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Inferring stability properties of chaotic systems on autoencoders' latent spaces

Elise Özalp, Luca Magri

TL;DR

It is shown that the CAE-ESN model infers the invariant stability properties and the geometry of the tangent space in the low-dimensional manifold through Lyapunov exponents and covariant Lyapunov vectors.

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.

Inferring stability properties of chaotic systems on autoencoders' latent spaces

TL;DR

It is shown that the CAE-ESN model infers the invariant stability properties and the geometry of the tangent space in the low-dimensional manifold through Lyapunov exponents and covariant Lyapunov vectors.

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.

Paper Structure

This paper contains 7 sections, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Learning stability properties in latent spaces
  3. Methods
  4. Convolutional autoencoder
  5. Echo state network
  6. Numerical tests
  7. Conclusions

Figures (3)

  • Figure 1: Illustration of the CAE-ESN with the Kuramoto-Sivashinsky data as an example.
  • Figure 2: (a) Autonomous prediction of the CAE-ESN on the latent manifold of dimension $N_{lat}=8$. (b) Convergence of the Lyapunov exponents of the CAE-ESN over $100LT$.
  • Figure 3: (a) The first $10$ Lyapunov exponents of the dynamical system (black squares) compared to the Lyapunov exponents across 10 different ESNs (red dots) after $100LT$. The shaded area indicates the standard deviation. (b - d) The angle distribution of the Kuramoto–Sivashinsky system for leading covariant Lyapunov vectors of the (b) unstable-neutral, (c) unstable-stable and (d) neutral-stable Lyapunov exponents.