Artificial intelligence-enhanced algebraic multigrid for 3D finite element simulations

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Artificial intelligence-enhanced algebraic multigrid for 3D finite element simulations

Damian Goik, Krzysztof Banaś*

AGH University of Krakow, al. Mickiewicza 30, 30-059 Krakow, Poland.

*corresponding author

DOI:

https://doi.org/10.7494/cmms.2026.1.1039

Abstract:

The paper presents preliminary investigations into a strategy for solving linear systems resulting from 3D finite element simulations, based on the algebraic multigrid (AMG) method, enhanced using artificial intelligence techniques. In particular, we adapt to 3D problems the algorithm presented in Luz et al. (2020) for using a graph neural network to create the prolongation and restriction operators in a way that will improve convergence. The process of training the network proceeds on the basis of a set of system matrices obtained for tasks much smaller in scale than the target problem to be solved. Learning is aimed at decreasing the spectral radius of the error propagation matrix after applying modified prolongation and restriction. We describe some implementation details of the solver developed based on the presented strategy and show several numerical examples of its application for medium-sized problems.

Cite as:

Goik, D., & Banaś, K. (2026). Artificial intelligence-enhanced algebraic multigrid for 3D finite element simulations . Computer Methods in Materials Science, 26(1), – . https://doi.org/10.7494/cmms.2026.1.1039

Article (PDF):

Accepted Manuscript – final pdf version coming soon

Keywords:

Graph neural networks, Algebraic multigrid, Finite element method

Publication dates:

Received: 19.03.2026, accepted: 14.04.2026, published: XX.04.2026

Publication type:

Original scientific paper

References:

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