  We consider the simulation of Bayesian statistical inverse problems governed
by large-scale linear and nonlinear partial differential equations (PDEs).
Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve
such problems. However, MCMC techniques are computationally challenging as they
require several thousands of forward PDE solves. The goal of this paper is to
introduce a fractional deep neural network based approach for the forward
solves within an MCMC routine. Moreover, we discuss some approximation error
estimates and illustrate the efficiency of our approach via several numerical
examples.
