Abstract
Abstract 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 simulation. 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 capture all aspects of the neutrino angular distribution in momentum space. In this paper, we develop a closure relation by using DNNs that take the neutrino energy density, flux, and the fluid velocity as the inputs 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 find that the diagonal component of the Eddington tensor is better reproduced 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 relation at large radii. In the comparison between the two DNNs, the TBNN displays slightly better performance than the CWNN. With these new closure relations at hand, based on DNNs that well reproduce the Eddington tensor at much lower costs, we have opened up a new possibility for the moment method.
Highlights
The core-collapse supernova (CCSN) is the explosive death of a massive star (Baade & Zwicky 1934)
Attempt of the sort, the main goal of this paper is to give a proof of principle, i.e., to demonstrate that the neural networks can be trained so that it could give an estimate of the Eddington tensor from the neutrino energy density, flux, and the fluid velocity; it could include other quantities, which is beyond the scope of this paper, though
The M1-Eddington tensor is first evaluated in the fluid-rest frame according to equations (4–9), and this is converted to the laboratory frame by the Lorentz transformation with the local fluid velocity
Summary
The core-collapse supernova (CCSN) is the explosive death of a massive star (Baade & Zwicky 1934). The explosion energy is ∼ 1051 erg and the ultimate energy source is the gravitational energy released when a stellar core collapses to form a neutron star. The recent discovery of the binary neutron star merger revealed that it is the site of the r-process nucleosynthesis (Tanaka et al 2017). To understand the chemical evolution of the universe in a coherent way, the understanding of the neutron star formation event, i.e., the CCSN is important. The leading hypothesis of the explosion mechanism of the CCSN is the neutrino heating mechanism (see, e.g., Janka 2012, for a review).
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