Analysis of chaotic dynamical systems with autoencoders

TL;DR

The paper addresses extracting minimal, information-rich representations of chaotic time series using chaotic autoencoders with an $\alpha$-guided sparse latent space. By combining delay-coordinate embedding with a symmetric encoder/decoder and optimizing a loss $L = \text{MSE} + \alpha |h(x)|$, the authors identify latent dimensions that faithfully capture dynamics, as evidenced by $LLE$ values closely matching those of the original systems. Tested on Rössler, Lorenz63, and Lorenz96, the approach reveals latent-space sizes that are typically smaller than the embedding dimension and demonstrates robust dynamical fidelity across varying input window lengths $W$. The work provides a data-driven, unsupervised method to quantify dynamical complexity and offers a practical path to reduced representations of chaotic systems while preserving essential chaotic characteristics. This has potential implications for efficient modeling, analysis, and interpretation of complex dynamical processes in physics and beyond.

Abstract

We focus on chaotic dynamical systems and analyze their time series with the use of autoencoders, i.e., configurations of neural networks that map identical output to input. This analysis results in the determination of the latent space dimension of each system and thus determines the minimal number of nodes necessary to capture the essential information contained in the chaotic time series. The constructed chaotic autoencoders generate similar maximal Lyapunov exponents as the original chaotic systems and thus encompass their essential dynamical information.

Figures (4)

• Figure 1: A schematic representation of the proposed model. The trajectories of a given attractor are transformed into input sequences, $x$, that pass through an encoder model, $h(x)$, get compressed into the latent space and pass through the decoder, $g(h)$, to reconstruct the original information, $\hat{x} = g(h(x))$.
• Figure 2: The evolution of the training (red dashed line) and the test set (solid black line) loss in logarithmic scale, as a function the number of epochs of training for the Lorenz63 system.
• Figure 3: The average number of non-zero nodes at the latent space (navy blue solid line), including the standard deviation of it (error bars) and the corresponding value of the loss function (red dashed line) of the test data as a function of the L1 regularization parameter, $\alpha$, for the three systems.
• Figure 4: The average number of non-zero nodes at the latent space (navy blue solid line), including the standard deviation of it (error bars) and the corresponding value of the loss function (red dashed line) of the test data as a function of the size of the input sequence, $W$, for the three systems.