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

arXiv: https://arxiv.org/abs/2411.02058 · HTML: https://mlsanity.com/paper/2411.02058/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 98 references that are in the map.
- First posted on arXiv 4 Nov 2024.

## Builds on (most cited first, 11 of 11)

- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [A Tutorial on Principal Component Analysis](https://mlsanity.com/paper/1404.1100/) (2014)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [FONDUE: an algorithm to find the optimal dimensionality of the latent representations of variational autoencoders](https://mlsanity.com/paper/2209.12806/) (2022)
- [An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds](https://mlsanity.com/paper/2308.02615/) (2023)

## Built on by (most cited first, 1 of 1)

- [Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model](https://mlsanity.com/paper/2601.19567/) (2026)

## Abstract (arXiv)

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.