  We introduced a novel method to solve Bayesian inverse problems governed by
PDE equations with a hybrid two-level MCMC where we took advantage of the AI
surrogate model speed and the accuracy of numerical models. We show
theoretically the potential to solve Bayesian inverse problems accurately with
only a small number of numerical samples when the AI surrogate model error is
small. Several numerical experiment results are included which demonstrates the
advantage of the hybrid method.
