Decoding neutron star observations: Revealing composition through Bayesian neural networks

We exploit the great potential offered by Bayesian neural networks (BNNs) to directly decipher the internal composition of neutron stars (NSs) based on their macroscopic properties. By analyzing a set of simulated observations, namely NS radius and tidal deformability, we leverage BNNs as effective tools for inferring the proton fraction and sound speed within NS interiors. To achieve this, several BNNs models were developed upon a dataset of $\sim 25\mathrm{K}$ nuclear equation of state within a relativistic mean-field framework, obtained through Bayesian inference that adheres to minimal low-density constraints. Unlike conventional neural networks, BNNs possess an exceptional quality: they provide a prediction uncertainty measure. To simulate the inherent imperfections present in real-world observations, we have generated four distinct training and testing datasets that replicate specific observational uncertainties. Our initial results demonstrate that BNNs successfully recover the composition with reasonable levels of uncertainty. Furthermore, using mock data prepared with the DD2, a different class of relativistic mean-field model utilized during training, the BNN model effectively retrieves the proton fraction and speed of sound for neutron star matter.

Figure 1

Illustration of $\mathit{Y}={\mathit{y}}_{\mathit{p}}(\mathit{n})$ for two EOSs: each EOS is represented in our datasets by the 15 points $y_p(n_k)$.

Figure 2

The $n_s=60$ mock observations generated for two EOSs in dataset 1 (left) and dataset 2 (right). The gray area represents the extremes of our EOS dataset. The two EOSs coincide with the ones used in Fig. 1.

Figure 3

The $n_s=60$ mock observations generated in the $\mathrm{\Lambda }-M$ diagram for two EOSs in dataset 3 (left) and dataset 4 (right). The gray area represents the extremes of our EOS dataset. The two EOSs coincide with the ones used in Fig. 1.

Figure 4

The BNNs predictions for $v_s^2(n)$ using one EOS of the test. The models trained on datasets 1 (blue) and 2 (orange) are in the upper figure while datasets 3 (purple) and 4 (green) models are in the lower figure. The prediction mean values (solid lines) and $2\sigma$ confidence intervals are shown. The true values are shown in black dots and the range of $v_s^2(n)$ from the train set is indicated by the gray region.

Figure 5

Median (solid line), 95.4% confidence interval (dashed and dotted lines), and extreme values (region) for $\mathrm{\Gamma }(n_k)=(v_s^2(n_k)-v_s^2(n_k{)}^{\mathrm{true}})/\sigma (n_k)$ (top) and $\mathrm{\Sigma }(n_k)=\sigma (n_k)$ (bottom) for each dataset.

Figure 6

Coverage probability calculated on the test set of the $v_s^2(n)$ BNNs models.

Figure 7

Prediction uncertainty deviation $\eta [a,b]$ between the $v_s^2(n)$ BNN models $a$ and $b$ (see text for details).

Figure 8

The BNNs predictions for $y_p(n)$ using one EOS of the test. The models trained on datasets 1 (blue) and 2 (orange) are in the upper figure while datasets 3 (purple) and 4 (green) models are in the lower figure. The predicted mean values (solid lines) and $2\sigma$ confidence intervals are shown. The true values are shown in black dots.

Figure 9

Median (solid line) and the 95.4% confidence interval (dashed and dotted lines) for $\delta (n_k)=y_p(n_k)-y_p(n_k{)}^{\mathrm{true}}$ (left), $\mathrm{\Sigma }(n_k)=\sigma (n_k)$ (center), and $\mathrm{\Gamma }(n_k)=(y_p(n_k)-y_p(n_k{)}^{\mathrm{true}})/\sigma (n_k)$ (right).

Figure 10

Some statistics of $y_p(n)$ calculated from the train dataset.

Figure 11

Prediction uncertainty deviation $\eta [a,b]$ between the $y_p(n)$ BNN models $a$ and $b$ (see text for details).

Figure 12

The pdf of $f_{\text{epist}}=({\widehat{\mathit{\sigma }}}_{\text{epist}}^2/{\widehat{\mathit{\sigma }}}^2)\times 100\%$ for $y_p(n)$ BNN models in sets 2 and 3 (left) and for set 1 model but trained in different training datasets with different number of mock observations $n_s$ (right).

Figure 13

The BNN model predictions, $v_s^2$ (upper) and $y_p$ (lower), for one mock observation ($n_s=1$) of the DD2 EOS, the blue area represents the 95% confidence interval, and the solid line the mean.

Figure 14

Correlation between $v_s^2(n)$ and $R(M)$ (left) or $\mathrm{\Lambda }(M)$ (right) for fixed mass values. In black squares we show the mean correlation value for $M/M_{\odot }\in [1,2.2]$.

Figure 15

Correlation between $y_p$ and $R(M)$ (left) or $\mathrm{\Lambda }(M)$ (right) for fixed mass values. In black squares we show the mean correlation value for $M/M_{\odot }\in [1,2.2]$.