  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.
