Bifurcations mark qualitative changes of long-term behavior in dynamical
systems and can often signal sudden ("hard") transitions or catastrophic events
(divergences). Accurately locating them is critical not just for deeper
understanding of observed dynamic behavior, but also for designing efficient
interventions. When the dynamical system at hand is complex, possibly noisy,
and expensive to sample, standard (e.g. continuation based) numerical methods
may become impractical. We propose an active learning framework, where Bayesian
Optimization is leveraged to discover saddle-node or Hopf bifurcations, from a
judiciously chosen small number of vector field observations. Such an approach
becomes especially attractive in systems whose state x parameter space
exploration is resource-limited. It also naturally provides a framework for
uncertainty quantification (aleatoric and epistemic), useful in systems with
inherent stochasticity.

定位动力系统的分歧点对于深入理解观察到的动态行为以及设计高效干预手段至关重要。在复杂、可能存在噪声且采样成本高昂的动力系统中，我们提出了一种主动学习框架，利用贝叶斯优化从少量选择的矢量场观察中发现鞍点或者 Hopf 分歧。在资源有限的系统的状态 - 参数空间探索中，这种方法尤为具有吸引力。它还自然地提供了不确定性量化的框架（aleatoric 和 epistemic），对于具有固有随机性的系统非常有用。