This work presents a new physically motivated supervised machine-learning method, HYDRO-BAM, to reproduce the three-dimensional Lyα forest field in real and redshift space, which learns from a reference hydrodynamic simulation and thereby saves about seven orders of magnitude in computing time. We show that our method is accurate up to k ~ 1 h Mpc-1 in the one- (probability distribution function), two- (power spectra), and three-point (bispectra) statistics of the reconstructed fields. When compared to the reference simulation including redshift-space distortions, our method achieves deviations of ≲2% up to k = 0.6 h Mpc-1 in the monopole and ≲5% up to k = 0.9 h Mpc-1 in the quadrupole. The bispectrum is well reproduced for triangle configurations with sides up to k = 0.8 h Mpc-1. In contrast, the commonly adopted Fluctuating Gunn-Peterson approximation shows significant deviations, already when peculiar motions are not included (real space) at configurations with sides of k = 0.2-0.4 h Mpc-1 in the bispectrum and is also significantly less accurate in the power spectrum (within 5% up to k = 0.7 h Mpc-1). We conclude that an accurate analysis of the Lyα forest requires considering the complex baryonic thermodynamical large-scale structure relations. Our hierarchical domain-specific machine-learning method can efficiently exploit this and is ready to generate accurate Lyα forest mock catalogs covering the large volumes required by surveys such as DESI and WEAVE. * Released on 2022 January 20.