用收敛动态替代数值匹配，首次实现数学常数公式自动聚类与发现。
Unsupervised Discovery of Formulas for Mathematical Constants
- 基于公式收敛动态设计新度量，突破传统数值误差限制。
- 在176万条多项式连分数中发现π、ln(2)等未知公式。
- 揭示公式模式可推广为无限族，助力生成式公式模型构建。
几十年来，人工智能在加速科学发现方面取得进展，但数学公式发现仍面临挑战，因每个公式需对无限位精度成立，近似公式无法提供有效线索。为此，本文提出系统性方法，通过公式收敛动态而非数值本身定义度量，实现首个数学公式自动化聚类。研究聚焦多项式连分数（PFC），其广泛关联数学常数并泛化多种函数结构。在1,768,900条公式中，识别出多个已知常数公式，并发现π、ln(2)、Gauss'常数及lemniscate常数的未知公式。这些模式可直接推广为无穷公式族，揭示丰富数学结构。该成果为构建满足特定数学性质的生成模型奠定基础，有望加速有用公式的发现进程。
原文摘要 · Abstract (English)
Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for $π$, $\ln(2)$, Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.
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