Estimating condition number with Graph Neural Networks

A new method uses graph neural networks to rapidly estimate the condition number of sparse matrices, a calculation critical for numerical stability in scientific computing, according to a preprint posted to arXiv [1]. The approach, authored by Xinye Chen, introduces a graph feature construction with a complexity of O(nnz + n), where nnz represents the number of non-zero elements and n the matrix dimension [1]. This design allows the graph neural network, or GNN, to process large sparse matrices without forming the computationally expensive matrix inverse [1]. The paper proposes two estimation schemes: one that decomposes the condition number and predicts the norm of the inverse matrix, and another that predicts the entire condition number directly [1]. The method can be extended to an arbitrary norm, and experiments on 1-norm and 2-norm condition numbers showed a significant speedup over traditional numerical estimation techniques [1]. The software has been made publicly available [1]. The preprint was first submitted on 10 March 2026 and last revised on 24 June 2026 [1]. The work appears on arXiv, an open-access repository that hosts electronic preprints in fields including computer science and mathematics and which, as of late 2024, receives about 24,000 new articles per month [10]. The repository does not conduct peer review before posting [10]. The paper’s reliance on GNNs places it within the broader machine-learning subfield of artificial intelligence, which Arthur Samuel defined in 1959 as a “field of study that gives computers the ability to learn without being explicitly programmed” [4]. Training such networks typically relies on optimization algorithms like gradient descent, a first-order iterative method first suggested by Augustin-Louis Cauchy in 1847 and now fundamental to deep learning [5]. The preprint’s abstract page also features arXivLabs, a framework launched by the repository to enable community-developed tools that add functionality for readers and authors while adhering to arXiv’s values of openness and user-data privacy [8].