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 ($zlesssim3$). 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.
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