math.NAApr 1, 2026
SaveMVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data
Organizations: Department of Mathematics University of California, Los Angeles Los Angeles, CA 90024, USA
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 & tables
Figure 1: Effect of training-data size on MVNN accuracy. (a) Time-averaged relative density error from autonomous rollouts. (b) Normalized error of the learned drift evaluated directly on reference state–measure pairs. The test set contains 30 independently generated held-out trajectories. Curves show the mean over three independently trained models, and the shaded regions show the corresponding uncertainty intervals.
Figure 2: 1D Motsch-Tadmor dynamics : Empirical density for the 1D Motsch-Tadmor dynamics: comparison between the reference -particle simulation (orange) and the MVNN-learned mean-field model (blue). Columns show ; rows correspond to three unseen initial distributions. Densities are estimated using Gaussian kernel density estimation. The error is computed between the KDE-smoothed densities.
Figure 3: Comparisons on 1D Motsch-Tadmor dynamics : Empirical density for the 1D Motsch-Tadmor dynamics: comparison between the reference -particle simulation (red), the MVNN-learned mean-field model trained on 16 agents, 9 trajectories, and 20 timesteps (orange), the MVNN-learned mean-field model trained on agents, 100 trajectories, and 200 timesteps (green) and the prediction from the Gaussian process model [ 88 ] (blue). Columns show ; rows correspond to three unseen initial distributions. Densities are estimated using Gaussian kernel density estimation.
Figure 4: Comparisons on 1D Motsch-Tadmor dynamics : error for the 1D Motsch-Tadmor dynamics: comparison of the error for the Gaussian process model (blue), the MVNN model trained on agents, trajectories, and timesteps (orange), and the MVNN model trained on agents, trajectories, and timesteps (green). The error is computed between the KDE-smoothed densities. Columns correspond to three unseen initial distributions.
Figure 5: Simulation Time Comparison : Comparison of average simulation times (seconds) against number of agents for the matched-data MVNN and Gaussian process models over 10 trials.
Figure 6: Comparisons on 1D Motsch-Tadmor dynamics : Empirical density for the 1D Motsch-Tadmor dynamics: comparison between the reference -particle simulation (red), the MVNN-learned mean-field model trained on agents, 100 trajectories, and 200 timesteps (blue), the prediction from mean-field WSINDy [ 56 ] (orange), and the prediction from online parameter estimation [ 51 ] (green). Columns show ; rows correspond to three unseen initial distributions. Densities are estimated using Gaussian kernel density estimation.
Figure 7: Stochastic Motsch-Tadmor dynamics ( ) : density evolution. Empirical density from the reference interacting-particle simulation (orange) and from the MVNN-learned McKean-Vlasov model (blue), shown at for an unseen initial distribution. Densities are estimated via Gaussian KDE, and the reported error is computed between the kernel-smoothed densities.
Figure 8: Comparisons on 1D Stochastic Motsch-Tadmor dynamics ( ) : Empirical density for the 1D Stochastic Motsch-Tadmor dynamics: comparison between the reference -particle simulation (red), the MVNN-learned mean-field model trained on agents, 100 trajectories, and 200 timesteps (blue), the prediction from mean-field WSINDy [ 56 ] (orange), and the prediction from online parameter estimation [ 51 ] (green). Columns show ; rows correspond to three unseen initial distributions. Densities are estimated using Gaussian kernel density estimation.
Figure 9: Comparison with WSINDy and online parameter estimation . The first three panels show the forecast densities at , , and for the learned MVNN dynamics, the B-spline mean-field WSINDy model, the local PDE-WSINDy model, and online parameter estimation. All methods use the same initial empirical distribution. The PDE models are initialized from its KDE reconstruction, and the same KDE observation operator is used to evaluate and plot the densities predicted by all methods. The rightmost panel shows the density forecast error versus time, averaged over all 36 held-out test trajectories. The shaded band spans the three independently trained MVNN models. The two additive-structure WSINDy baselines misplace and over-concentrate the bumps and the fitted parametric model displaces them, whereas MVNN tracks the reference throughout.
Figure 10: 2D aggregation model with ring-shaped initialization. The upper row displays the ground truth particle trajectories, while the lower row shows the evolution predicted by the learned mean-field dynamics. The model accurately preserves the topological structure of the ring over time.
Figure 11: 2D aggregation model with double-ring initialization. Comparison between the ground truth particle system (upper row) and the learned mean-field dynamics (lower row). The model correctly reproduces the contraction of both rings despite never seeing this topology during training.
Figure 12: 2D aggregation model with disk-shaped initialization. The learned dynamics (lower row) accurately capture the collapse of the uniform disk, matching the ground truth (upper row).
Figure 13: 2D aggregation model with binary asymmetric initialization. Evolution of a system initialized with heterogeneous densities (high density left, low density right). The learned model (lower row) preserves the density gradient and correctly predicts the asymmetric aggregation process.
Figure 14: Hierarchical dynamics: Initial condition 1. Evolution of the multi-group system initialized with spatially separated populations. The rows correspond to Group 1, Group 2, and Group 3. The learned model (blue) faithfully reproduces the reference particle dynamics (orange), capturing the directional information flow from Group 3 down to Group 1.
Figure 15: Hierarchical dynamics: Initial condition 2. Evolution of the multi-group system initialized with spatially separated populations. The rows correspond to Group 1, Group 2, and Group 3. The learned model (blue) accurately reproduces the reference particle dynamics (orange), capturing the directional information flow from Group 3 down to Group 1.
Figure 16: Second-order attraction-repulsion : Evolution of the second-order attraction-repulsion model initialized with ring-shaped initial position and Gaussian initial velocity. The upper rows display ground truth particle positions and velocities, while the lower rows show the positions and velocities predicted by the learned MVNN.
Figure 17: Second-order attraction-repulsion : Evolution of the second-order attraction-repulsion model initialized with double ring-shaped initial position and Gaussian initial velocity. The upper rows display ground truth particle positions and velocities, while the lower rows show the positions and velocities predicted by the learned MVNN.
Figure 18: Second-order attraction-repulsion : Evolution of the second-order attraction-repulsion model initialized with disk-shaped initial position and Gaussian initial velocity. The upper rows display ground truth particle positions and velocities, while the lower rows show the positions and velocities predicted by the learned MVNN.
Figure 19: Second-order attraction-repulsion : Evolution of the second-order attraction-repulsion model initialized with binary asymmetric initial position (high density left, low density right) and Gaussian initial velocity. The upper rows display ground truth particle positions and velocities, while the lower rows show the positions and velocities predicted by the learned MVNN.
Figure 20: Second-order Cucker-Smale : Evolution of the second-order Cucker-Smale model initialized with Gaussian initial position and Gaussian initial velocity. The upper rows display ground truth particle positions and velocities, while the lower rows show the positions and velocities predicted by the learned MVNN.
Figure 21: Second-order Cucker-Smale : Evolution of the second-order Cucker-Smale model initialized with a two-component Gaussian mixture initial position and Gaussian initial velocity. The upper rows display ground truth particle positions and velocities, while the lower rows show the positions and velocities predicted by the learned MVNN.
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