用机器学习发现费米-帕斯塔-乌拉姆系统轨迹的内在维度
Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach
- 通过主成分分析解析400万组轨迹数据，寻找低维流形
- 弱非线性时内在维度仅2~3，对应能量周期性回溯
- 适合研究混沌与守恒系统动力学的学者参考
本文提出一种基于无监督机器学习的数据驱动方法，用于推断费米-帕斯塔-乌拉姆-茨琴（FPUT）模型高维轨迹的内在维度 $m^{ ext{ast}}$。对包含 $n_s = 4,000,000$ 个数据点、$N = 32$ 个耦合振子的 FPUT $\beta$ 模型轨迹数据，应用主成分分析（PCA），揭示了 $m^{ ext{ast}}$ 与模型非线性强度之间的关键关系。通过参与度比、Kaiser准则和 Kneedle 算法等多种方法估计 $m^{ ext{ast}}$，发现其随非线性增强而增大。值得注意的是，在弱非线性情况下，若初始激发第一模态，参与度比估计 $m^{ ext{ast}} = 2, 3$，强烈表明准周期运动存在于低维黎曼流形上，这正是 FPUT 模型中观测到的能量周期性回溯现象的根源。
原文摘要 · Abstract (English)
A data-driven approach based on unsupervised machine learning is proposed to infer the intrinsic dimension $m^{\ast}$ of the high-dimensional trajectories of the Fermi-Pasta-Ulam-Tsingou (FPUT) model. Principal component analysis (PCA) is applied to trajectory data consisting of $n_s = 4,000,000$ datapoints, of the FPUT $β$ model with $N = 32$ coupled oscillators, revealing a critical relationship between $m^{\ast}$ and the model's nonlinear strength. By estimating the intrinsic dimension $m^{\ast}$ using multiple methods (participation ratio, Kaiser rule, and the Kneedle algorithm), it is found that $m^{\ast}$ increases with the model nonlinearity. Interestingly, in the weakly nonlinear regime, for trajectories initialized by exciting the first mode, the participation ratio estimates $m^{\ast} = 2, 3$, strongly suggesting that quasi-periodic motion on a low-dimensional Riemannian manifold underlies the characteristic energy recurrences observed in the FPUT model.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。