  Due to the importance of uncertainty quantification (UQ), Bayesian approach
to inverse problems has recently gained popularity in applied mathematics,
physics, and engineering. However, traditional Bayesian inference methods based
on Markov Chain Monte Carlo (MCMC) tend to be computationally intensive and
inefficient for such high dimensional problems. To address this issue, several
methods based on surrogate models have been proposed to speed up the inference
process. More specifically, the calibration-emulation-sampling (CES) scheme has
been proven to be successful in large dimensional UQ problems. In this work, we
propose a novel CES approach for Bayesian inference based on deep neural
network (DNN) models for the emulation phase. The resulting algorithm is not
only computationally more efficient, but also less sensitive to the training
set. Further, by using an Autoencoder (AE) for dimension reduction, we have
been able to speed up our Bayesian inference method up to three orders of
magnitude. Overall, our method, henceforth called \emph{Dimension-Reduced
Emulative Autoencoder Monte Carlo (DREAM)} algorithm, is able to scale Bayesian
UQ up to thousands of dimensions in physics-constrained inverse problems. Using
two low-dimensional (linear and nonlinear) inverse problems we illustrate the
validity this approach. Next, we apply our method to two high-dimensional
numerical examples (elliptic and advection-diffussion) to demonstrate its
computational advantage over existing algorithms.
