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Enforcing sparse structure within learning has led to significant advances inthe field of data-driven discovery of dynamical systems. However, such methodsrequire access not only to time-series of the state of the dynamical system,but also to the time derivative. In many applications, the data are availableonly in the form of time-averages such as moments and autocorrelationfunctions. We propose a sparse learning methodology to discover the vectorfields defining a (possibly stochastic or partial) differential equation, usingonly time-averaged statistics. Such a formulation of sparse learning naturallyleads to a nonlinear inverse problem to which we apply the methodology ofensemble Kalman inversion (EKI). EKI is chosen because it may be formulated interms of the iterative solution of quadratic optimization problems; sparsity isthen easily imposed. We then apply the EKI-based sparse learning methodology tovarious examples governed by stochastic differential equations (a noisy Lorenz63 system), ordinary differential equations (Lorenz 96 system and coalescenceequations), and a partial differential equation (the Kuramoto-Sivashinskyequation). The results demonstrate that time-averaged statistics can be usedfor data-driven discovery of differential equations using sparse EKI. Theproposed sparse learning methodology extends the scope of data-driven discoveryof differential equations to previously challenging applications anddata-acquisition scenarios.

Ensemble Kalman Inversion for Sparse Learning of Dynamical Systems from Time-Averaged Data