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.

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