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

TL;DR

This paper introduces a measure-valued neural network that learns interaction forces in collective biological systems from particle data, with theoretical guarantees and successful numerical experiments.

Contribution

It develops a neural network architecture operating on probability measures, with proven well-posedness, universal approximation, and demonstrated effectiveness on various systems.

Findings

Accurately predicts collective dynamics from particle data.

Proves well-posedness and propagation-of-chaos for the model.

Shows strong out-of-distribution generalization in experiments.

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.