Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach
This work addresses the understanding of complex dynamical systems in physics, specifically the FPUT model, by applying data-driven methods to reveal underlying manifold structures, though it is incremental as it adapts existing techniques like PCA to this domain.
The study tackled the problem of inferring the intrinsic dimensionality of high-dimensional trajectories in the Fermi-Pasta-Ulam-Tsingou model using unsupervised machine learning, finding that the intrinsic dimension increases with nonlinearity and suggesting quasi-periodic motion on low-dimensional manifolds (e.g., m* = 2, 3) in weakly nonlinear regimes.
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 $β$ 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.