Exploring and modeling rain generation mechanism is critical for augmenting paired data to ease training of rainy image processing models. Against this task, this study proposes a novel deep learning based rain generator, which fully takes the physical generation mechanism underlying rains into consideration and well encodes the learning of the fundamental rain factors (i.e., shape, orientation, length, width and sparsity) explicitly into the deep network. Its significance lies in that the generator not only elaborately design essential elements of the rain to simulate expected rains, like conventional artificial strategies, but also finely adapt to complicated and diverse practical rainy images, like deep learning methods. By rationally adopting filter parameterization technique, we first time achieve a deep network that is finely controllable with respect to rain factors and able to learn the distribution of these factors purely from data. Our unpaired generation experiments demonstrate that the rain generated by the proposed rain generator is not only of higher quality, but also more effective for deraining and downstream tasks compared to current state-of-the-art rain generation methods. Besides, the paired data augmentation experiments, including both in-distribution and out-of-distribution (OOD), further validate the diversity of samples generated by our model for in-distribution deraining and OOD generalization tasks.