Phase2vec: Dynamical systems embedding with a physics-informed convolutional network

TL;DR

Phase2vec is a physics-informed convolutional network that learns meaningful embeddings of 2D dynamical systems, capturing physical properties and enabling classification without supervision, with applications in meteorology.

Contribution

It introduces a novel embedding method that encodes physical properties of dynamical systems using a convolutional network trained with a physics-informed loss.

Findings

Embeddings encode stability, conservation laws, and flow incompressibility more accurately than existing methods.

The method can predict underlying equations of unseen data.

Embeddings reveal climatically relevant features in meteorological data.

Abstract

Dynamical systems are found in innumerable forms across the physical and biological sciences, yet all these systems fall naturally into universal equivalence classes: conservative or dissipative, stable or unstable, compressible or incompressible. Predicting these classes from data remains an essential open challenge in computational physics at which existing time-series classification methods struggle. Here, we propose, \texttt{phase2vec}, an embedding method that learns high-quality, physically-meaningful representations of 2D dynamical systems without supervision. Our embeddings are produced by a convolutional backbone that extracts geometric features from flow data and minimizes a physically-informed vector field reconstruction loss. In an auxiliary training period, embeddings are optimized so that they robustly encode the equations of unseen data over and above the performance of a per-equation fitting method. The trained architecture can not only predict the equations of unseen data, but also, crucially, learns embeddings that respect the underlying semantics of the embedded physical systems. We validate the quality of learned embeddings investigating the extent to which physical categories of input data can be decoded from embeddings compared to standard blackbox classifiers and state-of-the-art time series classification techniques. We find that our embeddings encode important physical properties of the underlying data, including the stability of fixed points, conservation of energy, and the incompressibility of flows, with greater fidelity than competing methods. We finally 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.