Machine Learning Moment Closure Models for the Radiative Transfer Equation II: Enforcing Global Hyperbolicity in Gradient-Based Closures
Abstract.
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 [J. Huang, Y. Cheng, A. J. Christlieb, and L. F. Roberts, J. Comput. Phys., 453 (2022), 110941], we proposed an approach to directly learn the spatial 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 [J. Huang, Y. Cheng, A. J. Christlieb, and L. F. Roberts, J. Comput. Phys., 453 (2022), 110941] 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 Knudsen 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.
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Acknowledgments.
We thank Michael M. Crockatt from Sandia National Laboratories for providing a numerical solver for the RTEs. We also would like to acknowledge the High Performance Computing Center at Michigan State University for providing computational resources that have contributed to the research results reported within this paper. We thank two anonymous reviewers for providing helpful comments on an earlier draft of the manuscript.
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 1 June 2021
Accepted: 16 November 2022
Published online: 25 April 2023
Keywords
MSC codes
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Funding Information
National Science Foundation (NSF): DMS-2011838
National Science Foundation (NSF): AST-2008004
Air Force Office of Scientific Research (AFOSR): FA9550-19-1-0281, FA9550-17-1-0394
National Science Foundation (NSF): DMS-1912183
National Key Research and Development Program of China: 2021YFA0719200
National Science Foundation (NSF): AST-2008004
U.S. Department of Energy (DOE): DE-SC0017955
Funding: The second author was supported by NSF grants DMS-2011838 and AST-2008004. The third author was supported by AFOSR grants FA9550-19-1-0281 and FA9550-17-1-0394 and by NSF grants DMS-1912183. The fifth author was supported by National Natural Science Foundation of China (grant 12071246) and National Key Research and Development Program of China (grant 2021YFA0719200). The second, third, and fourth authors were supported by NSF grant AST-2008004. The third and fourth authors were supported by DoE grant DE-SC0017955.
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