  Modeling the Point Spread Function (PSF) of wide-field surveys is vital for
many astrophysical applications and cosmological probes including weak
gravitational lensing. The PSF smears the image of any recorded object and
therefore needs to be taken into account when inferring properties of galaxies
from astronomical images. In the case of cosmic shear, the PSF is one of the
dominant sources of systematic errors and must be treated carefully to avoid
biases in cosmological parameters. Recently, forward modeling approaches to
calibrate shear measurements within the Monte-Carlo Control Loops ($MCCL$)
framework have been developed. These methods typically require simulating a
large amount of wide-field images, thus, the simulations need to be very fast
yet have realistic properties in key features such as the PSF pattern. Hence,
such forward modeling approaches require a very flexible PSF model, which is
quick to evaluate and whose parameters can be estimated reliably from survey
data. We present a PSF model that meets these requirements based on a fast
deep-learning method to estimate its free parameters. We demonstrate our
approach on publicly available SDSS data. We extract the most important
features of the SDSS sample via principal component analysis. Next, we
construct our model based on perturbations of a fixed base profile, ensuring
that it captures these features. We then train a Convolutional Neural Network
to estimate the free parameters of the model from noisy images of the PSF. This
allows us to render a model image of each star, which we compare to the SDSS
stars to evaluate the performance of our method. We find that our approach is
able to accurately reproduce the SDSS PSF at the pixel level, which, due to the
speed of both the model evaluation and the parameter estimation, offers good
prospects for incorporating our method into the $MCCL$ framework.
