Dynamically Meaningful Latent Representations of Dynamical Systems

Abstract

Dynamical systems are ubiquitous in the physical world and are often well-described by partial differential equations (PDEs). Despite their formally infinite-dimensional solution space, a number of systems have long time dynamics that live on a low-dimensional manifold. However, current methods to probe the long time dynamics require prerequisite knowledge about the underlying dynamics of the system. In this study, we present a data-driven hybrid modeling approach to help tackle this problem by combining numerically derived representations and latent representations obtained from an autoencoder. We validate our latent representations and show they are dynamically interpretable, capturing the dynamical characteristics of qualitatively distinct solution types. Furthermore, we probe the topological preservation of the latent representation with respect to the raw dynamical data using methods from persistent homology. Finally, we show that our framework is generalizable, having been successfully applied to both integrable and non-integrable systems that capture a rich and diverse array of solution types. Our method does not require any prior dynamical knowledge of the system and can be used to discover the intrinsic dynamical behavior in a purely data-driven way.