  Randomize-then-optimize (RTO) is widely used for sampling from posterior
distributions in Bayesian inverse problems. However, RTO may be computationally
intensive for complexity problems due to repetitive evaluations of the
expensive forward model and its gradient. In this work, we present a novel
strategy to substantially reduce the computation burden of RTO by using a
goal-oriented deep neural networks (DNN) surrogate approach. In particular, the
training points for the DNN-surrogate are drawn from a local approximated
posterior distribution, and it is shown that the resulting algorithm can
provide a flexible and efficient sampling algorithm, which converges to the
direct RTO approach. We present a Bayesian inverse problem governed by a
benchmark elliptic PDE to demonstrate the computational accuracy and efficiency
of our new algorithm (i.e., DNN-RTO). It is shown that with our algorithm, one
can significantly outperform the traditional RTO.
