  Cosmic microwave background radiation (CMB) is critical to the understanding
of the early universe and precise estimation of cosmological constants. Due to
the contamination of thermal dust noise in the galaxy, the CMB map that is an
image on the two-dimensional sphere has missing observations, mainly
concentrated on the equatorial region. The noise of the CMB map has a
significant impact on the estimation precision for cosmological parameters.
Inpainting the CMB map can effectively reduce the uncertainty of parametric
estimation. In this paper, we propose a deep learning-based variational
autoencoder --- CosmoVAE, to restoring the missing observations of the CMB map.
The input and output of CosmoVAE are square images. To generate training,
validation, and test data sets, we segment the full-sky CMB map into many small
images by Cartesian projection. CosmoVAE assigns physical quantities to the
parameters of the VAE network by using the angular power spectrum of the
Gaussian random field as latent variables. CosmoVAE adopts a new loss function
to improve the learning performance of the model, which consists of $\ell_1$
reconstruction loss, Kullback-Leibler divergence between the posterior
distribution of encoder network and the prior distribution of latent variables,
perceptual loss, and total-variation regularizer. The proposed model achieves
state of the art performance for Planck \texttt{Commander} 2018 CMB map
inpainting.
