Inference of interacting kernel in the mean-field regime

TL;DR

This paper compares particle-based and PDE-based models for reconstructing interaction kernels in agent systems, showing their estimators are close with a convergence rate of O(N^{-1/2}) under mild assumptions.

Contribution

It analyzes the relationship between particle and mean-field PDE formulations for kernel inference, providing theoretical convergence rates and numerical validation.

Findings

Particle and PDE estimators are close in a weak sense.

Convergence rate of the estimators is O(N^{-1/2}).

Numerical experiments confirm the theoretical rate.

Abstract

We study the problem of reconstructing interaction kernels in systems of interacting agents from macroscopic measurements when posed as an optimization problem. The reconstruction procedure depends on the formulation of the forward model, which may be given either by a finite-dimensional coupled ODE system tracking individual agent trajectories or by a mean-field PDE describing the evolution of the agent density. We investigate the similarities and differences between these two formulations in the mean-field regime. While the first variation derived from the particle system does not provide an unbiased estimator of the first variation associated with the limiting PDE, we prove that, under mild assumptions, the two are close in a weak sense with a convergence rate $\mathcal{O}(N^{-1/2})$. This rate is further confirmed by numerical evidences.