Ensemble Kalman inversion for sparse learning of dynamical systems from time-averaged data

Enforcing sparse structure within learning has led to significant advances in the field of data-driven discovery dynamical systems. However, such methods require access not only timeseries state system, but also time derivative. In many applications, data are available form time-averages as moments and autocorrelation functions. We propose a methodology discover vector fields defining (possibly stochastic or partial) differential equation, using time-averaged statistics. Such formulation naturally leads nonlinear inverse problem which we apply ensemble Kalman inversion (EKI). EKI is chosen because it may be formulated terms iterative solution quadratic optimization problems; sparsity then easily imposed. EKI-based various examples governed by equations (a noisy Lorenz 63 system), ordinary (Lorenz 96 system coalescence equations), partial equation (the Kuramoto-Sivashinsky equation). The results demonstrate that statistics can used for EKI. proposed extends scope previously challenging applications data-acquisition scenarios.