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) model is employed to infer the ID in the weakly nonlinear regime ($β\lesssim 1$). We find that the trajectories lie on a nonlinear manifold of dimension $m^{\ast} = 2$ embedded in a $64$-dimensional phase space. The DAE further reveals that this dimensionality increases to $m^{\ast} = 3$ at $β= 1.1$, coinciding with a symmetry breaking transition, in which additional energy modes with even wave numbers $k = 2, 4$ become excited. Finally, we discuss the limitations of the linear approach based on principal component analysis (PCA), which fails to capture the underlying structure of the data and therefore yields unreliable results in most cases.

翻译：我们研究了由 $n_s = 4\,000\,000$ 个数据点构成的高维轨迹的本征维数（ID），这些轨迹来自具有 $N = 32$ 个振子的费米-帕斯塔-乌拉姆-曾九（FPUT）$β$ 模型。为此，我们采用深度自编码器（DAE）模型来推断弱非线性区域（$β\lesssim 1$）中的本征维数。我们发现，轨迹位于一个嵌入在 $64$ 维相空间中的、维数为 $m^{\ast} = 2$ 的非线性流形上。DAE 进一步揭示，在 $β= 1.1$ 时，该维数增加到 $m^{\ast} = 3$，这与对称性破缺转变相吻合，在此转变中，具有偶数波数 $k = 2, 4$ 的额外能量模式被激发。最后，我们讨论了基于主成分分析（PCA）的线性方法的局限性，该方法未能捕捉数据的底层结构，因此在大多数情况下会产生不可靠的结果。