Atmospheric turbulence reduces the detection accuracy of orbital angular momentum (OAM) modes, which affects the performance of OAM optical communication. In this paper, we propose a method based on interferometry and a residual network (ResNet) to detect the OAM modes of ring Airy Gaussian vortex beams (RAGVBs) disturbed by atmospheric turbulence. The RAGVBs first interfere with spherical waves to obtain the sign features of the OAM modes, and then ResNet is employed to recognize OAM modes from the interferograms. The results demonstrate that the detection accuracy is higher than that of the OAM spectrum method under different turbulence strengths. The detection accuracy can even reach over 99% under strong fluctuations. Our research provides a reference for improving the performance of OAM optical communication through atmospheric turbulence.