# Learning to Solve Related Linear Systems

arXiv: https://arxiv.org/abs/2503.17265 · HTML: https://mlsanity.com/paper/2503.17265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 50 references that are in the map.
- First posted on arXiv 21 Mar 2025.

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

- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [Preconditioning for Scalable Gaussian Process Hyperparameter Optimization](https://mlsanity.com/paper/2107.00243/) (2021)
- [Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning](https://mlsanity.com/paper/1808.02229/) (2018)
- [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023)
- [Probabilistic Interpretation of Linear Solvers](https://mlsanity.com/paper/1402.2058/) (2014)
- [Computation-Aware Kalman Filtering and Smoothing](https://mlsanity.com/paper/2405.08971/) (2024)
- [Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference](https://mlsanity.com/paper/2411.01036/) (2024)
- [Calibrated Computation-Aware Gaussian Processes](https://mlsanity.com/paper/2410.08796/) (2024)
- [Probabilistic Gradients for Fast Calibration of Differential Equation Models](https://mlsanity.com/paper/2009.04239/) (2020)
- [Krylov Subspace Recycling for Fast Iterative Least-Squares in Machine Learning](https://mlsanity.com/paper/1706.00241/) (2017)
- [Accelerating Non-Conjugate Gaussian Processes By Trading Off Computation For Uncertainty](https://mlsanity.com/paper/2310.20285/) (2023)

## Abstract (arXiv)

Solving multiple parametrised related systems is an essential component of many numerical tasks, and learning from the already solved systems will make this process faster. In this work, we propose a novel probabilistic linear solver over the parameter space. This leverages information from the solved linear systems in a regression setting to provide an efficient posterior mean and covariance. We advocate using this as companion regression model for the preconditioned conjugate gradient method, and discuss the favourable properties of the posterior mean and covariance as the initial guess and preconditioner. We also provide several design choices for this companion solver. Numerical experiments showcase the benefits of using our novel solver in a hyperparameter optimisation problem.