# Let's do the time-warp-attend: Learning topological invariants of dynamical systems

arXiv: https://arxiv.org/abs/2312.09234 · HTML: https://mlsanity.com/paper/2312.09234/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 55 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Builds on (most cited first, 8 of 8)

- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Attractor and integrator networks in the brain](https://mlsanity.com/paper/2112.03978/) (2021)
- [Predicting discrete-time bifurcations with deep learning](https://mlsanity.com/paper/2303.09669/) (2023)
- [Learning the Koopman Eigendecomposition: A Diffeomorphic Approach](https://mlsanity.com/paper/2110.07786/) (2021)
- [Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems](https://mlsanity.com/paper/2009.08972/) (2020)
- [Deep Learning of Conjugate Mappings](https://mlsanity.com/paper/2104.01874/) (2021)
- [Phase2vec: Dynamical systems embedding with a physics-informed convolutional network](https://mlsanity.com/paper/2212.03857/) (2022)

## Built on by (most cited first, 2 of 2)

- [Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences](https://mlsanity.com/paper/2503.10336/) (2025)
- [MODE: Learning compositional representations of complex systems with Mixtures Of Dynamical Experts](https://mlsanity.com/paper/2510.09594/) (2025)

## Abstract (arXiv)

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.