  In this work, we propose a robust Bayesian sparse learning algorithm based on
Bayesian group Lasso with spike and slab priors for the discovery of partial
differential equations with variable coefficients. Using the samples draw from
the posterior distribution with a Gibbs sampler, we are able to estimate the
values of coefficients, together with their standard errors and confidence
intervals. Apart from constructing the error bars, uncertainty quantification
can also be employed for designing new criteria of model selection and
threshold setting. This enables our method more adjustable and robust in
learning equations with time-dependent coefficients. Three criteria are
introduced for model selection and threshold setting to identify the correct
terms: the root mean square, total error bar, and group error bar. Moreover,
three noise filters are integrated with the robust Bayesian sparse learning
algorithm for better results with larger noise. Numerical results demonstrate
that our method is more robust than sequential grouped threshold ridge
regression and group Lasso in noisy situations through three examples.
