# Learning incomplete factorization preconditioners for GMRES

arXiv: https://arxiv.org/abs/2409.08262 · HTML: https://mlsanity.com/paper/2409.08262/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 12 Sep 2024.

## Builds on (most cited first, 11 of 11)

- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems](https://mlsanity.com/paper/1906.06925/) (2019)
- [Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers](https://mlsanity.com/paper/2405.15557/) (2024)
- [Graph Neural Preconditioners for Iterative Solutions of Sparse Linear Systems](https://mlsanity.com/paper/2406.00809/) (2024)
- [Graph-based Neural Acceleration for Nonnegative Matrix Factorization](https://mlsanity.com/paper/2202.00264/) (2022)
- [Graph Neural Networks and Applied Linear Algebra](https://mlsanity.com/paper/2310.14084/) (2023)
- [Optimized Sparse Matrix Operations for Reverse Mode Automatic Differentiation](https://mlsanity.com/paper/2212.05159/) (2022)
- [Ising on the Graph: Task-specific Graph Subsampling via the Ising Model](https://mlsanity.com/paper/2402.10206/) (2024)
- [Generative modeling of Sparse Approximate Inverse Preconditioners](https://mlsanity.com/paper/2405.11007/) (2024)

## Built on by (most cited first, 9 of 9)

- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)
- [PEARL: Preconditioner Enhancement through Actor-critic Reinforcement Learning](https://mlsanity.com/paper/2501.10750/) (2025)
- [Learning to accelerate distributed ADMM using graph neural networks](https://mlsanity.com/paper/2509.05288/) (2025)
- [Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations](https://mlsanity.com/paper/2502.01397/) (2025)
- [Warm-starting active-set solvers using graph neural networks](https://mlsanity.com/paper/2511.13174/) (2025)
- [Factorization-in-Loop: Proximal Fill-in Minimization for Sparse Matrix Reordering](https://mlsanity.com/paper/2511.09093/) (2025)
- [Estimating Condition Number with Graph Neural Networks](https://mlsanity.com/paper/2603.10277/) (2026)
- [RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections](https://mlsanity.com/paper/2605.26854/) (2026)
- [CodeJeNN: A simple C++ neural network generator for physics applications](https://mlsanity.com/paper/2607.02746/) (2026)

## Abstract (arXiv)

Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to hyper-parameters. Instead, we replace the hand-engineered algorithm with a graph neural network that is trained to approximate the matrix factorization directly. To apply the output of the neural network as a preconditioner, we propose an output activation function that guarantees that the predicted factorization is invertible. Further, applying a graph neural network architecture allows us to ensure that the output itself is sparse which is desirable from a computational standpoint. We theoretically analyze and empirically evaluate different loss functions to train the learned preconditioners and show their effectiveness in decreasing the number of GMRES iterations and improving the spectral properties on synthetic data. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.