用树结构建模数学表达式,提升符号积分算法选择效率
Tree-Based Deep Learning for Ranking Symbolic Integration Algorithms
- 将数学表达式转为树结构,用两阶段模型筛选并排序可用算法
- 在7万条测试数据上准确率达近90%,优于传统方法
- 适合优化数学软件中算法选择问题的研究者和开发者
符号不定积分在计算机代数系统(如Maple)中需从多个可用方法中选择最优解法。并非所有方法都对每个问题有效,即使多个方法成功,结果虽数学等价但形式复杂度差异大。传统方法极少考虑具体问题实例,导致效率低下。本文提出基于树结构的深度学习两阶段框架:先识别适用于当前问题的方法,再按预测输出复杂度排序。实验表明,以树结构表示数学表达式显著优于序列化表示,且该框架优于其他机器学习方案。使用六个不同数据生成器构建的多样化数据集,模型在7万例独立测试集上达到近90%准确率。在Maple内部测试套件的分布外基准上,树形变换器模型仍保持强泛化能力,优于Maple内置选择器及先前机器学习方法。结果凸显数据表示与问题建模在符号计算中的关键作用,本方法有望推广至类似数学软件优化问题。
原文摘要 · Abstract (English)
Symbolic indefinite integration in Computer Algebra Systems such as Maple involves selecting the most effective algorithm from multiple available methods. Not all methods will succeed for a given problem, and when several do, the results, though mathematically equivalent, can differ greatly in presentation complexity. Traditionally, this choice has been made with minimal consideration of the problem instance, leading to inefficiencies. We present a machine learning (ML) approach using tree-based deep learning models within a two-stage architecture: first identifying applicable methods for a given instance, then ranking them by predicted output complexity. Furthermore, we find representing mathematical expressions as tree structures significantly improves performance over sequence-based representations, and our two-stage framework outperforms alternative ML formulations. Using a diverse dataset generated by six distinct data generators, our models achieve nearly 90% accuracy in selecting the optimal method on a 70,000 example holdout test set. On an independent out-of-distribution benchmark from Maple's internal test suite, our tree transformer model maintains strong generalisation, outperforming Maple's built-in selector and prior ML approaches. These results highlight the critical role of data representation and problem framing in ML for symbolic computation, and we expect our methodology to generalise effectively to similar optimisation problems in mathematical software.
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