Inference of interacting kernel in the mean-field regime
This addresses a theoretical challenge in statistical physics and machine learning for modeling collective behavior, but it is incremental as it builds on existing mean-field formulations.
The paper tackles the problem of reconstructing interaction kernels in systems of interacting agents from macroscopic measurements, showing that in the mean-field regime, the first variation from a particle system and a limiting PDE are close with a convergence rate of O(N^{-1/2}), as confirmed numerically.
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.