A Mean Field Games Perspective on Evolutionary Clustering

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.