  Accurate analyses of present and next-generation galaxy surveys require new
ways to handle effects of non-linear gravitational structure formation in data.
To address these needs we present an extension of our previously developed
algorithm for Bayesian Origin Reconstruction from Galaxies to analyse matter
clustering at non-linear scales in observations. This is achieved by
incorporating a numerical particle mesh model of structure formation into our
Bayesian inference framework. The algorithm simultaneously infers the 3D
primordial matter fluctuations from which present non-linear observations
formed and provides reconstructions of velocity fields and structure formation
histories. The physical forward modelling approach automatically accounts for
non-Gaussian features in evolved matter density fields and addresses the
redshift space distortion problem associated with peculiar motions of galaxies.
Our algorithm employs a hierarchical Bayes approach to jointly account for
observational effects, such as galaxy biases, selection effects, and noise.
Corresponding parameters are marginalized out via a sophisticated Markov Chain
Monte Carlo approach relying on a combination of a multiple block sampling
framework and a Hamiltonian Monte Carlo sampler. We demonstrate the performance
of the method by applying it to the 2M++ galaxy compilation, tracing the matter
distribution of the Nearby Universe. We show accurate and detailed inferences
of the 3D non-linear dark matter distribution of the Nearby Universe. As
exemplified in the case of the Coma cluster, we provide mass estimates that are
compatible with those obtained from weak lensing and X-ray observations. For
the first time, we reconstruct the vorticity of the non-linear velocity field
from observations. In summary, our method provides plausible and detailed
inferences of dark matter and velocity fields of our cosmic neighbourhood.
