Inference of interacting kernel in the mean-field regime

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.

Figures (9)

• Figure 1: First variation of the objective function with respect to the velocity field $a$.
• Figure 2: First variation of the objective function with respect to the interaction kernel $w$.
• Figure 3: Example 1. The two panels show the ground-true kernel $w^*$ and a forward PDE simulation.
• Figure 4: Example 1. (A) shows reconstructed $w(r)$ at three different iterations, and (B) shows the decay of error in optimization measured by $E^{(k)}$ following \ref{['eq:L_infty_error']}.
• Figure 5: Example 1: The three panels present the ground-true distribution evolution in time, error between the guessed and the true distributions at the initial iteration, and error between the reconstructed and the true distributions at $K=0$ (initial) and $K=78$-th iteration using a unified colormap.

Theorems & Definitions (27)

• Theorem 2.1
• Lemma 2.1
• Lemma 2.2
• proof : Proof for Theorem \ref{['thm:gradient_consistency']}
• Theorem 3.1
• Theorem 3.2
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem:continuous_dj/dw']}
• proof : Proof of Lemma \ref{['lem:particle_dj/dw']}