4min) $\pm 2$	$9577$ (33min) $\pm 18$	$77310$ (4hr) $\pm 207$

Table 4: The average number of iterations and evaluation time required for solving the two-dimensional integral equation.

Panel of Figure 10 — Figure 10: Difference between $\hat{E}_{l}$ and $\hat{E}_{l-1}$ for $l=1,2,3,4$ .

		$\alpha=10^{-4}$	$\alpha=10^{-5}$	$\alpha=10^{-6}$
trained by the level-wise losses ( 6 )	$L^{\prime}=2$	$13.5\pm 1.4$	$13.4\pm 2.0$	$18.5\pm 6.3$
	$L^{\prime}=3$	$12.5\pm 0.8$	$11.7\pm 1.3$	$12.6\pm 1.7$
	$L^{\prime}=4$	14.0 $\pm 1.2$	13.4 $\pm 1.7$	14.4 $\pm 1.0$
trained by the combined loss ( 4 )	$L^{\prime}=2$	$30.3\pm 12.2$	$32.6\pm 9.7$	$32.4\pm 5.8$
	$L^{\prime}=3$	$15.4\pm 6.7$	$24.7\pm 7.6$	$24.2\pm 5.1$
	$L^{\prime}=4$	$16.8\pm 5.5$	$19.4\pm 6.5$	$16.6\pm 6.3$

Table 5: The average number of cycles when training with different loss functions

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

problem dimension	$h$	$c$	$L$	$k$	kernel size
one-dimensional	$1/64$	64	4	16	5
	$1/128$	64	4	32	5
	$1/256$	64	5	64	7
	$1/512$	64	6	128	7
	$1/1024$	64	6	128	7
two-dimensional	$1/64$	64	4	16	3

Appendix

Table 6: Hyperparameters of neural smoothers

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