Laguerre-Gaussian (LG) beams possess radial quantum number p in addition to orbital angular momentum (OAM) dimension, and thus LG beams can provide more physical degrees of freedom for applications such as optical communication and optical computing. However, the recognition accuracy of the LG beam pattern detection method, which is commonly used by interference and diffraction mechanisms, is significantly reduced when it is disturbed by atmospheric turbulence (AT), which limits its practical application. We propose a diffraction neural network (DNN)-based LG beam recognition method that achieves p in the range of 1-3. Even in the case of strong turbulence intensity and diffraction distance of 5 m, the recognition accuracy still reaches more than 95%. This DNN method can provide an effective way to accurately identify LG beam patterns, and has potential applications in high-capacity OAM communication and high-dimensional quantum information processing.