Automatic modulation classification enables intelligent communications and it is of crucial importance in today’s and future wireless communication networks. Although many automatic modulation classification schemes have been proposed, they cannot tackle the intra-class diversity problem caused by the dynamic changes of the wireless communication environment. In order to overcome this problem, inspired by face recognition, a novel automatic modulation classification scheme is proposed by using the multi-scale network in this paper. Moreover, a novel loss function that combines the center loss and the cross entropy loss is exploited to learn both discriminative and separable features in order to further improve the classification performance. Extensive simulation results demonstrate that our proposed automatic modulation classification scheme can achieve better performance than the benchmark schemes in terms of the classification accuracy. The influence of the network parameters and the loss function with the two-stage training strategy on the classification accuracy of our proposed scheme are investigated.