Inferring stability properties of chaotic systems on autoencoders' latent spaces

TL;DR

This paper demonstrates that convolutional autoencoders combined with echo state networks can infer the stability properties and tangent space geometry of chaotic systems within their latent representations, enabling analysis of high-dimensional chaos.

Contribution

It introduces a method to infer stability and tangent space geometry of chaotic systems directly in the autoencoder's latent space using Lyapunov exponents and vectors.

Findings

CAE-ESN models accurately infer Lyapunov exponents.

The method reveals the geometry of the tangent space.

Stability properties are preserved in the latent space.

Abstract

The data-driven learning of solutions of partial differential equations can be based on a divide-and-conquer strategy. First, the high dimensional data is compressed to a latent space with an autoencoder; and, second, the temporal dynamics are inferred on the latent space with a form of recurrent neural network. In chaotic systems and turbulence, convolutional autoencoders and echo state networks (CAE-ESN) successfully forecast the dynamics, but little is known about whether the stability properties can also be inferred. We show that the CAE-ESN model infers the invariant stability properties and the geometry of the tangent space in the low-dimensional manifold (i.e. the latent space) through Lyapunov exponents and covariant Lyapunov vectors. This work opens up new opportunities for inferring the stability of high-dimensional chaotic systems in latent spaces.