用符号变压器生成训练数据，解决动态系统全局稳定性的长期难题
Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers
- 通过随机解生成合成数据训练序列模型
- 在多项式系统上超越算法求解器和人类表现
- 首次发现非多项式系统的新型李雅普诺夫函数
尽管语言模型在诸多任务中取得显著进展，但在复杂推理任务（如高等数学）上仍存在困难。本文聚焦数学领域一个长期未解的难题：寻找能保证动力系统全局稳定的李雅普诺夫函数。该问题尚无通用解法，现有算法求解器仅适用于部分小规模多项式系统。我们提出一种从随机解中生成合成训练样本的新方法，并证明基于此类数据集训练的序列到序列变换器，在多项式系统上的表现优于算法求解器和人类专家，且可成功发现非多项式系统的新型李雅普诺夫函数。
原文摘要 · Abstract (English)
Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.
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