Deep Learning of Conjugate Mappings
This work addresses a fundamental challenge in chaos theory for researchers studying dynamical systems, offering a novel computational tool, though it appears incremental as it builds on existing autoencoder and deep learning techniques.
The paper tackles the problem of obtaining explicit Poincaré mappings for chaotic dynamical systems, which are typically implicit, by using deep learning to learn invertible coordinate transformations that simplify the dynamics into conjugate representations; it demonstrates applications on systems like the Rössler and Lorenz equations, showing utility in classifying chaos through topological conjugacies.
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 Poincaré 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 Poincaré 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 Poincaré 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 Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto--Sivashinsky equation.