用分层注意力变压器加速多项式方程求解
HATSolver: Learning Groebner Bases with Hierarchical Attention Transformers
- 引入树状结构注意力机制，捕捉多项式数据层级关系
- 可处理更大规模方程组，比之前方法求解能力提升显著
- 适合计算机代数与符号计算方向研究者参考
在NeurIPS 2024上，Kera等人首次将Transformer用于计算格罗布纳基，这是计算机代数中的核心工具，具有广泛实际应用。本文通过引入分层注意力变换器（HATs），改进该方法，以格罗布纳基计算为基础求解多变量多项式方程组。HAT架构包含树状归纳偏置，能够有效建模数据中的层次关系，在计算效率上相比传统扁平注意力模型有显著提升。我们实现了任意深度的泛化，并进行了详细的计算成本分析。结合课程学习策略，本方法可求解规模远超Kera等（2024）所用实例的方程组。
原文摘要 · Abstract (English)
At NeurIPS 2024, Kera et al. introduced the use of transformers for computing Groebner bases, a central object in computer algebra with numerous practical applications. In this paper, we improve this approach by applying Hierarchical Attention Transformers (HATs) to solve systems of multivariate polynomial equations via Groebner bases computation. The HAT architecture incorporates a tree-structured inductive bias that enables the modeling of hierarchical relationships present in the data and thus achieves significant computational savings compared to conventional flat attention models. We generalize to arbitrary depths and include a detailed computational cost analysis. Combined with curriculum learning, our method solves instances that are much larger than those in Kera et al. (2024 Learning to compute Groebner bases)
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