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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 ourprevious work \cite{huang2021gradient}, we proposed an approach to learn thegradient of the unclosed high order moment, which performs much better thanlearning the moment itself and the conventional $P_N$ closure. However, whilethe ML moment closure has better accuracy, it is not able to guaranteehyperbolicity and has issues with long time stability. In our second paper\cite{huang2021hyperbolic}, we identified a symmetrizer which leads toconditions that enforce that the gradient based ML closure is symmetrizablehyperbolic and stable over long time. The limitation of this approach is thatin practice the highest moment can only be related to four, or fewer, lowermoments.  In this paper, we propose a new method to enforce the hyperbolicity of the MLclosure model. Motivated by the observation that the coefficient matrix of theclosure system is a lower Hessenberg matrix, we relate its eigenvalues to theroots of an associated polynomial. We design two new neural networkarchitectures based on this relation. The ML closure model resulting from thefirst neural network is weakly hyperbolic and guarantees the physicalcharacteristic speeds, i.e., the eigenvalues are bounded by the speed of light.The second model is strictly hyperbolic and does not guarantee the boundednessof the eigenvalues. Several benchmark tests including the Gaussian sourceproblem and the two-material problem show the good accuracy, stability andgeneralizability of our hyperbolic ML closure model.

Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds