# Acceleration of an algebraic multigrid pressure solver using graph neural networks

arXiv: https://arxiv.org/abs/2606.19251 · HTML: https://mlsanity.com/paper/2606.19251/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions](https://mlsanity.com/paper/2212.07564/) (2022)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh](https://mlsanity.com/paper/1910.08613/) (2019)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [Surrogate Modeling of Fluid Dynamics with a Multigrid Inspired Neural Network Architecture](https://mlsanity.com/paper/2105.03854/) (2021)
- [Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Network](https://mlsanity.com/paper/2109.09363/) (2021)
- [A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs](https://mlsanity.com/paper/2304.10832/) (2023)
- [PEARL: Preconditioner Enhancement through Actor-critic Reinforcement Learning](https://mlsanity.com/paper/2501.10750/) (2025)
- [Learning Relaxation for Multigrid](https://mlsanity.com/paper/2207.11255/) (2022)
- [Communication-Efficient Algorithms for Solving Pressure Poisson Equation for Multiphase Flows using Parallel Computers](https://mlsanity.com/paper/2202.10548/) (2022)

## Abstract (arXiv)

Solving the pressure-Poisson equation remains the primary computational bottleneck in incompressible unstructured flow solvers primarily due to the inherent sensitivity of traditional linear solvers to mesh irregularities. This work introduces a data-driven algebraic multigrid (AMG) smoother that uses a modified graph convolutional isomorphism network (GCIN). The graph neural network predicts optimal polynomial coefficients to construct a sparse pseudo-inverse operator across diverse grid topologies. The coefficients are optimized to reduce the residual after each V-cycle iteration. By directly capturing the algebraic structure of the system from the sparse coefficient matrix, the proposed method maintains the solver's linearity while adapting to local anisotropies in unstructured grids. Our framework demonstrates significant performance gains by reducing the number of V-cycles required for a given tolerance and delivering wall-clock speedups from 4% to 37% across diverse benchmarks. Notably, the model exhibits robust generalization by maintaining efficiency on meshes up to 128 times larger than those seen in training, and by accelerating the solver's convergence on unseen industry-relevant problems such as the AirfRANS dataset.