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) β 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 (β 1). We find that the trajectories lie on a nonlinear Riemannian manifold of dimension m^ = 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 β. Our DAE further reveals that the ID increases to m^ = 3 at β= 1.1, coinciding with a symmetry-breaking (SB) phenomenon characteristic of the β 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.

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