The analysis of superpositions of Orbital Angular Momentum (OAM) modes is a challenging problem, particularly when atmospheric turbulence is present or when the phase structure of the wavefront is not available. In such conditions it is not possible to correct the distortions and reconstruct the vorticial phase structure: the rings and petals that characterize the intensity profiles of such beams become deformed and may even lose integrity. These artifacts may compromise the possibility of establishing free-space optical links based on OAM superpositions. We propose using a particular selection of Laguerre-Gauss modes and convolutional neural networks for a reliable classification of superpositions of two modes. The network (based on a pre-trained network AlexNet that combines convolutional and fully-connected layers) is trained as a classifier based on 2-d intensity profiles that can be obtained from a digital camera. For illustrating the proposed method, we used simulations of light beams propagated through L = 1 km with three levels of turbulence: C-n(2) is an element of {2 x 10(-15), 9.24 x 10(-15), 2.9 x 10(-14)} m(-2/3). The emitted beams are made up of 2 different Laguerre-Gauss modes with OAM between -15 and +15, and radial indices between 0 and 3. Classification results show that the radial index can be used effectively to enlarge the set of information symbols.