PHASE2VEC: DYNAMICAL SYSTEMS EMBEDDING WITH A PHYSICS-INFORMED CONVOLUTIONAL NETWORK

Abstract

Dynamical systems are found in innumerable forms across the physical and biological sciences, yet all these systems fall naturally into equivalence classes: conservative or dissipative, stable or unstable, compressible or incompressible. Predicting these classes from data remains an essential open challenge in computational physics on which existing time-series classification methods struggle. Here, we propose, phase2vec, an embedding method that learns highquality, physically-meaningful representations of low-dimensional dynamical systems without supervision. Our embeddings are produced by a convolutional backbone that extracts geometric features from flow data and minimizes a physicallyinformed vector field reconstruction loss. The trained architecture can not only predict the equations of unseen data, but also produces embeddings that encode meaningful physical properties of input data (e.g. stability of fixed points, conservation of energy, and the incompressibility of flows) more faithfully than standard blackbox classifiers and state-of-the-art time series classification techniques. We additionally apply our embeddings to the analysis of meteorological data, showing we can detect climatically meaningful features. Collectively, our results demonstrate the viability of embedding approaches for the discovery of dynamical features in physical systems.

1. INTRODUCTION

The application of deep neural networks to the prediction (Lusch et al., 2018 ), control (Haluszczynski & Räth, 2021 ), and basic understanding (Raissi et al., 2019) of dynamical systems has spurred important advances across the sciences, from neuroscience (Sussillo et al., 2016) to physics (Karniadakis et al., 2021) , from earth and climate science (Reichstein et al., 2019; Fresca et al., 2020) to computational biology (Sapoval et al., 2022) . However, while deep learning has already contributed significantly to the analysis of single systems, its role in understanding the underlying principles of dynamical systems in general remains somewhat limited. The ability to predict or control one system, after all, tells us little about how to do so with another. Crucially, without the ability to generalize across behaviors of numerous dynamical systems, the problem of constructing new systems is quite difficult, and most current attempts rely on exhaustive search through system parameters (Hart et al., 2012; Scholes et al., 2019) .

