Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures
This work addresses stability issues in ML-based closures for radiative transfer, which is important for computational physics applications, but it is incremental as it builds directly on prior work in the series.
The paper tackles the problem of ensuring hyperbolicity and long-time stability in machine learning moment closure models for the radiative transfer equation by proposing a method to enforce global hyperbolicity, resulting in a model that shows good accuracy, stability, and generalizability in benchmark tests.
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 our previous work \cite{huang2021gradient}, we proposed an approach to directly 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, the ML moment closure model in \cite{huang2021gradient} is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model.