MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data

2026-03-31

AI Summary

This work addresses the problem of learning collective dynamics driven by probability measure–dependent interactions from particle trajectory data by introducing a measure-valued neural network. The proposed method employs an embedding network to learn cylindrical features that map probability measures to vector representations, thereby enabling the first end-to-end modeling of McKean–Vlasov–type dynamics. Theoretically, the authors establish well-posedness and propagation of chaos for the associated system and prove universal approximation capabilities of the network along with quantitative convergence rates under a low-dimensional measure dependence assumption. Numerical experiments demonstrate that the model achieves high predictive accuracy and strong out-of-distribution generalization across a range of first- and second-order systems, including Cucker–Smale, Motsch–Tadmor, two-dimensional attraction–repulsion, and multi-group hierarchical models.

Abstract

Collective behaviors that emerge from interactions are fundamental to numerous biological systems. To learn such interacting forces from observations, we introduce a measure-valued neural network that infers measure-dependent interaction (drift) terms directly from particle-trajectory observations. The proposed architecture generalizes standard neural networks to operate on probability measures by learning cylindrical features, using an embedding network that produces scalable distribution-to-vector representations. On the theory side, we establish well-posedness of the resulting dynamics and prove propagation-of-chaos for the associated interacting-particle system. We further show universal approximation and quantitative approximation rates under a low-dimensional measure-dependence assumption. Numerical experiments on first and second order systems, including deterministic and stochastic Motsch-Tadmor dynamics, two-dimensional attraction-repulsion aggregation, Cucker-Smale dynamics, and a hierarchical multi-group system, demonstrate accurate prediction and strong out-of-distribution generalization.

Problem

• McKean-Vlasov dynamics
• measure-valued learning
• interacting particle systems
• collective behavior
• drift estimation

Innovation

• measure-valued neural network
• McKean-Vlasov dynamics
• cylindrical features
• propagation of chaos
• distribution-to-vector embedding