用神经网络从粒子轨迹中学习群体交互力，能准确预测复杂系统行为。
MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data
- 构建测度值神经网络，直接从粒子轨迹推断依赖分布的交互力。
- 在多类动力系统上实现高精度预测与强泛化能力，包括随机与确定性模型。
- 适合研究群体智能、生物集群、多智能体系统等方向的学者参考。
由相互作用引发的集体行为是众多生物系统的核心特征。为从观测数据中学习此类相互作用力，本文提出一种测度值神经网络，可直接从粒子轨迹观测中推断依赖测度的交互项（漂移项）。该架构通过学习柱状特征，将概率测度映射为可扩展的向量表示，从而将标准神经网络推广至测度空间。理论上，我们证明了所导出动力学的适定性，并建立了关联粒子系统的混沌传播性质。在低维测度依赖假设下，还证明了通用逼近性及量化逼近速率。数值实验涵盖一阶与二阶系统，包括确定性与随机的Motsch-Tadmor模型、二维吸引-排斥聚集、Cucker-Smale动力学以及分层多组系统，结果表明该方法具有高精度预测能力和出色的分布外泛化性能。
原文摘要 · Abstract (English)
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.
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