Learning the intrinsic dimensionality of Fermi-Pasta-Ulam-Tsingou trajectories: A nonlinear approach using a deep...

We address the intrinsic dimensionality (ID) of high-dimensional trajectories, comprising ns = 4 000 000 data points, generated by the Fermi-Pasta-Ulam-Tsingou (FPUT) β model with N = 32 oscillators. To this end, a deep autoencoder (DAE) is employed to infer the ID in the weakly nonlinear regime, where energy recurrences are observed (β≲1). Our results indicate that the trajectories lie on a two-dimensional Riemannian manifold (m* = 2) embedded in a 64-dimensional phase space. This finding is independently corroborated by a Fourier analysis of the time series associated with the normal mode coordinates and their corresponding energies, which reveals two independent frequencies, consistent with quasi-periodic motion on a two-dimensional invariant torus T2. By contrast, principal component analysis (PCA) can provide only a reasonable upper bound on the intrinsic dimensionality. Furthermore, the DAE predicts that the intrinsic dimensionality increases to m* = 3 at β = 1.1, coinciding with the onset of a symmetry-breaking (SB) phenomenon characteristic of the β model, in which additional modes with even wave numbers (k = 2, 4) become excited. Notably, this SB transition is not detected by the linear PCA-based approach.