Data-driven stability analysis of chaotic systems in latent spaces

The spatio-temporal dynamics of turbulent and chaotic systems are inherently unstable, which makes the design of accurate reduced-order models for forecasting challenging. We employ a data-driven framework to separate observations into spatial and temporal components, using a convolutional autoencoder (CAE) to compute a latent space representing the chaotic dynamics. An echo state network (ESN) predicts the temporal evolution of the latent representation. With the CAE-ESN, we perform stability analysis in the latent manifold from data only. The Lyapunov spectrum, Kaplan-Yorke dimension, and covariant Lyapunov vectors are computed to analyse the chaotic properties and geometric structure of the latent manifold. The analysis is performed on the Kuramoto-Sivashinsky equation, where it produces a latent space that preserves key properties of the chaotic system, thus retaining the geometric structure of the attractor in the latent space. Finally, the method is applied to a direct numerical simulation of the turbulent 2D Kolmogorov flow. This work opens new opportunities for analysing the stability properties of chaotic systems from data only.