  We present a dimension-reduced KRnet map approach (DR-KRnet) for
high-dimensional Bayesian inverse problems, which is based on an explicit
construction of a map that pushes forward the prior measure to the posterior
measure in the latent space. Our approach consists of two main components:
data-driven VAE prior and density approximation of the posterior of the latent
variable. In reality, it may not be trivial to initialize a prior distribution
that is consistent with available prior data; in other words, the complex prior
information is often beyond simple hand-crafted priors. We employ variational
autoencoder (VAE) to approximate the underlying distribution of the prior
dataset, which is achieved through a latent variable and a decoder. Using the
decoder provided by the VAE prior, we reformulate the problem in a
low-dimensional latent space. In particular, we seek an invertible transport
map given by KRnet to approximate the posterior distribution of the latent
variable. Moreover, an efficient physics-constrained surrogate model without
any labeled data is constructed to reduce the computational cost of solving
both forward and adjoint problems involved in likelihood computation. With
numerical experiments, we demonstrate the accuracy and efficiency of DR-KRnet
for high-dimensional Bayesian inverse problems.
