  The properties of ground state of spin-$\frac{1}{2}$ kagome antiferromagnetic
Heisenberg (KAFH) model have attracted considerable interest in the past few
decades, and recent numerical simulations reported a spin liquid phase. The
nature of the spin liquid phase remains unclear. For instance, the interplay
between symmetries and $Z_2$ topological order leads to different types of
$Z_2$ spin liquid phases. In this paper, we develop a numerical simulation
method based on symmetric projected entangled-pair states (PEPS), which is
generally applicable to strongly correlated model systems in two spatial
dimensions. We then apply this method to study the nature of the ground state
of the KAFH model. Our results are consistent with that the ground state is a
$U(1)$ Dirac spin liquid rather than a $Z_2$ spin liquid.
