Metasurfaces composed of meta-atoms provide promising platforms for manipulating amplitude, phase, and polarization of light. However, the traditional design methods of metasurfaces are time consuming and laborious. Here, we propose a bidirectional cascaded deep neural network with a pretrained autoencoder for rapid design of dielectric metasurfaces in the range of 450 nm to 850 nm. The forward model realizes a prediction of amplitude and phase responses with a mean absolute error of 0.03. Meanwhile, the backward model can retrieve patterns of meta-atoms in an inverse-design manner. The availability of this model is demonstrated by database establishment, model evaluation, and generalization testing. Furthermore, we try to reveal the mechanism behind the model in a visualization way. The proposed approach is beneficial to reduce the cost of computation burden and improve nanophotonic design efficiency for solving electromagnetic on-demand design issues automatically.