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This is the second paper in a series in which we develop machine learning(ML) moment closure models for the radiative transfer equation (RTE). In ourprevious work \cite{huang2021gradient}, we proposed an approach to directlylearn the gradient of the unclosed high order moment, which performs muchbetter than learning the moment itself and the conventional $P_N$ closure.However, the ML moment closure model in \cite{huang2021gradient} is not able toguarantee hyperbolicity and long time stability. We propose in this paper amethod to enforce the global hyperbolicity of the ML closure model. The mainidea is to seek a symmetrizer (a symmetric positive definite matrix) for theclosure system, and derive constraints such that the system is globallysymmetrizable hyperbolic. It is shown that the new ML closure system inheritsthe dissipativeness of the RTE and preserves the correct diffusion limit as theKnunsden number goes to zero. Several benchmark tests including the Gaussiansource problem and the two-material problem show the good accuracy, long timestability and generalizability of our globally hyperbolic ML closure model.

Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures