  Enforcing sparse structure within learning has led to significant advances in
the field of data-driven discovery of dynamical systems. However, such methods
require 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 available
only in the form of time-averages such as moments and autocorrelation
functions. We propose a sparse learning methodology to discover the vector
fields defining a (possibly stochastic or partial) differential equation, using
only time-averaged statistics. Such a formulation of sparse learning naturally
leads to a nonlinear inverse problem to which we apply the methodology of
ensemble Kalman inversion (EKI). EKI is chosen because it may be formulated in
terms of the iterative solution of quadratic optimization problems; sparsity is
then easily imposed. We then apply the EKI-based sparse learning methodology to
various examples governed by stochastic differential equations (a noisy Lorenz
63 system), ordinary differential equations (Lorenz 96 system and coalescence
equations), and a partial differential equation (the Kuramoto-Sivashinsky
equation). The results demonstrate that time-averaged statistics can be used
for data-driven discovery of differential equations using sparse EKI. The
proposed sparse learning methodology extends the scope of data-driven discovery
of differential equations to previously challenging applications and
data-acquisition scenarios.
