This paper demonstrates a deep learning based methodology for the rapid simulation of planar microwave circuits based on their layouts. We train convolutional neural networks to compute the scattering parameters of general, twoport circuits consisting of a metallization layer printed on a grounded dielectric substrate, by processing the metallization pattern along with the thickness and dielectric permittivity of the substrate. This approach harnesses the efficiency of convolutional neural networks with pattern recognition tasks and extends previous efforts to employ neural networks for the simulation of parameterized circuit geometries. Training is based on full-wave simulation data generated via the Finite-Difference Time-Domain (FDTD) method over a target frequency range. To accelerate the generation of such data, we build a hybrid neural network including recursive neural network modules, to compensate numerical dispersion errors in coarse-grid FDTD. This novel dispersion compensation scheme allows us to generate accurate FDTD training data from fast, coarse-grid simulations.