  Recent Lyman-$\alpha$ forest tomography measurements of the intergalactic
medium (IGM) have revealed a wealth of cosmic structures at high redshift
($z\sim 2.5$). In this work, we present the Tomographic Absorption
Reconstruction and Density Inference Scheme (TARDIS), a new chrono-cosmographic
analysis tool for understanding the formation and evolution of these observed
structures. We use maximum likelihood techniques with a fast non-linear
gravitational model to reconstruct the initial density field of the observed
regions. We find that TARDIS allows accurate reconstruction of smaller scale
structures than standard Wiener filtering techniques. Applying this technique
to mock Lyman-$\alpha$ forest data sets that simulate ongoing and future
surveys such as CLAMATO, Subaru-PFS or the ELTs, we are able to infer the
underlying matter density field at observed redshift and classify the cosmic
web structures. We find good agreement with the underlying truth both in the
characteristic eigenvalues and eigenvectors of the pseudo-deformation tensor,
with the eigenvalues inferred from 30m-class telescopes correlated at $r=0.95$
relative to the truth. As an output of this method, we are able to further
evolve the inferred structures to late time ($z=0$), and also track the
trajectories of coeval $z=2.5$ galaxies to their $z=0$ cosmic web environments.
