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.

Figures (17)

• Figure 1: 1D Motsch-Tadmor dynamics: Empirical density $\rho(x,t)$ for the 1D Motsch-Tadmor dynamics: comparison between the reference $N$-particle simulation (orange) and the MVNN-learned mean-field model (blue). Columns show $t=0,1,2$; rows correspond to three unseen initial distributions. Densities are estimated using Gaussian kernel density estimation. The $L^2$ error is computed between the KDE-smoothed densities.
• Figure 2: Comparisons on 1D Motsch-Tadmor dynamics: Empirical density $\rho(x,t)$ for the 1D Motsch-Tadmor dynamics: comparison between the reference $N$-particle simulation (green), the MVNN-learned mean-field model (orange), and the prediction from the Gaussian process model feng2023learning (blue). Columns show $t=0,0.5,1$; rows correspond to three unseen initial distributions. Densities are estimated using Gaussian kernel density estimation.
• Figure 3: Comparisons on 1D Motsch-Tadmor dynamics: $L^2$ error for the 1D Motsch-Tadmor dynamics: comparison of the $L^2$ error for the Gaussian process model (blue), the MVNN model trained on $16$ agents, $9$ trajectories, and $20$ timesteps (orange), and the MVNN model trained on $16,000$ agents, $100$ trajectories, and $200$ timesteps (green). The $L^2$ error is computed between the KDE-smoothed densities. Columns correspond to three unseen initial distributions.
• Figure 4: Simulation Time Comparison: Comparison of average simulation times (seconds) against number of agents $N$ for the MVNN and Gaussian process models over 10 trials.
• Figure 5: Stochastic Motsch-Tadmor dynamics ($\sigma=0.1$): density evolution. Empirical density $\rho(x,t)$ from the reference interacting-particle simulation (orange) and from the MVNN-learned McKean-Vlasov model (blue), shown at $t=0,1,2$ for an unseen initial distribution. Densities are estimated via Gaussian KDE, and the reported $L^2$ error is computed between the kernel-smoothed densities.

Theorems & Definitions (14)

• Proposition 2.1: Well-Posedness of MVNN-Induced McKean--Vlasov Dynamics
• Proposition 2.2: Mean-Field Convergence and Propagation of Chaos for the Learned Particle System
• Theorem 2.3: Universal Approximation for Measure Valued Neural Network
• Theorem 2.4: Quantitative Approximation via Finite-Dimensional Measure Embeddings
• Remark 2.5
• Remark 2.6
• Theorem 2.7: Approximation Rate of MVNN under Low-Dimensional Assumption
• proof
• proof
• Theorem C.1