Constraining Symmetry Energy from Neutron Star Observables via Machine Learning

Neutron stars (NSs) provide a unique astrophysical laboratory for probing the symmetry energy E sym ( ρ ), a fundamental quantity that remains elusive at suprasaturation densities. To address the inverse mapping from NS observables to E sym ( ρ ), a likelihood-weighted neural network reconstruction framework is established. The proposed architecture synergizes a neural network (NN) model with a likelihood-based inference scheme. The NN model is optimized to reconstruct the symmetry energy using 15 mass–radius ( M – R ) pairs as inputs to ensure reliable inference from limited observations. Concurrently, likelihood weights are assigned to these M – R curves to quantify their consistency with observations from the Neutron Star Interior Composition Explorer (NICER). These weights are mapped via the trained NN model to constrain E sym ( ρ ). A joint analysis incorporates the earlier NICER observations of PSR J0030+0451 and PSR J0740+6620, alongside the recent measurements of PSR J0437-4715, PSR J1231-1411, and PSR J0614-3329. The inclusion of recent observations favors empirical symmetry energy constraints. Specifically, the slope parameter L and the symmetry energy at twice saturation density E sym (2 ρ 0 ) are constrained to L = 66.3 5 − 24.81 + 39.09 MeV and E sym ( 2 ρ 0 ) = 41.5 4 − 3.13 + 24.95 MeV , respectively. A systematic evaluation of individual pulsars reveals that L and E sym (2 ρ 0 ) exhibit a strong positive correlation with the stellar radius R . For canonical-mass neutron stars, the reduced uncertainties in PSR J0437-4715 and PSR J0614-3329 yield significantly tighter constraints compared to PSR J0030+0451. Beyond the present constraints, this framework provides a predictive tool for assessing the impact of future astronomical observations and offers crucial insights into the interior structure of NSs.