  Soon, the combination of electromagnetic and gravitational signals will open
the door to a new era of gravitational-wave (GW) cosmology. It will allow us to
test the propagation of tensor perturbations across cosmic time and study the
distribution of their sources over large scales. In this work, we show how
machine learning techniques can be used to reconstruct new physics by
leveraging the spatial correlation between GW mergers and galaxies. We explore
the possibility of jointly reconstructing the modified GW propagation law and
the linear bias of GW sources, as well as breaking the slight degeneracy
between them by combining multiple techniques. We show predictions roughly
based on a network of Einstein Telescopes combined with a high-redshift galaxy
survey ($z\lesssim3$). Moreover, we investigate how these results can be
re-scaled to other instrumental configurations. In the long run, we find that
obtaining accurate and precise luminosity distance measurements (extracted
directly from the individual GW signals) will be the most important factor to
consider when maximizing constraining power.
