  Weak gravitational lensing of distant galaxies provides a powerful probe of
dark energy. The aim of this study is to investigate the application of
convolutional neural networks (CNNs) to precision shear estimation. In
particular, using a shallow CNN, we explore the impact of point spread function
(PSF) misestimation and `galaxy population bias' (including `distribution bias'
and `morphology bias'), focusing on the accuracy requirements of next
generation surveys. We simulate a population of noisy disk and elliptical
galaxies and adopt a PSF that is representative of a Euclid-like survey. We
quantify the accuracy achieved by the CNN assuming a linear relationship
between the estimated and true shears and measure the multiplicative ($m$) and
additive ($c$) biases. We make use of an unconventional loss function to
mitigate the effects of noise bias and measure $m$ and $c$ when we use either:
(i) an incorrect galaxy ellipticity distribution or size-magnitude relation, or
the wrong ratio of morphological types, to describe the population of galaxies
(distribution bias); (ii) an incorrect galaxy light profile (morphology bias);
or (iii) a PSF with size or ellipticity offset from its true value (PSF
misestimation). We compare our results to the Euclid requirements on the
knowledge of the PSF model shape and size. Finally, we outline further work to
build on the promising potential of CNNs in precision shear estimation.
