# SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

arXiv: https://arxiv.org/abs/2510.24170 · HTML: https://mlsanity.com/paper/2510.24170/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 69 references that are in the map.
- First posted on arXiv 28 Oct 2025.

## Builds on (most cited first, 25 of 25)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Learning to Execute](https://mlsanity.com/paper/1410.4615/) (2014)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [OptiTree: Hierarchical Thoughts Generation with Tree Search for LLM Optimization Modeling](https://mlsanity.com/paper/2510.22192/) (2025)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023)
- [MILP-StuDio: MILP Instance Generation via Block Structure Decomposition](https://mlsanity.com/paper/2410.22806/) (2024)
- [Apollo-MILP: An Alternating Prediction-Correction Neural Solving Framework for Mixed-Integer Linear Programming](https://mlsanity.com/paper/2503.01129/) (2025)
- [Weakly-Supervised Deep Learning of Heat Transport via Physics Informed Loss](https://mlsanity.com/paper/1807.11374/) (2018)
- [Accelerating PDE Data Generation via Differential Operator Action in Solution Space](https://mlsanity.com/paper/2402.05957/) (2024)
- [Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers](https://mlsanity.com/paper/2405.15557/) (2024)
- [Promoting Generalization for Exact Solvers via Adversarial Instance Augmentation](https://mlsanity.com/paper/2310.14161/) (2023)
- [Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling](https://mlsanity.com/paper/2401.09516/) (2024)
- [Deep Symbolic Optimization for Combinatorial Optimization: Accelerating Node Selection by Discovering Potential Heuristics](https://mlsanity.com/paper/2406.09740/) (2024)
- [Machine Learning Insides OptVerse AI Solver: Design Principles and Applications](https://mlsanity.com/paper/2401.05960/) (2024)
- [Exploiting Edited Large Language Models as General Scientific Optimizers](https://mlsanity.com/paper/2503.09620/) (2025)

## Built on by (most cited first, 5 of 5)

- [CoCo-MILP: Inter-Variable Contrastive and Intra-Constraint Competitive MILP Solution Prediction](https://mlsanity.com/paper/2511.09209/) (2025)
- [RoME: Domain-Robust Mixture-of-Experts for MILP Solution Prediction across Domains](https://mlsanity.com/paper/2511.02331/) (2025)
- [Learning Neural Operators from Partial Observations via Latent Autoregressive Modeling](https://mlsanity.com/paper/2601.15547/) (2026)
- [Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers](https://mlsanity.com/paper/2511.01927/) (2025)
- [HGATSolver: A Heterogeneous Graph Attention Solver for Fluid-Structure Interaction](https://mlsanity.com/paper/2601.09251/) (2026)

## Abstract (arXiv)

Matrix preconditioning is a critical technique to accelerate the solution of linear systems, where performance heavily depends on the selection of preconditioning parameters. Traditional parameter selection approaches often define fixed constants for specific scenarios. However, they rely on domain expertise and fail to consider the instance-wise features for individual problems, limiting their performance. In contrast, machine learning (ML) approaches, though promising, are hindered by high inference costs and limited interpretability. To combine the strengths of both approaches, we propose a symbolic discovery framework-namely, Symbolic Matrix Preconditioning (SymMaP)-to learn efficient symbolic expressions for preconditioning parameters. Specifically, we employ a neural network to search the high-dimensional discrete space for expressions that can accurately predict the optimal parameters. The learned expression allows for high inference efficiency and excellent interpretability (expressed in concise symbolic formulas), making it simple and reliable for deployment. Experimental results show that SymMaP consistently outperforms traditional strategies across various benchmarks.