Applying Two-Grid Preconditioner for Subsurface Flow Simulation using Attention-enhanced Hybrid Network to Accelerate Multiscale Discretization in High-contrast Media

Peiqi Li; Jie Chen; Shubin Fu

cs.CEApr 15, 2026

Applying Two-Grid Preconditioner for Subsurface Flow Simulation using Attention-enhanced Hybrid Network to Accelerate Multiscale Discretization in High-contrast Media

Authors: Peiqi Li, Jie Chen, Shubin Fu

Organizations: Department of Applied Mathemaitcs, Xi’an Jiaotong-Liverpool University, 215123, Suzhou, China · Department of Mathematical Sciences, University of Liverpool, L69 3BX, Liverpool, China · Eastern Institute for Advanced Study, Eastern Institute of Technology, 315200, Ningbo, China

Abstract

In this paper, we study the efficient numerical solution of Darcy equations in strongly heterogeneous media with high-contrast permeability and propose a hybrid framework that combines learning with multiscale numerical methods. The learning component is used for the prediction of multiscale basis functions in the mixed generalized multiscale finite element method (mixed GMsFEM), with the goal of reducing the repeated local computations required in the offline stage. Once these basis functions are predicted, the global system is assembled and the pressure field is computed by a two-grid preconditioned solver. The resulting method accelerates the costly local basis-construction stage while retaining the multiscale discretization and preconditioned iterative structure of the underlying solver. Numerical experiments on two-dimensional heterogeneous Darcy problems show that the proposed framework yields more accurate final pressure reconstruction than several representative learning-based methods and remains stable under strong heterogeneity and high-contrast coefficients. In comparison with the traditional mixed GMsFEM, its main advantage lies in the efficiency of the basis-generation stage, while the quality of the global solve is still ensured by the two-grid preconditioner. These results indicate that accelerating multiscale basis construction through learning, while preserving a mature numerical solver for the global problem, provides a viable approach for high-resolution Darcy-type simulations.

Figures & tables

Figure 1: Schematic illustration of the overall technical pipeline of the proposed framework.

Figure 2: Example of our dataset. Right: high-contrast fractured permeability field. The yellow lines refer to the fractures. Right: total multiscale basis functions. Totally we have 12 basis functions, and the first (basis 1) is piecewise constant (1 or -1).

Hyperparameter	Value
Truncated modes of Fourier operator	16 (x-axis); 16 (y-axis)
Fine grid resolution	$128\times 128$
Coarse grid resolution	$4\times 4$
Gradient loss weight	0.1
Optimizer	Adam
Learning rate	0.001

Table 1: Experimental Hyperparameter Settings

Figure 3: Schematic diagram of multiscale basis functions (2-12). For each two lines, the top denotes the ground truth, also the reference solutions, while the bottom refers to the predicted solutions.

Figure 4: Learning curves of multiscale basis functions.

mm	MSE	R 2	mm	MSE	R 2
			7	0.0030	0.9594
2	0.0056	0.9790	8	0.0023	0.9762
3	0.0081	0.9664	9	0.0013	0.9529
4	0.0079	0.5540	10	0.0016	0.9232
5	0.0038	0.9657	11	0.0014	0.9499
6	0.0028	0.9641	12	0.0008	0.9835

Table 2: Performance metrics for different multiscale basis functions

Figure 5: Schematic diagrams of sparse matrix in the linear system $Ap=F$ . Top: case using 6 basis functions; bottom:case using 12 basis functions. $nz$ refers to the number of non-sparse points.

Figure 6: Comparison of our model and baseline models. Left: reference solution; Right Top: pressure solutions; bottom: abstract error. In this case, ’ours’ also refers to that we use $mm=6$ to solve the linear system.

Figure 7: Relative L2 error of ours and baseline models

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Schematic diagram of the solution based on GMsFEM. Upper part: Pressure solutions using different numbers of multi-scale basis functions (which we refer to as multiscale solutions) and the corresponding absolute errors. Lower part, left side: Histogram of relative L2 error; Bottom right part: Specific values of relative L2 error. MS Sol: Multiscale Solution; Abs Err: Absolute Error; $mm$ : Number of used multiscale basis functions.

Figure 9: Schematic diagram of the solutions of iterative two-grid algorithm using $\hat{R}_{\mathrm{off}}$ . Upper part: Pressure solutions using different numbers of predicted multi-scale basis functions in two-grid preconditioner and the corresponding absolute errors. Lower part, left side: Histogram of relative L2 error; Bottom right part: Specific values of relative L2 error. Abs Err: Absolute Error; $mm$ : Number of used multiscale basis functions. In this figure, the part of ’ $mm=1$ ’ means using only one basis function that is not learning using networks, so it keep the same as the one in A . Meanwhile, the part ’ $mm=6$ ’ is shown in Section 6 .

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McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

Jiwei Jia, Xinliang Liu, Juntao Wang, Jinchao Xu

School of Mathematics, Jilin University, Changchun 130012, China · Shenzhen Loop Area Institute, Shenzhen 518038, China · Ocean University of China, Qingdao 266100, China+1

Neural PDE Solvers PDE Solving PDE Operator Learning

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Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Abstract

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