  We use TNG and EAGLE hydrodynamic simulations to investigate the central
galaxy - dark matter halo relations that are needed for a halo-based empirical
model of star formation in galaxies. Using a linear dimension reduction
algorithm and a model ensemble method, we find that for both star-forming and
quenched galaxies, the star formation history (SFH) is tightly related to the
halo mass assembly history (MAH). The quenching of a low-mass galaxy is mainly
due to the infall-ejection process related to a nearby massive halo, while the
quenching of a high-mass galaxy is closely related to the formation of a
massive progenitor in its host halo. The classification of star-forming and
quenched populations based solely on halo properties contains contamination
produced by sample imbalance and overlapping distributions of the two
populations. Guided by the results from hydrodynamic simulations, we build an
empirical model to predict the SFH of central galaxies based on the MAH of
their host halos, and we model the star-forming and quenched populations
separately. Our model is based on the idea of adopting star formation templates
from hydrodynamic simulations to reduce model complexity. We use various tests
to demonstrate that the model can recover star formation histories of
individual galaxies, and can statistically reproduce the galaxy bimodal
distribution, stellar mass - halo mass and star formation rate - halo mass
relations from low to high redshift, and assembly bias. Our study provides a
framework of using hydrodynamic simulations to discover, and to motivate the
use of, key ingredients to model galaxy formation using halo properties.
