RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections

2026-05-26

AI Summary

This work addresses the longstanding challenge in algebraic multigrid (AMG) methods of balancing sparsity and convergence quality in coarse-grid operators. The authors propose RAPNet, the first graph neural network framework that directly learns nongalerkin coarse operators from sparse algebraic systems. Employing a hierarchical training strategy, RAPNet generalizes from small subgraphs to problems with millions of unknowns, achieving an effective trade-off among sparsity, convergence, and generalization while preserving computational efficiency during the solve phase. Experimental results demonstrate that RAPNet significantly outperforms classical nongalerkin baselines across a range of discretized partial differential equations and graph Laplacian problems, substantially accelerating multi-query tasks such as eigenvalue computations, time-dependent simulations, inverse problems, and design optimization.

Abstract

The scalable solution of large sparse linear systems is a bottleneck in scientific computing and graph analysis. While algebraic multigrid (AMG) offers optimal linear scaling, its performance is severely constrained by the trade-off between the sparsity and convergence quality of coarse-grid operators. Classical AMG heuristics struggle to balance these objectives, often sacrificing stability or performance for sparsity. We propose RAPNet, a graph neural network (GNN) framework that resolves this trade-off by learning to generate sparse, robust coarse operators directly from the sparse algebraic system. Key to our approach is a level-wise training strategy that enables learning from small subgraphs and generalization to million-node domains, bypassing the bottlenecks of prior neural AMG attempts. RAPNet executes exclusively during the solver setup phase, ensuring that the solve phase retains its favorable computational properties. We show that our method outperforms classical non-Galerkin baselines on diverse PDE discretizations and graph Laplacians, making it particularly effective for multi-query tasks such as eigenproblems, time-dependent simulations, and inverse or design problems.

Problem

• algebraic multigrid
• sparse linear systems
• coarse-grid operators
• sparsity-convergence trade-off
• scientific computing

Innovation

• RAPNet
• algebraic multigrid
• graph neural network
• sparse correction
• coarse-grid operator