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

TL;DR

The paper addresses the intrinsic dimensionality of high-dimensional FPUT $\beta$-model trajectories using a nonlinear approach based on a deep autoencoder (DAE). This method overcomes the limitations of linear PCA by uncovering a low-dimensional nonlinear manifold, with an intrinsic dimension $m^*=2$ in the weakly nonlinear regime $\beta\lesssim 1$ and a rise to $m^*=3$ at $\beta=1.1$ due to symmetry breaking that excites even Fourier modes. The results demonstrate that the DAE provides more accurate reconstructions and can reveal dynamical transitions that PCA misses, highlighting the geometric and topological structure of FPUT trajectories. The approach offers a data-driven lens on nonlinear lattice dynamics and has implications for understanding ergodicity, KAM-related structures, and energy transfer in nonlinear chains.

Abstract

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) model is employed to infer the ID in the weakly nonlinear regime ($β\lesssim 1$). We find that the trajectories lie on a nonlinear manifold of dimension $m^{\ast} = 2$ embedded in a $64$-dimensional phase space. The DAE further reveals that this dimensionality increases to $m^{\ast} = 3$ at $β= 1.1$, coinciding with a symmetry breaking transition, in which additional energy modes with even wave numbers $k = 2, 4$ become excited. Finally, we discuss the limitations of the linear approach based on principal component analysis (PCA), which fails to capture the underlying structure of the data and therefore yields unreliable results in most cases.

Figures (10)

• Figure 1: The FPUT chain consists in a discrete system of equal mass points connected by springs. The coordinate $q_i$ represents the displacement of $i$-th point from equilibrium.
• Figure 2: Schematic view of a DAE model with five hidden layers. The input and output layers contain $n$ nodes, while the bottleneck layer has $m$ nodes ($m \ll n$). All hidden nodes use ReLU activation function, as indicated by the symbol inside the nodes. The output layer's nodes employ a linear activation function which is appropriate for regression.
• Figure 3: Training and validation MSE loss curves as functions of epoch for bottleneck sizes $m=1, 2$ and trajectory data with $\beta= 0.3$. (Inset) Validation MSE loss curves as functions of epoch for $m=3, 4, 5$ and same data. Throughout training, the deep autoencoder employs an early stopping with a patience of $30$.
• Figure 4: (Top panel) DAE test reconstruction error curves of the test data as functions of the dimension $m$ for $\beta= 0.1, 0.4, 0.7, 0.8, 0.9, 1.0$, and $1.1$. The PCA reconstruction error curve, corresponding to data with $\beta= 1.1$, is rescaled by a factor of $100$ for proper visualization. (Bottom panel) DAE reconstruction error curve for the test data corresponding to $\beta= 0.1$.
• Figure 5: Estimated intrinsic dimension $m^{\ast}$ as function of $\beta \in [0.1, 1.1]$, using DAE model with Kneedle algorithm (KA) and PCA with the following heuristics: Participation Ratio (PR), Kneedle algorithm (KA), and Kaiser criterion (KC).