  Discovering governing equations from data is important to many scientific and
engineering applications. Despite promising successes, existing methods are
still challenged by data sparsity and noise issues, both of which are
ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty
quantification and/or are costly in training. To overcome these limitations, we
propose a novel equation discovery method based on Kernel learning and BAyesian
Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target
function, which is flexible, expressive, and more robust to data sparsity and
noises. We combine it with a Bayesian spike-and-slab prior -- an ideal Bayesian
sparse distribution -- for effective operator selection and uncertainty
quantification. We develop an expectation-propagation expectation-maximization
(EP-EM) algorithm for efficient posterior inference and function estimation. To
overcome the computational challenge of kernel regression, we place the
function values on a mesh and induce a Kronecker product construction, and we
use tensor algebra to enable efficient computation and optimization. We show
the advantages of KBASS on a list of benchmark ODE and PDE discovery tasks.
