Mean-Field Theory

Mean-field theory replaces a large collection of interacting units with dynamics for their shared distribution, making systems with many neurons, agents, or particles easier to analyze. In machine learning, it describes infinite-width neural-network training and dynamic mean-field behavior; related approaches use state evolution for approximate message passing and McKean–Vlasov equations for interacting particles, control, and games. Researchers derive these limits with gradient-flow, kinetic, and cavity methods, then test how well finite systems follow them using propagation-of-chaos bounds, particle simulations, and learning benchmarks. A central challenge is quantifying when the simplified population dynamics remain accurate for finite systems, especially with non-exchangeable interactions, common noise, and transitions between lazy and feature-learning regimes.