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

Abstract

This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents, an area of growing interest across various scientific and engineering fields. The main focus is identifying data-dependent function spaces where a quadratic loss functional possesses a unique minimizer. We consider two data-adaptive $L^2$ spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. In each $L^2$ space, we show that the function space of identifiability is the closure of the RKHS associated with the integral operator of inversion. Alongside prior research, our study completes a full characterization of identifiability in interacting particle systems with either finite or infinite particles, highlighting critical differences between these two settings. Moreover, the identifiability analysis has important implications for computational practice. It shows that the inverse problem is ill-posed, necessitating regularization. Our numerical demonstrations show that the weighted $L^2$ space is preferable over the unweighted $L^2$ space, as it yields more accurate regularized estimators.

References in corpus (7)

• Collective motion
• Nonparametric inference of interaction laws in systems of agents from trajectory data
• A general constitutive model for dense, fine particle suspensions validated in many geometries
• Learning interaction kernels in heterogeneous systems of agents from multiple trajectories
• Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles
• Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation
• On the coercivity condition in the learning of interacting particle systems