Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning

A data-driven approach based on unsupervised machine learning is proposed to infer the intrinsic dimensions $m^{\ast}$ 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 $\beta$ model with $N = 32$ coupled oscillators, revealing a critical relationship between $m^{\ast}$ and the model's nonlinear strength. For weak nonlinearities, $m^{\ast} \ll n$, where $n = 2N$. In contrast, for strong nonlinearities, $m^{\ast} \rightarrow n - 1$, consistently with the ergodic hypothesis. Furthermore, one of the potential limitations of PCA is addressed through an analysis with t-distributed stochastic neighbor embedding ($t$-SNE). Accordingly, we found strong evidence suggesting that the datapoints lie near or on a curved low-dimensional manifold for weak nonlinearities.