  This is the third paper in a series in which we develop machine learning (ML)
moment closure models for the radiative transfer equation (RTE). In our
previous work \cite{huang2021gradient}, we proposed an approach to learn the
gradient of the unclosed high order moment, which performs much better than
learning the moment itself and the conventional $P_N$ closure. However, while
the ML moment closure has better accuracy, it is not able to guarantee
hyperbolicity and has issues with long time stability. In our second paper
\cite{huang2021hyperbolic}, we identified a symmetrizer which leads to
conditions that enforce that the gradient based ML closure is symmetrizable
hyperbolic and stable over long time. The limitation of this approach is that
in practice the highest moment can only be related to four, or fewer, lower
moments.
  In this paper, we propose a new method to enforce the hyperbolicity of the ML
closure model. Motivated by the observation that the coefficient matrix of the
closure system is a lower Hessenberg matrix, we relate its eigenvalues to the
roots of an associated polynomial. We design two new neural network
architectures based on this relation. The ML closure model resulting from the
first neural network is weakly hyperbolic and guarantees the physical
characteristic speeds, i.e., the eigenvalues are bounded by the speed of light.
The second model is strictly hyperbolic and does not guarantee the boundedness
of the eigenvalues. Several benchmark tests including the Gaussian source
problem and the two-material problem show the good accuracy, stability and
generalizability of our hyperbolic ML closure model.
