  We trained deep neural networks (DNNs) as a function of the neutrino energy
density, flux, and the fluid velocity to reproduce the Eddington tensor for
neutrinos obtained in our first-principles core-collapse supernova (CCSN)
simulations. Although the moment method, which is one of the most popular
approximations for neutrino transport, requires a closure relation, none of the
analytical closure relations commonly employed in the literature captures all
aspects of the neutrino angular distribution in momentum space. In this paper,
we developed a closure relation by using the DNN that takes the neutrino energy
density, flux, and the fluid velocity as the input and the Eddington tensor as
the output. We consider two kinds of DNNs: a conventional DNN named a
component-wise neural network (CWNN) and a tensor-basis neural network (TBNN).
We found that the diagonal component of the Eddington tensor is reproduced
better by the DNNs than the M1-closure relation especially for low to
intermediate energies. For the off-diagonal component, the DNNs agree better
with the Boltzmann solver than the M1 closure at large radii. In the comparison
between the two DNNs, the TBNN has slightly better performance than the CWNN.
With the new closure relations at hand based on the DNNs that well reproduce
the Eddington tensor with much smaller costs, we opened up a new possibility
for the moment method.
