Extracting nuclear matter properties from the neutron star matter equation of state using deep neural networks

The extraction of the nuclear matter properties from neutron star (NS) observations is nowadays an important issue, in particular, the properties that characterize the symmetry energy which are essential to describe correctly asymmetric nuclear matter. We use deep neural networks (DNNs) to map the relation between cold $\beta$-equilibrium NS matter and the nuclear matter properties. Assuming a quadratic dependence on the isospin asymmetry for the energy per particle of homogeneous nuclear matter and using a Taylor expansion up to fourth order in the isoscalar and isovector contributions, we generate a dataset of different realizations of $\beta$-equilibrium NS matter and the corresponding nuclear matter properties. The DNN model was successfully trained, attaining great accuracy in the test set. Finally, a real-case scenario was used to test the DNN model, where a set of 33 nuclear models, obtained within a relativistic mean-field approach or a Skyrme force description, were fed into the DNN model and the corresponding nuclear matter parameters recovered with considerable accuracy; in particular, the standard deviations $\sigma (L_{\mathrm{sym}})=12.85\mathrm{MeV}$ and $\sigma (K_{\mathrm{sat}})=41.02\mathrm{MeV}$ were obtained, respectively, for the slope of the symmetry energy and the nuclear matter incompressibility at saturation.

Figure 1

Schematic and simplified representation of the DNN model, where only a fraction of the connections between nodes is shown. A rigorous representation would display all connections (weights) between any neuron in a given layer and all neurons of the previous layer, i.e., a densely connected neural network.

Figure 2

Model residuals (top) and percentage (bottom) for the saturation density $n_0$.

Figure 3

Model residuals for the isovector properties, $E_{\mathrm{sym}}$ (top), $L_{\mathrm{sym}}$ (middle), and $K_{\mathrm{sym}}$ (bottom).

Figure 4

Model residuals for the isoscalar properties, $E_{\mathrm{sat}}$ (top) and $K_{\mathrm{sat}}$ (bottom).

Figure 5

Scatter plot of the model residuals $\mathrm{\Delta }{L}_{\mathrm{sym}}$ vs $\mathrm{\Delta }{n}_0$ (top), $\mathrm{\Delta }{K}_{\mathrm{sym}}$ vs $\mathrm{\Delta }{n}_0$ (middle), and $\mathrm{\Delta }{K}_{\mathrm{sat}}$ vs $\mathrm{\Delta }{n}_0$ (bottom). To make the top plot easier to read, the DDME2 residual $(\mathrm{\Delta }{L}_{\mathrm{sym}},\mathrm{\Delta }{n}_0)=(59.62872\mathrm{MeV},0.0153{\mathrm{fm}}^{-3})$ is not shown.