Identifiability of interaction kernels in mean-field equations of interacting particles

Systems of interacting particles or agents are widely used in many areas of science and engineering (see [2, 29, 24, 1] and the references therein). Motivated by these applications, there has been increasing interests in inferring the interaction kernel (or the interaction potential) in a nonparametric fashion for generality. When the system has finitely many particles, the recent efforts [4, 20, 18, 19, 17, 16] provide systematical tools for the inference of the kernel from multiple trajectories of all particles. When the number of particles is large, it becomes impractical to collect trajectories of all particles, but one can often observe the population density, i.e., the solution of the mean-field equations. This leads to the inverse problem of inferring the interaction kernel of the mean-field equation from data.