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

A data-driven approach based on unsupervised machine learning is proposed to infer the intrinsic dimension mast of the high-dimensional trajectories of the Fermi-Pasta-Ulam-Tsingou (FPUT) model. Principal component analysis (PCA) is applied to trajectory data consisting of ns = 4,000,000 datapoints, of the FPUT β model with N = 32 coupled oscillators, revealing a critical relationship between mast and the model's nonlinear strength. By estimating the intrinsic dimension mast using multiple methods (participation ratio, Kaiser rule, and the Kneedle algorithm), it is found that mast increases with the model nonlinearity. Interestingly, in the weakly nonlinear regime, for trajectories initialized by exciting the first mode, the participation ratio estimates mast = 2, 3, strongly suggesting that quasi-periodic motion on a low-dimensional Riemannian manifold underlies the characteristic energy recurrences observed in the FPUT model.