arXiv:2607.15472math.DScs.LG2026-07被引 1

用机器学习发现通用动力时钟,揭示复杂振荡的统一相位规律。

Ptolemy's Equant Equates to a Universal Dynamical Clock via Machine Learning

论文配图:Ptolemy's Equant Equates to a Universal Dynamical Clock via Machine Learning
图 1 · 摘自论文原文
  • 基于托勒密均轮思想,通过机器学习构建非线性坐标下的均匀旋转模型。
  • 在细菌群体、基因回路等系统中验证了超线性尺度律和临界预警信号。
  • 适用于任意维度振荡系统,适合研究复杂网络的动力学机制。

非线性高维振荡普遍存在,但如何识别物理可解释的相位与相位动力学仍是未解难题。本文提出通用动力时钟原理:任意维度与几何的振荡均可通过托勒密均轮启发的非线性视角坐标,等价表示为均匀旋转,其核心是面积均匀性原则,类似开普勒第二定律。利用机器学习框架,我们证明了这类均轮在广泛振荡系统中存在,并构建了含噪声、周期扰动及耦合下的动力时钟与相位动力学。该方法成功揭示四项新物理现象:(i) 大肠杆菌种群集体振荡服从2004年未解的超线性尺度律;(ii) 工程化基因回路对基因表达与环境变化的响应机制;(iii) 经典力学中自然涌现类贝里几何相位;(iv) 最优均轮非均匀性提供几何早期预警信号,可预测临界参数。该动力时钟从数据直接构建,具备操作性和系统无关性,支持振荡系统的分类、比较与控制,为理解网络系统中不同动力学态如何支撑特定功能开辟新路径。

原文摘要 · Abstract (English)

Oscillatory dynamics arise ubiquitously in nonlinear systems, yet identifying a physically interpretable phase and phase dynamics in nonlinear, high-dimensional oscillations remains a central unresolved problem. Here we establish the principle of a universal dynamical clock, a physical perspective in which oscillations of arbitrary dimensionality and geometry are equivalently represented as uniform rotation through an equant-induced nonlinear viewing coordinate, inspired by Ptolemy's equant and formalised through an areal-uniformity principle reminiscent of Kepler's second law. Using a machine-learning framework, we demonstrate the existence of such an equant for a broad class of oscillatory dynamics and construct the associated dynamical clock and phase dynamics under additive forces, including noise, periodic perturbations, and coupling. Its value in uncovering new physical rules and phenomena is demonstrated by four findings: (i) collective oscillations in Escherichia coli populations obey a previously unexplained superlinear scaling law, resolving a long-standing open problem posed in 2004; (ii) the response mechanisms of engineered genetic circuits to changes in gene expression and environmental conditions; (iii) a classical-mechanics counterpart of the Berry geometric phase emerges naturally from the phase of the dynamical clock; and (iv) optimal equant non-uniformity provides a geometric early-warning signal for critical transitions and enables prediction of critical parameters. By providing operational and system-agnostic phase dynamics that can be constructed directly from data, the dynamical clock enables principled classification, comparison, and control of oscillatory systems, and offers a new route to understanding how specific dynamical regimes support distinct functional behaviours in networked systems.

动力时钟机器学习振荡系统相位分析

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