Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds
Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds
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DOI：
10.1007/s10915-022-02056-7
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发表时间：
2021-09
影响因子：
2.5
通讯作者：
Juntao Huang;Yingda Cheng;A. Christlieb;L. Roberts
中科院分区：
文献类型：
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作者：
Juntao Huang;Yingda Cheng;A. Christlieb;L. Roberts
这是我们为辐射传输方程开发机器学习（ML）矩封闭模型的系列论文中的第三篇。在我们之前的工作中（Huang et al. in J Comput Phys 453：110941，2022），我们提出了一种学习未闭合高阶矩的梯度的方法，其性能比学习矩本身和常规闭包要好得多。然而，虽然ML矩闭合具有更好的精度，但它不能保证双曲性并且具有长时间稳定性的问题。在我们的第二篇论文（Huang et al.，在：机器学习矩封闭模型的辐射传输方程II：加强全球双曲在梯度为基础的封闭，2021年。arXiv：2105.14410），我们确定了一个对称化器，它导致了强制基于梯度的ML闭包是可对称化的双曲型和长时间稳定的条件。这种方法的局限性在于，在实践中，最高时刻只能与四个或更少的较低时刻相关。在本文中，我们提出了一种新的方法来加强ML闭包模型的双曲性。出于观察，封闭系统的系数矩阵是一个较低的Hessenberg矩阵，我们将其特征值的根相关联的多项式。基于这种关系，我们设计了两种新的神经网络结构。第一个神经网络产生的ML闭包模型是弱双曲线的，并保证物理特征速度，即，本征值受光速限制。第二个模型是严格双曲型的，不能保证特征值的有界性。包括高斯源问题和双材料问题在内的多个基准测试表明，我们的双曲ML闭包模型具有良好的精度，稳定性和推广性。
This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation. In our previous work (Huang et al. in J Comput Phys 453:110941, 2022), we proposed an approach to learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventionalclosure. However, while the ML moment closure has better accuracy, it is not able to guarantee hyperbolicity and has issues with long time stability. In our second paper (Huang et al., in: Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures, 2021. arXiv:2105.14410), we identified a symmetrizer which leads to conditions that enforce that the gradient based ML closure is symmetrizable hyperbolic and stable over long time. The limitation of this approach is that in practice the highest moment can only be related to four, or fewer, lower moments. In this paper, we propose a new method to enforce the hyperbolicity of the ML closure model. Motivated by the observation that the coefficient matrix of the closure system is a lower Hessenberg matrix, we relate its eigenvalues to the roots of an associated polynomial. We design two new neural network architectures based on this relation. The ML closure model resulting from the first neural network is weakly hyperbolic and guarantees the physical characteristic speeds, i.e., the eigenvalues are bounded by the speed of light. The second model is strictly hyperbolic and does not guarantee the boundedness of the eigenvalues. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, stability and generalizability of our hyperbolic ML closure model.