Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914403958390784 |
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| author | Marchetti, Gionni |
| author_facet | Marchetti, Gionni |
| contents | We address the intrinsic dimensionality (ID) of high-dimensional trajectories, comprising $n_s = 4\,000\,000$ 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 $m^{\ast} = 2$ 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 $m^{\ast} = 3$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19567 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model Marchetti, Gionni Statistical Mechanics Machine Learning We address the intrinsic dimensionality (ID) of high-dimensional trajectories, comprising $n_s = 4\,000\,000$ 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 $m^{\ast} = 2$ 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 $m^{\ast} = 3$ 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. |
| title | Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model |
| topic | Statistical Mechanics Machine Learning |
| url | https://arxiv.org/abs/2601.19567 |