Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures
Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures
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DOI：
10.1137/21m1423956
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发表时间：
2021-05
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影响因子：
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通讯作者：
Juntao Huang;Yingda Cheng;A. Christlieb;L. Roberts;W. Yong
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作者：
Juntao Huang;Yingda Cheng;A. Christlieb;L. Roberts;W. Yong
这是我们建立辐射传递方程(RTE)的机器学习(ML)矩闭合模型的系列文章中的第二篇。在我们以前的工作中，我们提出了一种直接学习非闭合高阶矩的梯度的方法，该方法比学习矩本身和传统的$P_N闭包要好得多。然而，CITE中的ML矩闭合模型不能保证双曲性和长期稳定性。本文提出了一种增强ML闭包模型全局双曲性的方法。其主要思想是寻找闭包系统的对称化(对称正定矩阵)，并导出约束使得系统是全局对称化的双曲的。结果表明，新的ML闭包系统继承了RTE的耗散性，并在Knunsden数趋于零时保持了正确的扩散极限。包括高斯源问题和双材料问题在内的几个基准测试表明，我们的全局双曲ML闭合模型具有良好的精度、长期稳定性和泛化能力。
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.