Universal Approximation of Mean-Field Models via Transformers

Abstract: This paper investigates the use of transformers to approximate the mean-field dynamics of interacting particle systems exhibiting collective behavior. Such systems are fundamental in modeling phenomena across physics, biology, and engineering, including opinion formation, biological networks, and swarm robotics. The key characteristic of these systems is that the particles are indistinguishable, leading to permutation-equivariant dynamics. First, we empirically demonstrate that transformers are well-suited for approximating a variety of mean field models, including the Cucker-Smale model for flocking and milling, and the mean-field system for training two-layer neural networks. We validate our numerical experiments via mathematical theory. Specifically, we prove that if a finite-dimensional transformer effectively approximates the finite-dimensional vector field governing the particle system, then the $L_2$ distance between the \textit{expected transformer} and the infinite-dimensional mean-field vector field can be uniformly bounded by a function of the number of particles observed during training. Leveraging this result, we establish theoretical bounds on the distance between the true mean-field dynamics and those obtained using the transformer.