用深度自编码器发现费米-帕斯塔-乌拉姆系统轨迹的内在维度为2
Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model
- 用深度自编码器探测高维轨迹的非线性结构
- 在弱非线性下轨迹嵌入二维流形，β=1.1时升至三维
- 可捕捉线性方法无法识别的对称性破缺现象
我们研究了包含 $n_s = 4\,000\,000$ 个数据点的费米-帕斯塔-乌拉姆-茨基努（FPUT）$β$ 模型（$N = 32$ 个振子）在弱非线性区（$β\lesssim 1$）的轨迹内在维度（ID）。采用深度自编码器（DAE）发现轨迹位于64维相空间中的二维非线性黎曼流形上。相比之下，主成分分析（PCA）结合参与比（PR）仅能提供每个 $β$ 值下的合理上限估计。当 $β = 1.1$ 时，DAE揭示内在维度增至 $m^\ast = 3$，与 $β$ 模型的对称性破缺（SB）现象一致，表现为偶数波数 $k = 2, 4$ 的额外能量模被激发。值得注意的是，这一现象无法通过线性方法如PCA检测到。
原文摘要 · Abstract (English)
We address the intrinsic dimensionality (ID) of high-dimensional trajectories, comprising $n_s = 4\,000\,000$ data points, of the Fermi-Pasta-Ulam-Tsingou (FPUT) $β$ model with $N = 32$ oscillators. To this end, a deep autoencoder (DAE) is used to infer the ID in the weakly nonlinear regime where energy recurrences are observed ($β\lesssim 1$). We find that the trajectories lie on a nonlinear Riemannian manifold of dimension $m^{\ast} = 2$ embedded in a $64$-dimensional phase space. By contrast, principal component analysis (PCA) together with the Participation Ratio (PR) method provides only a reasonable upper bound on the ID for each value of $β$. Our DAE further reveals that the ID increases to $m^{\ast} = 3$ at $β= 1.1$, coinciding with a symmetry-breaking (SB) phenomenon characteristic of the $β$ model, in which additional energy modes with even wave numbers $k = 2, 4$ become excited. Notably, the SB phenomenon cannot be detected by the linear approach provided by PCA.
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