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：跨科学领域的动力系统，从电路到生态网络，当其底层参数越过阈值时，会经历定性且通常是灾难性的行为变化，称为分岔。现有方法能预测单个系统中的即将发生的灾难，但主要基于时间序列，难以对跨不同系统的定性动力机制进行分类，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、基于物理信息的深度学习框架，用于对动力机制进行分类，并基于拓扑不变特征的提取来刻画分岔边界。我们聚焦于超临界霍普夫分岔这一典型情况，该模型广泛应用于模拟周期动力学。我们的卷积注意方法通过数据增强进行训练，鼓励学习可用于检测未见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。我们进一步展示了该方法在分析真实数据中的应用，即基于单细胞数据，在基因表达空间中恢复胰腺内分泌发生轨迹上不同的增殖和分化动力学。我们的方法为理解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。