# Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model

arXiv: https://arxiv.org/abs/2601.19567 · HTML: https://mlsanity.com/paper/2601.19567/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 94 references that are in the map.
- First posted on arXiv 27 Jan 2026.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
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- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [The dynamics of representation learning in shallow, non-linear autoencoders](https://mlsanity.com/paper/2201.02115/) (2022)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds](https://mlsanity.com/paper/2308.02615/) (2023)
- [Learning Geometry and Topology via Multi-Chart Flows](https://mlsanity.com/paper/2505.24665/) (2025)
- [Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach](https://mlsanity.com/paper/2411.02058/) (2024)

## Abstract (arXiv)

We address the intrinsic dimensionality (ID) of high-dimensional trajectories, comprising ns = 4,000,000 data points, of the Fermi-Pasta-Ulam-Tsingou (FPUT) β model with N = 32 oscillators. To this end, a deep autoencoder (DAE) is used to infer the ID in the weakly nonlinear regime where energy recurrences are observed (β lesssim 1). We find that the trajectories lie on a nonlinear Riemannian manifold of dimension mast = 2 embedded in a 64-dimensional phase space. By contrast, principal component analysis (PCA) together with the Participation Ratio (PR) method provides only a reasonable upper bound on the ID for each value of β. Our DAE further reveals that the ID increases to mast = 3 at β = 1.1, coinciding with a symmetry-breaking (SB) phenomenon characteristic of the β model, in which additional energy modes with even wave numbers k = 2, 4 become excited. Notably, the SB phenomenon cannot be detected by the linear approach provided by PCA.