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This work presents a new physically-motivated supervised machine learningmethod, Hydro-BAM, to reproduce the three-dimensional Lyman-$\alpha$ forestfield in real and in redshift space learning from a reference hydrodynamicsimulation, thereby saving about 7 orders of magnitude in computing time. Weshow that our method is accurate up to $k\sim1\,h\,\rm{Mpc}^{-1}$ in the one-(PDF), two- (power-spectra) and three-point (bi-spectra) statistics of thereconstructed fields. When compared to the reference simulation includingredshift space distortions, our method achieves deviations of $\lesssim2\%$ upto $k=0.6\,h\,\rm{Mpc}^{-1}$ in the monopole, $\lesssim5\%$ up to$k=0.9\,h\,\rm{Mpc}^{-1}$ in the quadrupole. The bi-spectrum is well reproducedfor triangle configurations with sides up to $k=0.8\,h\,\rm{Mpc}^{-1}$. Incontrast, the commonly-adopted Fluctuating Gunn-Peterson approximation showssignificant deviations already neglecting peculiar motions at configurationswith sides of $k=0.2-0.4\,h\,\rm{Mpc}^{-1}$ in the bi-spectrum, being alsosignificantly less accurate in the power-spectrum (within 5$\%$ up to$k=0.7\,h\,\rm{Mpc}^{-1}$). We conclude that an accurate analysis of theLyman-$\alpha$ forest requires considering the complex baryonic thermodynamicallarge-scale structure relations. Our hierarchical domain specific machinelearning method can efficiently exploit this and is ready to generate accurateLyman-$\alpha$ forest mock catalogues covering large volumes required bysurveys such as DESI and WEAVE.

Mapping Lyman-alpha forest three-dimensional large scale structure in real and redshift space