用符号表达式自动学习线性方程求解的高效预条件参数。
SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning
- 通过神经网络搜索符号表达式，自动发现最优预条件参数。
- 在多个基准测试中显著优于传统方法，且推理效率高。
- 结果可解释性强，适合对效率与可靠性要求高的工程场景。
矩阵预条件是加速求解线性系统的关键技术，其性能高度依赖于预条件参数的选择。传统方法通常为特定场景设定固定常数，但依赖领域知识，未能考虑具体问题实例的特征，限制了性能表现。相比之下，机器学习方法虽具潜力，却受限于高推理开销和低可解释性。为此，我们提出符号发现框架——符号矩阵预条件（SymMaP），用于学习高效的符号表达式来表示预条件参数。具体而言，采用神经网络在高维离散空间中搜索能准确预测最优参数的表达式。所学表达式具备高推理效率和优秀可解释性（以简洁符号公式表示），便于部署与信任。实验结果表明，SymMaP在多个基准测试中持续优于传统策略。
原文摘要 · Abstract (English)
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.
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