  [Abridged] Galaxy clusters are the most massive gravitationally-bound systems
in the universe and are widely considered to be an effective cosmological
probe. We propose the first Machine Learning method using galaxy cluster
properties to derive unbiased constraints on a set of cosmological parameters,
including Omega_m, sigma_8, Omega_b, and h_0. We train the machine learning
model with mock catalogs including "measured" quantities from Magneticum
multi-cosmology hydrodynamical simulations, like gas mass, gas bolometric
luminosity, gas temperature, stellar mass, cluster radius, total mass, velocity
dispersion, and redshift, and correctly predict all parameters with
uncertainties of the order of ~14% for Omega_m, ~8% for sigma_8, ~6% for
Omega_b, and ~3% for h_0. This first test is exceptionally promising, as it
shows that machine learning can efficiently map the correlations in the
multi-dimensional space of the observed quantities to the cosmological
parameter space and narrow down the probability that a given sample belongs to
a given cosmological parameter combination. In the future, these ML tools can
be applied to cluster samples with multi-wavelength observations from surveys
like LSST, CSST, Euclid, Roman in optical and near-infrared bands, and eROSITA
in X-rays, to constrain both the cosmology and the effect of the baryonic
feedback.
