  One of the most significant challenges involved in efforts to understand the
equation of state of dense neutron-rich matter is the uncertain density
dependence of the nuclear symmetry energy. Because of its broad impact, pinning
down the density dependence of the nuclear symmetry energy has been a
longstanding goal of both nuclear physics and astrophysics. Recent observations
of neutron stars, in both electromagnetic and gravitational-wave spectra, have
already constrained significantly the nuclear symmetry energy at high
densities. Training deep neural networks to learn a computationally efficient
representation of the mapping between astrophysical observables of neutron
stars, such as masses, radii, and tidal deformabilities, and the nuclear
symmetry energy allows its density dependence to be determined reliably and
accurately. In this work we use a deep learning approach to determine the
nuclear symmetry energy as a function of density directly from observational
neutron star data. We show for the first time that artificial neural networks
can precisely reconstruct the nuclear symmetry energy from a set of available
neutron star observables, such as, masses and radii as those measured by, e.g.,
the NICER mission, or masses and tidal deformabilities as measured by the
LIGO/VIRGO/KAGRA gravitational-wave detectors. These results demonstrate the
potential of artificial neural networks to reconstruct the symmetry energy, and
the equation of state, directly from neutron star observational data, and
emphasize the importance of the deep learning approach in the era of
Multi-Messenger Astrophysics.
