Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds

This is the third paper in a series which we develop machine learning (ML) moment closure models for radiative transfer equation. In our previous work (Huang et al. J Comput Phys 453:110941, 2022), proposed an approach to learn gradient of unclosed high order moment, performs much better than itself and conventional $$P_N$$ closure. However, while ML has accuracy, it not able guarantee hyperbolicity issues with long time stability. second al., in: Machine equation II: enforcing global based closures, 2021. arXiv:2105.14410 ), identified symmetrizer leads conditions that enforce symmetrizable hyperbolic stable over time. The limitation this practice highest can only be related four, or fewer, lower moments. paper, propose new method model. Motivated by observation coefficient matrix system Hessenberg matrix, relate its eigenvalues roots associated polynomial. We design two neural network architectures on relation. model resulting from first weakly guarantees physical characteristic speeds, i.e., are bounded speed light. strictly does boundedness eigenvalues. Several benchmark tests including Gaussian source problem two-material show good stability generalizability