Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model

Abstract

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) $\beta$ 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 ($\beta \lesssim 1$). 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 $\beta$. Our DAE further reveals that the ID increases to $m^{\ast} = 3$ at $\beta = 1.1$, coinciding with a symmetry-breaking (SB) phenomenon characteristic of the $\beta$ 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.