Utilizing fractional vortex beams (FVBs) as information carriers can significantly enhance the capacity of communication systems. However, the small gap difference between adjacent fractional orbital angular momentum (FOAM) modes makes FVBs highly sensitive to atmospheric turbulence. Therefore, precise measurement of distorted FOAM modes is crucial for practical FVBs-based communication systems. To fully utilize the beam intensity information and the triangular diffraction pattern information, we propose a dual-channel deep learning model with a hybrid architecture combining convolutional neural network (CNN) and vision transformer (ViT). The beam intensity information is extracted using the CNN, while the diffraction pattern information is extracted using the ViT. Then, by combining the complementary feature information from the intensity distribution of FVBs and their triangular diffraction patterns, this model can effectively identify the FOAM modes. The results show that the proposed model only requires a relatively small number of samples to reach convergence, namely 100 sets of data under weak turbulence and 400 sets of data under strong turbulence. Moreover, within a transmission distance of 1000 m, the proposed model can identify 101 FOAM modes with a mode spacing of 0.1 with an accuracy of 100% under weak and moderate turbulences, and maintains 98.12% accuracy under strong turbulence. Furthermore, the model can expand the detection range of turbulence intensity with only a minimal loss in accuracy, exhibiting strong generalization ability under unknown atmospheric turbulence strengths, thus providing a novel approach for accurately identifying FOAM modes.