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.

我们提出了一个基于数据驱动的、物理上可行的深度学习框架，用于分类动力学区域和表征分支边界，基于提取拓扑不变特征。我们还演示了该方法在分析真实数据中的使用，通过基于单细胞数据，在基因表达空间中恢复胰岛内分泌发育轨迹上的不同增殖和分化动态。我们的方法为各种动力学系统的定性长期行为提供了有价值的洞察，并能检测大规模物理和生物系统中的分叉或灾变转变。