Machine Learning Moment Closure Models for the Radiative Transfer Equation II: Enforcing Global Hyperbolicity in Gradient-Based Closures

This is the second paper in a series which we develop machine learning (ML) moment closure models for radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, proposed an approach to directly learn gradient of unclosed high order moment, performs much better than itself and conventional $P_N$ closure. However, ML model \cite{huang2021gradient} not able guarantee hyperbolicity long time stability. We propose this method enforce global model. The main idea seek symmetrizer (a symmetric positive definite matrix) system, derive constraints such that system globally symmetrizable hyperbolic. It shown new inherits dissipativeness RTE preserves correct diffusion limit as Knunsden number goes zero. Several benchmark tests including Gaussian source problem two-material show good accuracy, stability generalizability hyperbolic