Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations

Abstract

Graph Neural Networks (GNNs) have been proposed as a tool for learning sparse matrix preconditioners, which are key components in accelerating linear solvers. This position paper argues that message-passing GNNs are fundamentally incapable of approximating sparse triangular factorizations. We demonstrate that message-passing GNNs fundamentally fail to approximate sparse triangular factorizations for classes of matrices for which high-quality preconditioners exist but require non-local dependencies. To illustrate this, we construct a set of baselines using both synthetic matrices and real-world examples from the SuiteSparse collection. Across a range of GNN architectures, including Graph Attention Networks and Graph Transformers, we observe severe performance degradation compared to exact or K-optimal factorizations, with cosine similarity dropping below 0.6 in key cases. Our theoretical and empirical results suggest that architectural innovations beyond message-passing are necessary for applying GNNs to scientific computing tasks such as matrix factorization. Experiments demonstrate that overcoming non-locality alone is insufficient. Tailored architectures are necessary to capture the required dependencies since even a completely non-local Graph Transformer fails to match the proposed baselines.

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Reproductions