Applied Math Colloquium

Abstract: We consider the problem of learning interaction kernels from data for stochastic interacting particle systems and their mean-field limit, described by the McKean–Vlasov PDE. We study two measurement models: observation of a single particle trajectory, and access to discrete noisy space-time measurements of the solution to the mean-field PDE. We treat both the parametric and fully nonparametric settings. We construct estimators based on the spectral analysis of the linearized McKean–Vlasov operator and present a fully Bayesian nonparametric analysis of the problem. Finally, we consider a Bayesian inverse problem for a coupled drift-diffusion–Poisson system, and show how both MCMC methods and the Ensemble Kalman filter can be used to infer the permittivity from noisy measurements of the Fokker–Planck equation.