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

TL;DR

This study uses unsupervised machine learning, specifically PCA, to determine the intrinsic dimensionality of high-dimensional FPUT trajectories, revealing a relationship between nonlinearity and dimensionality, and suggesting low-dimensional quasi-periodic dynamics.

Contribution

It introduces a data-driven PCA-based method to estimate the intrinsic dimension of FPUT trajectories and links it to the system's nonlinearity, providing new insights into the underlying dynamics.

Findings

Intrinsic dimension increases with nonlinearity.

Low-dimensional manifolds underlie weakly nonlinear trajectories.

Quasi-periodic motion is characterized by 2-3 dimensions.

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.