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：跨越各科学领域的动力系统，从电路到生态网络，当其基本参数跨越阈值时，会经历行为的定性且往往是灾难性的变化，称为分岔。现有方法可预测单个系统中的逼近灾难，但主要基于时间序列，且难以对跨不同系统的定性动力学状态进行分类，也难以推广到真实数据。为应对这一挑战，我们提出了一种数据驱动、物理信息的深度学习框架，用于基于拓扑不变特征的提取来分类动力学状态并刻画分岔边界。我们聚焦于超临界霍普夫分岔这一典范情形，该情形被广泛用于模拟周期动力学。我们的卷积注意力方法通过数据增强进行训练，鼓励学习可用于检测未知系统中分岔边界的拓扑不变量，并设计如振荡基因调控网络等生物系统模型。我们进一步通过基于单细胞数据恢复基因表达空间中胰腺内分泌发生轨迹上的独特增殖与分化动力学，展示了该方法在分析真实数据中的应用。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，并能检测大规模物理和生物系统中的分岔或灾难性转变。