Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach

Abstract

A data-driven approach based on unsupervised machine learning is proposed to infer the intrinsic dimension $m^{\ast }$ 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. By estimating the intrinsic dimension $m^{\ast }$ using multiple methods (participation ratio, Kaiser rule, and the Kneedle algorithm), it is found that $m^{\ast }$ increases with the model nonlinearity. Interestingly, in the weakly nonlinear regime, for trajectories initialized by exciting the first mode, the participation ratio estimates $m^{\ast }=2,3$, strongly suggesting that quasi-periodic motion on a low-dimensional Riemannian manifold underlies the characteristic energy recurrences observed in the FPUT model.

arXiv comment14 pages, 15 figures. This version matches the article published in Chaos 35, 103118 (2025)

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