Intrinsic dimensionality of Fermi-Pasta-Ulam-Tsingou high-dimensional trajectories through manifold learning: A...

A data-driven approach based on unsupervised machine learning is proposed to infer the intrinsic dimensionality of high-dimensional trajectories in the Fermi-Pasta-Ulam-Tsingou (FPUT) model. Principal component analysis is applied to trajectory data accurately computed using a symplectic integrator, comprising ns=4000000 data points from the FPUT β model with N=32 coupled harmonic oscillators. By estimating the intrinsic dimension m∗ using multiple methods (participation ratio, Kaiser rule, and the Kneedle algorithm), it is found that m∗ increases with the model's nonlinearity. Interestingly, in the weakly nonlinear regime (β≲1.1), for trajectories initialized by exciting the first mode (k=1), the participation ratio estimates m∗=2,3, strongly suggesting that quasi-periodic motion on a low-dimensional Riemannian manifold underlies the characteristic energy recurrences observed in the FPUT model.