We study the permeation dynamics of helium and carbon dioxide through an atomistically detailed model of a polymer of intrinsic microporosity, PIM-1, via non-equilibrium molecular dynamics (NEMD) simulations. This work presents the first explicit molecular modeling of gas permeation through a high free-volume polymer sample, and it demonstrates how permeability and solubility can be obtained coherently from a single simulation. Solubilities in particular can be obtained to a very high degree of confidence and within experimental inaccuracies. Furthermore, the simulations make it possible to obtain very specific information on the diffusion dynamics of penetrant molecules and yield detailed maps of gas occupancy, which are akin to a digital tomographic scan of the polymer network. In addition to determining permeability and solubility directly from NEMD simulations, the results shed light on the permeation mechanism of the penetrant gases, suggesting that the relative openness of the microporous topology promotes the anomalous diffusion of penetrant gases, which entails a deviation from the pore hopping mechanism usually observed in gas diffusion in polymers.