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Deep learning accelerated algebraic multigrid methods for polytopal discretizations of second-order differential problems.

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Title: Deep learning accelerated algebraic multigrid methods for polytopal discretizations of second-order differential problems.
Authors: Antonietti, Paola F.1 (AUTHOR) paola.antonietti@polimi.it, Caldana, Matteo1 (AUTHOR) matteo.caldana@polimi.it, Gentile, Lorenzo Bruno Donato1 (AUTHOR) lorenzo3.gentile@mail.polimi.it, Verani, Marco1 (AUTHOR) marco.verani@polimi.it
Source: Mathematical Models & Methods in Applied Sciences. Jul2026, Vol. 36 Issue 8, p1649-1677. 29p.
Subjects: Algebraic multigrid methods, Deep learning, Numerical solutions to partial differential equations, Mathematical optimization, Partial differential equations, Galerkin methods
Abstract: Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for Partial Differential Equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e. the relaxation methods used on the fine grid to damp out high-frequency components of the error. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother's performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element (VEM) methods, AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. In this paper, we focus on the linear systems of equations stemming from polytopal discretizations of second-order elliptic problems. We propose a novel deep learning approach that automatically tunes the strong threshold parameter and the smoother choice in AMG solvers, thereby maximizing AMG performance. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to 2 7 % with minimal changes to existing PolyDG and VEM software libraries. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Models & Methods in Applied Sciences is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Deep learning accelerated algebraic multigrid methods for polytopal discretizations of second-order differential problems.
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  Data: <searchLink fieldCode="DE" term="%22Algebraic+multigrid+methods%22">Algebraic multigrid methods</searchLink><br /><searchLink fieldCode="DE" term="%22Deep+learning%22">Deep learning</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+to+partial+differential+equations%22">Numerical solutions to partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink>
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  Data: Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for Partial Differential Equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e. the relaxation methods used on the fine grid to damp out high-frequency components of the error. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother's performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element (VEM) methods, AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. In this paper, we focus on the linear systems of equations stemming from polytopal discretizations of second-order elliptic problems. We propose a novel deep learning approach that automatically tunes the strong threshold parameter and the smoother choice in AMG solvers, thereby maximizing AMG performance. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to 2 7 % with minimal changes to existing PolyDG and VEM software libraries. [ABSTRACT FROM AUTHOR]
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  Label:
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  Data: <i>Copyright of Mathematical Models & Methods in Applied Sciences is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1142/S0218202526420029
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      – Code: eng
        Text: English
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        PageCount: 29
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    Subjects:
      – SubjectFull: Algebraic multigrid methods
        Type: general
      – SubjectFull: Deep learning
        Type: general
      – SubjectFull: Numerical solutions to partial differential equations
        Type: general
      – SubjectFull: Mathematical optimization
        Type: general
      – SubjectFull: Partial differential equations
        Type: general
      – SubjectFull: Galerkin methods
        Type: general
    Titles:
      – TitleFull: Deep learning accelerated algebraic multigrid methods for polytopal discretizations of second-order differential problems.
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            NameFull: Antonietti, Paola F.
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            NameFull: Caldana, Matteo
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            NameFull: Gentile, Lorenzo Bruno Donato
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            NameFull: Verani, Marco
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            – D: 01
              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
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              Value: 36
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