MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data
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| Format: | Preprint |
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2026
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| _version_ | 1866911571112886272 |
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| author | Lyu, Liyao Yu, Xinyue Schaeffer, Hayden |
| author_facet | Lyu, Liyao Yu, Xinyue Schaeffer, Hayden |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00333 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data Lyu, Liyao Yu, Xinyue Schaeffer, Hayden Numerical Analysis Machine Learning Computational Physics 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. |
| title | MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data |
| topic | Numerical Analysis Machine Learning Computational Physics |
| url | https://arxiv.org/abs/2604.00333 |