  Zeeman-Doppler Imaging (ZDI) is a powerful inversion method to reconstruct
stellar magnetic surface fields. The reconstruction process is usually solved
by translating the inverse problem into a regularized least-square or
optimization problem. In this contribution we will emphasize that ZDI is an
inherent non-linear problem and the corresponding regularized optimization is,
like many non-linear problems, potentially prone to local minima. We show how
this problem will be exacerbated by using an inadequate forward model. To
facilitate a more consistent full radiative transfer driven approach to ZDI we
describe a two-stage strategy that consist of a principal component analysis
(PCA) based line profile reconstruction and a fast approximate polarized
radiative transfer method to synthesize local Stokes profiles. Moreover, we
introduce a novel statistical inversion method based on artificial neural
networks (ANN) which provide a fast calculation of a first guess model and
allows to incorporate better physical constraints into the inversion process.

Solar and Stellar Astrophysics