  Upcoming 21-cm intensity surveys will use the hyperfine transition in
emission to map out neutral hydrogen in large volumes of the universe.
Unfortunately, large spatial scales are completely contaminated with spectrally
smooth astrophysical foregrounds which are orders of magnitude brighter than
the signal. This contamination also leaks into smaller radial and angular modes
to form a foreground wedge, further limiting the usefulness of 21-cm
observations for different science cases, especially cross-correlations with
tracers that have wide kernels in the radial direction. In this paper, we
investigate reconstructing these modes within a forward modeling framework.
Starting with an initial density field, a suitable bias parameterization and
non-linear dynamics to model the observed 21-cm field, our reconstruction
proceeds by combining the likelihood of a forward simulation to match the
observations (under given modeling error and a data noise model) with the
Gaussian prior on initial conditions and maximizing the obtained posterior. For
redshifts $z=2$ and $4$, we are able to reconstruct 21cm field with cross
correlation, $r_c > 0.8$ on all scales for both our optimistic and pessimistic
assumptions about foreground contamination and for different levels of thermal
noise. The performance deteriorates slightly at $z=6$. The large-scale
line-of-sight modes are reconstructed almost perfectly. We demonstrate how our
method also reconstructs baryon acoustic oscillations, outperforming standard
methods on all scales. We also describe how our reconstructed field can provide
superb clustering redshift estimation at high redshifts, where it is otherwise
extremely difficult to obtain dense spectroscopic samples, as well as open up
cross-correlation opportunities with projected fields (e.g. lensing) which are
restricted to modes transverse to the line of sight.
