  Regressing the vector field of a dynamical system from a finite number of
observed states is a natural way to learn surrogate models for such systems. A
simple and interpretable way to learn a dynamical system from data is to
interpolate its vector-field with a data-adapted kernel which can be learned by
using Kernel Flows. The method of Kernel Flows is a trainable machine learning
method that learns the optimal parameters of a kernel based on the premise that
a kernel is good if there is no significant loss in accuracy if half of the
data is used. The objective function could be a short-term prediction or some
other objective for other variants of Kernel Flows). However, this method is
limited by the choice of the base kernel. In this paper, we introduce the
method of \emph{Sparse Kernel Flows } in order to learn the ``best'' kernel by
starting from a large dictionary of kernels. It is based on sparsifying a
kernel that is a linear combination of elemental kernels. We apply this
approach to a library of 132 chaotic systems.
