We study the tipping point collective dynamics of an adaptive susceptible-infected-susceptible (SIS) epidemiological network in a data-driven, machine learning-assisted manner. We identify a parameter-dependent effective stochastic differential equation (eSDE) in terms of physically meaningful coarse mean-field variables through a deep-learning ResNet architecture inspired by numerical stochastic integrators. We construct an approximate effective bifurcation diagram based on the identified drift term of the eSDE and contrast it with the mean-field SIS model bifurcation diagram. We observe a subcritical Hopf bifurcation in the evolving network's effective SIS dynamics, that causes the tipping point behavior; this takes the form of large amplitude collective oscillations that spontaneously -- yet rarely -- arise from the neighborhood of a (noisy) stationary state. We study the statistics of these rare events both through repeated brute force simulations and by using established mathematical/computational tools exploiting the right-hand-side of the identified SDE. We demonstrate that such a collective SDE can also be identified (and the rare events computations also performed) in terms of data-driven coarse observables, obtained here via manifold learning techniques, in particular Diffusion Maps. The workflow of our study is straightforwardly applicable to other complex dynamics problems exhibiting tipping point dynamics.

通过机器学习辅助的数据驱动方法，研究了自适应易感-感染-易感（SIS）流行病学网络的临界点集体动力学。使用深度学习ResNet架构识别出参数依赖的有效随机微分方程(eSDE)，并构建了基于eSDE漂移项的近似有效分支图，与均场SIS模型的分支图进行对比。观察到演化网络的有效SIS动力学中存在亚临界Hopf分支，导致临界点行为；这形成了大幅振荡的集体现象，很少从（嘈杂的）稳定状态附近自发出现。通过重复蛮力模拟和利用已建立的数学/计算工具分析识别的SDE的右侧进行稀有事件统计分析。还展示了通过基于流形学习技术（特别是扩散映射）获得的数据驱动的粗粒度可观测量也可以识别出这样的集体SDE（并进行稀有事件计算）。我们的研究工作流程可以直接应用于其他具有临界点动力学的复杂系统。

流行病学网络的临界转折点：机器学习辅助的数据驱动有效建模