We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by a variational cost functional rather than predefined statistical shapes, this continuous-time formulation provides a flexible basis for non-parametric cluster evolution. To validate the framework, we analyze the setting of time-dependent Gaussian mixtures, showing that the MFG dynamics recover the trajectories of the classical Expectation-Maximization (EM) algorithm while ensuring mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to regularize temporal fluctuations. Numerical experiments illustrate the stability of our approach and suggest a path toward more general non-parametric clustering applications where traditional EM methods may face limitations.

翻译：我们提出一种基于平均场博弈(Mean Field Games, MFG)的控制论框架，用于演化聚类问题。该框架突破了静态或启发式方法的局限，将问题建模为由耦合的Hamilton-Jacobi-Bellman方程和Fokker-Planck方程刻画的种群动力学博弈。该连续时间形式化方法由变分代价泛函驱动，而非依赖预定义的统计形态，从而为非参数化聚类演化提供灵活基础。为验证该框架，我们分析了时变高斯混合模型场景，证明MFG动力学在保证质量守恒的同时，能够恢复经典期望最大化(EM)算法的轨迹。此外，我们引入时间平均对数似然泛函以正则化时序波动。数值实验展示了该方法的稳定性，并揭示了其在传统EM方法受限场景下拓展至更通用非参数聚类应用的可能性。