arXiv:2411.02058v3 Announce Type: replace
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.