In this paper, a new convolution long short-term memory networks model (CNN-LSTM), rotor finite element mimetic neural network (RFEMNN), is proposed and used for the diagnostics of rotor unbalance and shaft crack faults. RFEMNN aims to accomplish the recognition and localization of faults in the rotor structure. In this context, "Mimetic" refers to two levels: the topology of the structural division of the rotor finite element model through custom layers design and the topology of finite element solution process data flow by designing interlayer connections. The "Mimetic" theory is a new paradigm of physics-informed structure that enhances the physics consistency of machine learning (ML) and does not require complete and analytic physical knowledge with all known parameters. To train RFEMNN, this paper proposes a multi-label and multi-task supervised learning approach with one-hot encoded fault type labels, fault location labels, and vibration behavior labels. These labels are also involved in the training process of other tasks through the proposed physics-informed structure. The effectiveness of the proposed model is validated through a series of experimental platform tests on different rotor layouts and fault combination conditions. Several evaluation metrics are proposed to calculate the RFEMNN performance in a hierarchical 10-fold validation of the experimental data. The average test results show that the comprehensive diagnostics accuracy (on fault identification and error location aspects) is 94.7%, which is better than the benchmark models in the literature.