We present a data-driven approach to construct entropy-based closures for the moment system from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the space-time discretization of the moment system and specific problem configurations such as initial and boundary conditions. With convex and $C^2$ approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, optimization-based M$_N$ closures. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than the M$_N$ closures.

中文翻译：

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动力学方程的基于熵的矩闭包的数据驱动、结构保持近似

我们提出了一种数据驱动的方法，从动力学方程为矩系统构建基于熵的闭包。所提出的闭包通过拟合矩和矩系统的熵之间的映射来学习熵函数，因此不依赖于矩系统的时空离散化和特定的问题配置，例如初始和边界条件。使用凸函数和 $C^2$ 近似值，这种数据驱动的闭包继承了基于熵的闭包的几个结构特性，例如熵耗散、双曲性和 H 定理。我们使用凸样条和神经网络构建 Maxwell-Boltzmann 熵的凸近似，在平板几何中线性传输的平面源基准问题上测试它们，并将结果与​​标准的、基于优化的 M$_N$ 闭包进行比较。