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

**Authors:** Eric Chillón, Artur K. Lidtke, Nguyen Anh Khoa Doan, Bernat Font  
**arXiv:** [2606.19251](https://arxiv.org/abs/2606.19251)  
**Date:** 2026-06-17  
**Summary page:** https://www.summarizepaper.com/en/paper/2606.19251v1/

## Key points

- Authors Eric Chillón, Artur K. Lidtke, Nguyen Anh Khoa Doan, and Bernat Font introduce a data-driven algebraic multigrid (AMG) smoother using a modified graph convolutional isomorphism network (GCIN).
- The key innovation is the use of a graph neural network to predict optimal polynomial coefficients for constructing a sparse pseudo-inverse operator across diverse grid topologies.
- This method improves solver performance by optimizing coefficients to reduce the residual at each V-cycle iteration.
- The approach maintains linearity while adapting to local anisotropies in unstructured grids.
- Extensive testing shows significant performance gains with notable wall-clock speedups ranging from 4% to 37%.
- The model exhibits robust generalization capabilities on larger meshes and accelerates convergence on industry-relevant problems like the AirfRANS dataset.
- This research represents a major advancement in computational fluid dynamics by addressing challenges with data-driven approaches and neural networks, enhancing solver efficiency, adaptability, and scalability across various grid configurations.

## Summary

Authors Eric Chillón, Artur K. Lidtke, Nguyen Anh Khoa Doan, and Bernat Font present a groundbreaking approach to accelerating the solution of the pressure-Poisson equation in incompressible unstructured flow solvers. Traditional linear solvers have long struggled with mesh irregularities, leading to computational bottlenecks in solving this critical equation. In their work, the authors introduce a data-driven algebraic multigrid (AMG) smoother that leverages a modified graph convolutional isomorphism network (GCIN). The key innovation lies in the use of a graph neural network to predict optimal polynomial coefficients for constructing a sparse pseudo-inverse operator across diverse grid topologies. These coefficients are then optimized to reduce the residual at each V-cycle iteration, effectively improving solver performance. By directly capturing the algebraic structure of the system from the sparse coefficient matrix, this method maintains linearity while adapting to local anisotropies present in unstructured grids. Through extensive testing on diverse benchmarks, the authors demonstrate significant performance gains achieved by reducing the number of V-cycles required to reach a given tolerance level. This results in notable wall-clock speedups ranging from 4% to 37%. Impressively, their model exhibits robust generalization capabilities by maintaining efficiency on meshes up to 128 times larger than those encountered during training. Furthermore, it accelerates convergence on unseen industry-relevant problems such as the AirfRANS dataset. Overall, this research represents a major advancement in computational fluid dynamics by addressing a longstanding challenge with innovative techniques rooted in data-driven approaches and neural networks. The proposed framework not only enhances solver efficiency but also showcases adaptability and scalability across various grid configurations, paving the way for more effective simulations in complex fluid dynamics scenarios.