Graphene, as a widely used nanomaterial, has shown great flexibility in designing optically transparent microwave metasurfaces with broadband absorption. However, the design of graphene-based microwave metasurfaces relies on cumbersome parameter sweeping as well as the expertise of researchers. In this paper, we propose a machine-learning network which enables the forward prediction of reflection spectra and inverse design of versatile microwave absorbers. Techniques such as the normalization of input and transposed convolution layers are introduced in the machine-learning network to make the model lightweight and efficient. Particularly, the tunable conductivity of graphene enables a new degree in the intelligent design of metasurfaces. The inverse design system based on the optimization method is proposed for the versatile design of microwave absorbers. Representative cases are demonstrated, showing very promising performances on satisfying various absorption requirements. The proposed machine-learning network has significant potential for the intelligent design of graphene-based metasurfaces for various microwave applications.