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

Gionni Marchetti

TL;DR

This work addresses how the intrinsic dimensionality of high-dimensional FPUT trajectories depends on nonlinearity. It employs PCA-based reconstruction error and multiple estimators of $m^{\ast}$ on $n_s=4{,}000{,}000$ samples from the $N=32$ FPUT-$\beta$ model, complemented by $t$-SNE visualizations to reveal nonlinear structure. The findings show that $m^{\ast}$ increases with $\beta$ and energy density $\epsilon$, with weak nonlinearity yielding $m^{\ast}\approx 2$–$3$ (consistent with a low-dimensional quasi-periodic manifold) and strong nonlinearity yielding $m^{\ast}\approx 37}$, indicating a progression toward thermalization. The study highlights both the utility and limits of linear PCA for probing Hamiltonian dynamics and motivates nonlinear manifold learning and topological/geometric data analysis for a more complete understanding of the FPUT dynamics.

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 $β$ 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.

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

TL;DR

This work addresses how the intrinsic dimensionality of high-dimensional FPUT trajectories depends on nonlinearity. It employs PCA-based reconstruction error and multiple estimators of on samples from the FPUT- model, complemented by -SNE visualizations to reveal nonlinear structure. The findings show that increases with and energy density , with weak nonlinearity yielding – (consistent with a low-dimensional quasi-periodic manifold) and strong nonlinearity yielding , indicating a progression toward thermalization. The study highlights both the utility and limits of linear PCA for probing Hamiltonian dynamics and motivates nonlinear manifold learning and topological/geometric data analysis for a more complete understanding of the FPUT dynamics.

Abstract

A data-driven approach based on unsupervised machine learning is proposed to infer the intrinsic dimension of the high-dimensional trajectories of the Fermi-Pasta-Ulam-Tsingou (FPUT) model. Principal component analysis (PCA) is applied to trajectory data consisting of datapoints, of the FPUT model with coupled oscillators, revealing a critical relationship between and the model's nonlinear strength. By estimating the intrinsic dimension using multiple methods (participation ratio, Kaiser rule, and the Kneedle algorithm), it is found that increases with the model nonlinearity. Interestingly, in the weakly nonlinear regime, for trajectories initialized by exciting the first mode, the participation ratio estimates , strongly suggesting that quasi-periodic motion on a low-dimensional Riemannian manifold underlies the characteristic energy recurrences observed in the FPUT model.

Paper Structure

This paper contains 9 sections, 12 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The Fermi-Pasta-Ulam-Tsingou $\beta$ Model
  3. Methods
  4. PCA and the Reconstruction Error
  5. t-Distributed Stochastic Neighbor Embedding
  6. Results and Discussion
  7. Conclusion
  8. Initial Conditions and Simulations
  9. PCA Results

Figures (15)

  • Figure 1: The energy $E_k$ of modes for $k=1, 3, 5$ as a function of time $t$ in units of recurrence time $t_r$ ($t_r = 2 \times 10^{5}$) for $\beta$ model with $\beta=0.3$, assuming $N=32$. The system's equations of motion were numerically integrated with size step $h=0.05$. The initial condition is set to provide the energy $\mathcal{E}_1 \approx 0.45$ to the first normal mode ($k=1, A=10$).
  • Figure 2: The energy of modes $E_k$ for $k = 1, 2, 3, 4$ as a function of time $t$ for $\beta$ model with $\beta=3$, assuming $N=32$. The system's equations of motion were numerically integrated with size step $h=0.05$. The initial condition is set to provide the energy $\mathcal{E}_1 \approx 0.45$ to the first normal mode ($k=1, A=10$).
  • Figure 3: $t$-SNE embeddings of the entire trajectories of early-stage dynamics, with $n_s = 10,000$ and $n_s = 2,000$ data points corresponding to $\beta = 0.1$ and $\beta = 0.5, 1$, respectively. The trajectory data were generated with the initial condition $k=1$, $A=10$. The top panels (a), (b), (c) and bottom panels (d), (e), (f) show embeddings obtained using Euclidean distance and Cosine distance, respectively. PCA initialization was used throughout, and $\tau_p = 50$.
  • Figure 4: Reconstruction error $J_m$ in percentage ($\%$) as a function of the dimension $m$ of the best-fitting subspace $U$ for $\beta \in [0.1, 3]$, using trajectories of $N=32$ coupled oscillators, consisting of $n_s = 4, 000, 000$ data points, assuming the initial condition equivalent to giving the energy $\mathcal{E}_1 \approx 0.45$ to the first mode ($k=1, A=10$). Note that the zero of the horizontal axis is set at $m =1$. (Inset) The same plot for $m \in [60, 63]$ shows the curves corresponding to $\beta \in [2.4, 3]$.
  • Figure 5: Reconstruction error $J_m$ in percentage ($\%$) as a function of the dimension $m$ of the best-fitting subspace $U$ for $\beta \in [0.1, 3]$, using trajectories of $N=32$ coupled oscillators, consisting of $n_s = 4, 000, 000$ data points, assuming the initial condition equivalent to giving the energy $\mathcal{E}_1 \approx 1.8$ to the second mode ($k=2, A=10$). Note that the zero of the horizontal axis is set at $m =1$. (Inset) The same plot for $m \in [60, 63]$ shows the curves corresponding to $\beta \in [2.4, 3]$.
  • ...and 10 more figures