SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

TL;DR

SymMaP introduces a symbolic learning framework that combines neural network search with symbolic expressions to improve the efficiency and interpretability of preconditioning in linear solvers, outperforming traditional methods.

Contribution

It proposes a novel symbolic discovery framework using neural networks to learn interpretable preconditioning expressions, enhancing efficiency and performance.

Findings

SymMaP outperforms traditional preconditioning strategies on multiple benchmarks.

The learned symbolic expressions enable fast and reliable parameter prediction.

The approach combines interpretability with high inference efficiency.

Abstract

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.