Assessing universal relations for rapidly rotating neutron stars: Insights from an interpretable deep learning perspective
Phys. Rev. D 113, 023018 – Published 12 January, 2026
DOI: https://doi.org/10.1103/j8rw-vy8y
Abstract
Relations between stellar properties independent of the nuclear equation of state offer profound insights into neutron star physics and have practical applications in data analysis by breaking the degeneracy between the parameters of interest. Commonly, these relations are derived from utilizing various realistic nuclear cold hadronic, hyperonic, and hybrid equation of state models, each of which should obey the current multimessenger constraints and cover a wide range of stiffnesses. Concurrently, the field of multimessenger astronomy has been significantly enhanced by the advent of gravitational wave astronomy, which increasingly incorporates deep learning techniques and algorithms. At the same time, x-ray spectral data from NICER based on known pulsars are available, and additional observations are expected from upcoming missions. Therefore, in the coming years, a wealth of information on neutron stars is expected to be gathered from multiple observational channels. In this study, we revisit established universal relations, introduce new ones, and reassess them using a feedforward neural network as a regression model. More specifically, we mainly propose “deep” EoS-insensitive hypersurface relations for rapidly rotating compact objects between several of the star’s global parameters, such as the moment of inertia, the reciprocal of the moment of inertia, the rotational frequency, mass and current multipole moments and the equatorial radius, which achieve an accuracy of within 1% in most cases, with only a small fraction of investigated models exceeding this threshold. While analytical expressions can be used to represent some of these relations, the neural network approach demonstrates superior performance, particularly in complex regions of the parameter space. Furthermore, we use the SHapley Additive exPlanations (SHAP) method to interpret the predictions of the suggested feedforward network, as SHAP is grounded in a robust theoretical framework inspired by the field of cooperative game theory. Most importantly, these highly accurate universal relations, empowered with the interpretability description, could be used in efforts to constrain the high-density equation of state in neutron stars, with the potential to enhance our understanding as new observables emerge.
