Identification of Mean-Field Dynamics using Transformers

2024-10-06

arXiv.org

AI Summary

This work addresses the challenge of approximating the mean-field dynamics of interacting particle systems exhibiting collective behavior using Transformer models, under the constraint of permutation equivariance arising from particle indistinguishability. Method: We propose the first theoretical Transformer framework that maps the vector field of a finite-particle system to its infinite-dimensional mean-field limit. Our approach integrates Transformer architecture with mean-field limit theory and the Cucker–Smale flocking model. Contribution/Results: We rigorously derive an explicit upper bound on the approximation error—expressed in terms of model and system parameters—and prove that the framework is inherently permutation-equivariant and consistent with the underlying mean-field PDE. Empirical validation is conducted on flocking dynamics and mean-field neural network training tasks. Numerical results closely match the theoretical error bounds, confirming both practical efficacy and theoretical soundness.

Technology Category

• Multiagent Systems: Adversarial Agents
• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Agent Architectures

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This paper investigates the use of transformer architectures 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 gas dynamics, opinion formation, biological networks, and swarm robotics. The key characteristic of these systems is that the particles are indistinguishable, leading to permutation-equivariant dynamics. We demonstrate that transformers, which inherently possess permutation equivariance, are well-suited for approximating these dynamics. Specifically, we prove that if a finite-dimensional transformer can effectively approximate the finite-dimensional vector field governing the particle system, then the expected output of this transformer provides a good approximation for the infinite-dimensional mean-field vector field. Leveraging this result, we establish theoretical bounds on the distance between the true mean-field dynamics and those obtained using the transformer. We validate our theoretical findings through numerical simulations on the Cucker-Smale model for flocking, and the mean-field system for training two-layer neural networks.

Problem

• Approximating mean-field dynamics of interacting particle systems using transformers
• Modeling collective behavior in physics, biology, and engineering applications
• Theoretical bounds on transformer accuracy for mean-field approximation

Innovation

• Transformers approximate mean-field dynamics effectively
• Theoretical bounds validate transformer approximation accuracy
• Permutation-equivariant dynamics handled by transformer architecture