This is the third 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 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, 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 \cite{huang2021hyperbolic}, 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.
翻译:这是一系列中的第三页, 我们在这个系列中为辐射传输方程式开发机器学习( ML) 时间关闭模式。 在先前的工作 \ cite{ huang2021gradient} 中, 我们提出了一个方法来学习未关闭高顺序时刻的梯度, 这比学习时间本身和常规的$P_N美元关闭要好得多。 但是, 虽然 ML 关闭时间的准确性更高, 但是它无法保证超偏斜性, 并且存在长期稳定性问题 。 在我们的第二份文件 \ cite{ huang2021- hyperblic} 中, 我们找到了一个交错性交错的交错点 。 我们找到了一个交错系统系数矩阵不是较轻的 Hesenberg 标准值, 严格到基于 modalbality 格式的精度基准值, 我们将这个网络的稳定性与这个模型的精度测试联系起来。