Methodology study of machine learning for the neutron star equation of state

We discuss a methodology of machine learning to deduce the neutron star equation of state from a set of mass-radius observational data. We propose an efficient procedure to deal with a mapping from finite data points with observational errors onto an equation of state. We generate training data and optimize the neural network. Using independent validation data (mock observational data) we confirm that the equation of state is correctly reconstructed with precision surpassing observational errors. We finally discuss the relation between our method and Bayesian analysis with an emphasis put on generality of our method for underdetermined problems.

Figure 1

Feedforward neural network.

Figure 2

Loss function estimated for the training data (dashed lines) and the validation data (solid lines) as functions of the epoch. The red line represents the result for the actual training data set, and the blue line the result for a reduced data set for the purpose of demonstrating the problematic learning with overfitting. The observation number per EoS is denoted by $n_s$.

Figure 3

Two examples of the randomly generated EoSs (dashed lines) and the machine learning outputs (solid lines) reconstructed from one observation of 15 $M\text{−}R$ points [see Fig. 4 for actual ($M_i$, $R_i$)].

Figure 4

Randomly sampled 15 data points and the $M\text{−}R$ relations with the reconstructed EoS (solid lines) and the original EoS (dashed lines). The red and blue colors correspond to two EoSs shown with the same color in Fig. 3.