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

arXiv: https://arxiv.org/abs/2212.03857 · HTML: https://mlsanity.com/paper/2212.03857/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 7 Dec 2022.

## Builds on (most cited first, 17 of 17)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Generalizing to Unseen Domains: A Survey on Domain Generalization](https://mlsanity.com/paper/2103.03097/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Learning to Adapt in Dynamic, Real-World Environments Through Meta-Reinforcement Learning](https://mlsanity.com/paper/1803.11347/) (2018)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Context-aware Dynamics Model for Generalization in Model-Based Reinforcement Learning](https://mlsanity.com/paper/2005.06800/) (2020)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Generalizing to New Physical Systems via Context-Informed Dynamics Model](https://mlsanity.com/paper/2202.01889/) (2022)
- [LEADS: Learning Dynamical Systems that Generalize Across Environments](https://mlsanity.com/paper/2106.04546/) (2021)
- [LFADS - Latent Factor Analysis via Dynamical Systems](https://mlsanity.com/paper/1608.06315/) (2016)
- [High-dimensional Convolutional Networks for Geometric Pattern Recognition](https://mlsanity.com/paper/2005.08144/) (2020)
- [Deep learning-based reduced order models in cardiac electrophysiology](https://mlsanity.com/paper/2006.03040/) (2020)
- [Controlling nonlinear dynamical systems into arbitrary states using machine learning](https://mlsanity.com/paper/2102.12969/) (2021)
- [Deep Learning of Conjugate Mappings](https://mlsanity.com/paper/2104.01874/) (2021)
- [Input-output relations in biological systems: measurement, information and the Hill equation](https://mlsanity.com/paper/1312.2838/) (2013)
- [Recurrent Neural Network Based Hybrid Model of Gene Regulatory Network](https://mlsanity.com/paper/1408.5405/) (2014)

## Built on by (most cited first, 5 of 5)

- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Let's do the time-warp-attend: Learning topological invariants of dynamical systems](https://mlsanity.com/paper/2312.09234/) (2023)
- [Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences](https://mlsanity.com/paper/2503.10336/) (2025)
- [Detangled: A Framework for Creating, Editing, and Inferencing Feature Rich Hair Strands](https://mlsanity.com/paper/2607.09811/) (2026)

## Abstract (arXiv)

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, textttphase2vec, 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.