SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

Hong Wang, Jie Wang, Minghao Ma, Haoran Shao, Haoyang Liu

Keywords: Matrix Preconditioning, Symbolic Learning, Linear System Solver

Abstract:

Matrix preconditioning is a crucial modern technique for accelerating the solving of linear systems.
Its effectiveness heavily depends on the choice of preconditioning parameters.
Traditional methods often depend on domain expertise to define a set of fixed constants for specific scenarios.
However, the characteristics of each problem instance also affect the selection of optimal parameters, while fixed constants do not account for specific instance characteristics and may lead to performance loss.
In this paper, we propose **Sym**bolic **Ma**trix **P**reconditioning (**SymMaP**), a novel framework based on Recurrent Neural Networks (RNNs) for automatically generating symbolic expressions to compute efficient preconditioning parameters.
Our method begins with a grid search to identify optimal parameters according to task-specific performance metrics.
SymMaP then performs a risk-seeking search over the high-dimensional discrete space of symbolic expressions, using the best-found expression as the evaluation criterion.
The resulting symbolic expressions are seamlessly integrated into modern linear system solvers to improve computational efficiency.
Experimental results demonstrate that SymMaP consistently outperforms traditional algorithms across various benchmarks. The learned symbolic expressions can be easily embedded into existing specialized solvers with negligible computational overhead. Furthermore, the high interpretability of these concise mathematical expressions facilitates deeper understanding and further optimization of matrix preconditioning strategies.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 8948