Kinetic approaches are generally accurate in dealing with microscale plasma physics problems but are computationally expensive for large-scale or multiscale systems. One of the long-standing problems in plasma physics is the integration of kinetic physics into fluid models, which is often achieved through sophisticated analytical closure terms. In this paper, we successfully construct a multi-moment fluid model with an implicit fluid closure included in the neural network using machine learning. The multi-moment fluid model is trained with a small fraction of sparsely sampled data from kinetic simulations of Landau damping, using the physics-informed neural network (PINN) and the gradient-enhanced physics-informed neural network (gPINN). The multi-moment fluid model constructed using either PINN or gPINN reproduces the time evolution of the electric field energy, including its damping rate, and the plasma dynamics from the kinetic simulations. In addition, we introduce a variant of the gPINN architecture, namely, gPINN$p$ to capture the Landau damping process. Instead of including the gradients of all the equation residuals, gPINN$p$ only adds the gradient of the pressure equation residual as one additional constraint. Among the three approaches, the gPINN$p$-constructed multi-moment fluid model offers the most accurate results. This work sheds light on the accurate and efficient modeling of large-scale systems, which can be extended to complex multiscale laboratory, space, and astrophysical plasma physics problems.

翻译：动理学方法在处理微观等离子体物理问题时通常精确，但在大规模或多尺度系统中计算成本高昂。等离子体物理中长期存在的难题之一是将动理学物理融入流体模型，这通常通过复杂的解析闭合项实现。本文利用机器学习成功构建了一个多矩流体模型，该模型在神经网络中包含了隐式流体闭合项。基于稀疏采样的朗道阻尼动理学模拟数据，采用物理信息神经网络（PINN）和梯度增强型物理信息神经网络（gPINN）对多矩流体模型进行训练。使用PINN或gPINN构建的多矩流体模型均能再现动理学模拟中电场能量的时间演化（包括其阻尼率）以及等离子体动力学行为。此外，我们引入了一种改进的gPINN架构——gPINN$p$以捕捉朗道阻尼过程。与包含所有方程残差的梯度不同，gPINN$p$仅将压力方程残差的梯度作为额外约束。在三种方法中，基于gPINN$p$构建的多矩流体模型提供了最精确的结果。本研究为大规模系统的精确高效建模提供了新思路，并可推广至实验室、空间和天体物理中的复杂多尺度等离子体物理问题。