  Analyses of cosmic shear typically condense weak lensing information over a
range of scales to a single cosmological parameter, $S_8$. This paper presents
a method to extract more information from Stage-IV cosmic shear measurements by
directly reconstructing the matter power spectrum from linear to non-linear
scales. We demonstrate that cosmic shear surveys will be sensitive to the shape
of the matter power spectrum on non-linear scales. We show that it should be
possible to distinguish between different models of baryonic feedback and we
investigate the impact of intrinsic alignments and observational systematics on
forecasted constraints. In addition to providing important information on
galaxy formation, power spectrum reconstruction should provide a definitive
answer to the question of whether weak lensing measurements of $S_8$ on linear
scales are consistent with the $\textit{Planck}$ $\Lambda$CDM cosmology. In
addition, power spectrum reconstruction may lead to new discoveries on the
composition of the dark sector.
