# A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs

arXiv: https://arxiv.org/abs/2304.10832 · HTML: https://mlsanity.com/paper/2304.10832/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 84 references that are in the map.
- First posted on arXiv 21 Apr 2023.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size](https://mlsanity.com/paper/1602.07360/) (2016)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [A machine learning framework for data driven acceleration of computations of differential equations](https://mlsanity.com/paper/1807.09519/) (2018)
- [Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods](https://mlsanity.com/paper/2102.05738/) (2021)
- [Agglomeration of Polygonal Grids using Graph Neural Networks with applications to Multigrid solvers](https://mlsanity.com/paper/2210.17457/) (2022)
- [Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods](https://mlsanity.com/paper/2202.12654/) (2022)
- [A data-driven approach for the closure of RANS models by the divergence of the Reynolds Stress Tensor](https://mlsanity.com/paper/2203.16944/) (2022)

## Built on by (most cited first, 6 of 6)

- [Discovering Artificial Viscosity Models for Discontinuous Galerkin Approximation of Conservation Laws using Physics-Informed Machine Learning](https://mlsanity.com/paper/2402.16517/) (2024)
- [A Greedy PDE Router for Blending Neural Operators and Classical Methods](https://mlsanity.com/paper/2509.24814/) (2025)
- [NeuraLSP: A Neural Spectral Preconditioner for Accelerating PDE Solvers](https://mlsanity.com/paper/2601.20174/) (2026)
- [Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms](https://mlsanity.com/paper/2603.17641/) (2026)
- [RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections](https://mlsanity.com/paper/2605.26854/) (2026)
- [Acceleration of an algebraic multigrid pressure solver using graph neural networks](https://mlsanity.com/paper/2606.19251/) (2026)

## Abstract (arXiv)

Algebraic multigrid (AMG) methods are among the most efficient solvers for linear systems of equations and they are widely used for the solution of problems stemming from the discretization of Partial Differential Equations (PDEs). The most severe limitation of AMG methods is the dependence on parameters that require to be fine-tuned. In particular, the strong threshold parameter is the most relevant since it stands at the basis of the construction of successively coarser grids needed by the AMG methods. We introduce a novel Deep Learning algorithm that minimizes the computational cost of the AMG method when used as a finite element solver. We show that our algorithm requires minimal changes to any existing code. The proposed Artificial Neural Network (ANN) tunes the value of the strong threshold parameter by interpreting the sparse matrix of the linear system as a black-and-white image and exploiting a pooling operator to transform it into a small multi-channel image. We experimentally prove that the pooling successfully reduces the computational cost of processing a large sparse matrix and preserves the features needed for the regression task at hand. We train the proposed algorithm on a large dataset containing problems with a highly heterogeneous diffusion coefficient defined in different three-dimensional geometries and discretized with unstructured grids and linear elasticity problems with a highly heterogeneous Young's modulus. When tested on problems with coefficients or geometries not present in the training dataset, our approach reduces the computational time by up to 30%.