  In Bayesian inverse problems, surrogate models are often constructed to speed
up the computational procedure, as the parameter-to-data map can be very
expensive to evaluate. However, due to the curse of dimensionality and the
nonlinear concentration of the posterior, traditional surrogate approaches
(such us the polynomial-based surrogates) are still not feasible for large
scale problems. To this end, we present in this work an adaptive multi-fidelity
surrogate modeling framework based on deep neural networks (DNNs), motivated by
the facts that the DNNs can potentially handle functions with limited
regularity and are powerful tools for high dimensional approximations. More
precisely, we first construct offline a DNNs-based surrogate according to the
prior distribution, and then, this prior-based DNN-surrogate will be adaptively
\& locally refined online using only a few high-fidelity simulations. In
particular, in the refine procedure, we construct a new shallow neural network
that view the previous constructed surrogate as an input variable -- yielding a
composite multi-fidelity neural network approach. This makes the online
computational procedure rather efficient. Numerical examples are presented to
confirm that the proposed approach can obtain accurate posterior information
with a limited number of forward simulations.
