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

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.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Computational Physics (physics.comp-ph)

arXiv:2604.00333 [math.NA]

(or arXiv:2604.00333v3 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2604.00333 Focus to learn more arXiv-issued DOI via DataCite

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