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 $n_s=4000000$ 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.

This version matches the published one in Chaos Journal. Preliminary results were presented in November 2025 at the IUPAP Conference on Computational Physics, CP2025 XXXVI, Oak Ridge National Laboratory in Oak Ridge