Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles

Abstract

We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for 1st-order systems of interacting particles. The data consist of discrete space-time observations of the solution. By least squares with regularization, the algorithm learns the kernel on data-adaptive hypothesis spaces efficiently. A key ingredient is a probabilistic error functional derived from the likelihood of the mean-field equation's diffusion process. The estimator converges, in a reproducing kernel Hilbert space and an L2 space under an identifiability condition, at a rate optimal in the sense that it equals the numerical integrator's order. We demonstrate our algorithm on three typical examples: the opinion dynamics with a piecewise linear kernel, the granular media model with a quadratic kernel, and the aggregation-diffusion with a repulsive-attractive kernel.

References in corpus (6)

• On the Dynamics of Large Particle Systems in the Mean Field Limit
• Nonparametric inference of interaction laws in systems of agents from trajectory data
• Learning interaction kernels in heterogeneous systems of agents from multiple trajectories
• Inverse problem robustness for multi-species mean field spin models
• On the identifiability of interaction functions in systems of interacting particles
• Cluster Prediction for Opinion Dynamics from Partial Observations

Cited by in corpus (4)

• Learning Mean-Field Equations from Particle Data Using WSINDy
• Identifiability of interaction kernels in mean-field equations of interacting particles
• On the coercivity condition in the learning of interacting particle systems
• Cluster Prediction for Opinion Dynamics from Partial Observations