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.

翻译：从电路到生态网络，各科学领域的动力系统在底层参数跨越阈值时，会经历定性且往往灾难性的行为变化，称为分岔。现有方法能预测单个系统的即将发生的灾难，但主要基于时间序列，难以对不同系统的定性动力体制进行分类，也难以泛化到真实数据。为应对这一挑战，我们提出了一种数据驱动、基于物理信息的深度学习框架，通过提取拓扑不变特征来分类动力体制并表征分岔边界。我们聚焦于超临界霍普夫分岔这一典范情形，该分岔广泛用于模拟多种应用中的周期动力学。我们的卷积注意力方法通过数据增广进行训练，鼓励学习可用于检测未知系统分岔边界的拓扑不变量，并设计生物系统模型，如振荡基因调控网络。我们进一步展示了该方法在分析真实数据中的应用，基于单细胞数据，在基因表达空间中恢复了胰腺内分泌发生轨迹中不同的增殖和分化动态。我们的方法为广泛动力系统的定性长期行为提供了宝贵见解，并能在大规模物理和生物系统中检测分岔或灾难性转变。