Replicating chaotic characteristics of non-linear dynamics by machine
learning (ML) has recently drawn wide attentions. In this work, we propose that
a ML model, trained to predict the state one-step-ahead from several latest
historic states, can accurately replicate the bifurcation diagram and the
Lyapunov exponents of discrete dynamic systems. The characteristics for
different values of the hyper-parameters are captured universally by a single
ML model, while the previous works considered training the ML model
independently by fixing the hyper-parameters to be specific values. Our
benchmarks on the one- and two-dimensional Logistic maps show that variational
quantum circuit can reproduce the long-term characteristics with higher
accuracy than the long short-term memory (a well-recognized classical ML
model). Our work reveals an essential difference between the ML for the chaotic
characteristics and that for standard tasks, from the perspective of the
relation between performance and model complexity. Our results suggest that
quantum circuit model exhibits potential advantages on mitigating over-fitting,
achieving higher accuracy and stability.

利用机器学习模型，通过训练预测离散动力系统多个历史状态后一步的状态，可以准确地复制分叉图和 Lyapunov 指数；我们的结果表明，量子电路模型在减少过拟合、提高准确性和稳定性方面具有潜在优势。

经典和量子机器学习对混沌特性的通用复现

Universal replication of chaotic characteristics by classical and  quantum machine learning

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

Tipping Points of Evolving Epidemiological Networks: Machine  Learning-Assisted, Data-Driven Effective Modeling

Deriving closed-form, analytical expressions for reduced-order models, and
judiciously choosing the closures leading to them, has long been the strategy
of choice for studying phase- and noise-induced transitions for agent-based
models (ABMs). In this paper, we propose a data-driven framework that pinpoints
phase transitions for an ABM in its mean-field limit, using a smaller number of
variables than traditional closed-form models. To this end, we use the manifold
learning algorithm Diffusion Maps to identify a parsimonious set of data-driven
latent variables, and show that they are in one-to-one correspondence with the
expected theoretical order parameter of the ABM. We then utilize a deep
learning framework to obtain a conformal reparametrization of the data-driven
coordinates that facilitates, in our example, the identification of a single
parameter-dependent ODE in these coordinates. We identify this ODE through a
residual neural network inspired by a numerical integration scheme (forward
Euler). We then use the identified ODE -- enabled through an odd symmetry
transformation -- to construct the bifurcation diagram exhibiting the phase
transition.

通过降阶模型、数据驱动框架、流形学习算法、深度学习框架和分叉图表，本文提出了一种定位基于代理模型 (ABMs) 的平均场极限相变的方法。