Deep Learning of Conjugate Mappings

TL;DR

This paper introduces a deep learning approach to explicitly construct Poincaré maps for chaotic systems by learning invertible conjugate transformations, enabling better understanding and classification of chaos.

Contribution

It proposes a novel neural network method using autoencoders to find conjugate mappings, facilitating explicit Poincaré map construction for complex chaotic systems.

Findings

Successfully applied to low-dimensional systems like Rössler and Lorenz.

Extended the approach to infinite-dimensional systems such as Kuramoto--Sivashinsky.

Demonstrated improved understanding of chaotic dynamics through learned conjugacies.

Abstract

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincar\'e first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincar\'e map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincar\'e mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the R\"ossler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto--Sivashinsky equation.