In this study, we use machine learning to classify and interpolate the phase structure of the Vicsek flocking model across the three-dimensional parameter space $(η,ρ,v_0)$. We construct a dataset of simulated parameter points and characterize each point using long-time dynamical observables. These observables are then used as inputs to a K-Means clustering procedure, which assigns each point to a disorder, order, or coexistence phase. Using these clustered labels, we train a neural-network classifier to learn the mapping from model parameters to phase behavior, achieving a classification accuracy of 0.92. The resulting phase map resolves a narrow coexistence region separating the ordered and disordered phases and extends the inferred phase boundaries beyond the originally sampled simulation points. More broadly, this approach provides a systematic way to convert sparse simulation data into a global phase diagram for collective-motion models.

翻译：在本研究中，我们利用机器学习方法在三维参数空间$(η,ρ,v_0)$中对维切克 flocking 模型的相结构进行分类和插值。我们构建了一个模拟参数点数据集，并通过长时间动力学观测量对每个点进行表征。这些观测量随后作为K均值聚类过程的输入，将每个点归类为无序相、有序相或共存相。利用这些聚类标签，我们训练了一个神经网络分类器，学习从模型参数到相行为的映射，分类准确率达到0.92。由此得到的相图解析出了分隔有序相和无序相的狭窄共存区域，并将推断出的相边界扩展到了原始采样模拟点之外。更广泛而言，这种方法为将稀疏模拟数据转化为集体运动模型的全局相图提供了一种系统途径。