Collective oscillation of quasi-free electrons on the surface of metallic plasmonic nanoparticles (NPs) in the ultraviolet to near-infrared (NIR) region induces a strong electromagnetic enhancement around the NPs, which leads to numerous important applications. These interesting far- and near-field optical characteristics of the plasmonic NPs can be typically obtained from numerical simulations for theoretical guidance of NP design. However, traditional numerical simulations encounter irreconcilable conflicts between the accuracy and speed due to the high demand of computing power. In this work, we utilized the machine learning method, specifically the deep neural network (DNN), to establish mapping between the far-field spectra/near-field distribution and dimensional parameters of three types of plasmonic NPs including nanospheres, nanorods, and dimers. After the training process, both the forward prediction of far-field optical properties and the inverse prediction of on-demand dimensional parameters of NPs can be accomplished accurately and efficiently with the DNN. More importantly, we have achieved for the first time ultrafast and accurate prediction of two-dimensional on-resonance electromagnetic enhancement distributions around NPs by greatly reducing the amount of electromagnetic data via screening and resampling methods. These near-field predictions can be realized typically in less than 10-2 seconds on a laptop, which is 6 orders faster than typical numerical simulations implemented on a server. Therefore, we demonstrate that the DNN is an ultrafast, highly efficient, and computing resource-saving tool to investigate the far- and near-field optical properties of plasmonic NPs, especially for a number of important nano-optical applications such as surface-enhanced Raman spectroscopy, photocatalysis, solar cells, and metamaterials.