Acceleration of an algebraic multigrid pressure solver using graph neural networks

2026-06-17

AI Summary

This work addresses the limited efficiency of traditional algebraic multigrid (AMG) pressure solvers on unstructured grids, which stems from their sensitivity to grid irregularities. The study introduces, for the first time, a Graph Convolutional Isomorphism Network (GCIN) into the AMG framework to directly learn the algebraic structure of smoothers. By predicting optimal polynomial coefficients, the method constructs a sparse pseudoinverse operator that adaptively handles local anisotropy while preserving linear computational complexity. The proposed approach demonstrates strong generalization across scales and problem types, reducing V-cycle iteration counts and achieving practical speedups of 4%–37% on various benchmarks. Notably, it maintains robust convergence on grids up to 128 times larger than those used in training and on unseen industrial cases such as AirfRANS.

Abstract

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.

Problem

• pressure-Poisson equation
• algebraic multigrid
• unstructured grids
• computational bottleneck
• mesh irregularities

Innovation

• graph neural networks
• algebraic multigrid
• pressure-Poisson solver
• data-driven smoother
• unstructured grids