Ever pondered why ducks fly in a V-formation or why fish school together? These are instances of self-organization, a process where a system initially in disarray evolves into an ordered pattern through the individual interactions of its constituent agents, without any central coordination.

Typically, such behavior emerges when the system comprises a large number of agents, and the system’s local fluctuations can be overlooked in favor of macroscopic dynamics. As the number of agents approaches infinity, this limit can be expressed using a non-linear Partial Differential Equation (PDE), often referred to as the mean-field limit.

This project aims to quantitatively understand when the mean-field limit serves as an accurate descriptor of the system and the microscopic system’s fluctuations around this limit. We hope to apply this theory to characterize the convergence of certain machine learning algorithms quantitatively.