Application of neural networks for the reconstruction of supernova neutrino energy spectra following fast neutrino flavor conversions

Neutrinos can undergo fast flavor conversions (FFCs) within extremely dense astrophysical environments, such as core-collapse supernovae (CCSNe) and neutron star mergers (NSMs). In this study, we explore FFCs in a multienergy neutrino gas, revealing that when the FFC growth rate significantly exceeds that of the vacuum Hamiltonian, all neutrinos (regardless of energy) share a common survival probability dictated by the energy-integrated neutrino spectrum. We then employ physics-informed neural networks (PINNs) to predict the asymptotic outcomes of FFCs within such a multienergy neutrino gas. These predictions are based on the first two moments of neutrino angular distributions for each energy bin, typically available in state-of-the-art CCSN and NSM simulations. Our PINNs achieve errors as low as $\lesssim 6\%$ and $\lesssim 18\%$ for predicting the number of neutrinos in the electron channel and the relative absolute error in the neutrino moments, respectively.

Figure 1

Comparison of the spanned range of spatially averaged survival probabilities for neutrinos with different $\omega$ [gray shaded area; see Eq. (10)], with the survival probability averaged over space and $\omega$ [black dashed curve; see Eq. (9)], at the final time of the simulation for a system with $\omega \ll \lambda$. Also shown is the analytical prescription for $⟨P_{\mathrm{sur}}(\mu ,\omega ){⟩}_f$ (blue dotted curve) described in Eq. (16). Note that the gray-shaded area basically overlaps with the black dashed line, implying that the survival probabilities are nearly independent of the neutrino energy.

Figure 2

Schematic architecture of our NNs. The green zone shows the implementation of the extra features, ${\mu }_c$, and $E_{RL}$, which are obtained through an extra layer of regression, using linear and logistic regressions, respectively. Here, ${\mu }_c$ is the crossing direction, and $E_{RL}$ is a binary, which is 1 if the equilibrium occurs for ${\mu }_c\le \mu$, and 0 otherwise. The blue zone represents energy-integrated inputs, while the orange zone displays inputs for specific energy bins. Note that the neutrino number densities in the particular energy bin must be smaller than the corresponding energy-integrated values. In our basic NN, referred to as the NN with no extra features, the NN only takes the inputs highlighted in Eq. (18). However and in our PINN, we provide our NN with the extra features ${\mu }_c$ and $E_{RL}$.

Figure 3

Performance evaluation of the PINN and the basic NN with no extra features. We present the relative absolute error in the output parameters, along with the relative error in the total number of neutrinos within the electron channel, $N_{{\nu }_e+{\overline{\nu }}_e}$. It is evident that the PINN can well outperform the basic NN with no extra features. Here, an epoch refers to a single pass through the entire training dataset during the training phase.

Figure 4

Performance evaluation of our PINN and the basic NN with no extra features (on the validation set) as a function of the number of neurons in the hidden layer. It is evident that the NNs have achieved their best performance on the validation set once $n_h\gtrsim 150$. The labels and NN models are the same as those in Fig. 3.

Figure 5

Absolute relative error in the output of our PINN (red curve) vs the relative error in the number of neutrinos in electron channel, i.e., $N_{{\nu }_e+{\overline{\nu }}_e}$ (blue curve). The inclusion of a few thousand data points in the training set leads to the disappearance of error variations between the validation and training sets. Note that we do not display the absolute relative error in the training set. This is due to the presence of small $I^{\text{′}}s$ in the training set (which are removed from the test set), causing a significantly larger relative error. Hence, any direct comparisons between the absolute relative errors in the training and test sets would be unfair.

Figure 6

Performance evaluation of our PINNs in reconstructing neutrino energy spectra. The upper panels exhibit the initial spectra characteristics, including the neutrino initial energy spectra, angular distributions of the energy-integrated neutrino spectra as a function of $\mu$, and the corresponding FFC survival probability. Here, we have assumed $F_{{\nu }_e}=0.5$, $F_{{\overline{\nu }}_e}=0.7$, and $F_{{\overline{\nu }}_e}=0.8$. The lower panels illustrate the post-FFCs final neutrino spectra for ${\nu }_e$, ${\overline{\nu }}_e$, and ${\nu }_x$ (${\overline{\nu }}_x$), respectively. The results are shown for both the PINN approach and an exact method, assuming having access to the full neutrino angular distributions. As one can see, the prediction errors for antineutrinos are much smaller than those concerning neutrinos.