Information transfer rates in optical communications may be dramatically increased by making use of spatially non-Gaussian states of light. Here, we demonstrate the ability of deep neural networks to classify numerically generated, noisy Laguerre-Gauss modes of up to 100 quanta of orbital angular momentum with near-unity fidelity. The scheme relies only on the intensity profile of the detected modes, allowing for considerable simplification of current measurement schemes required to sort the states containing increasing degrees of orbital angular momentum. We also present results that show the strength of deep neural networks in the classification of experimental superpositions of Laguerre-Gauss modes when the networks are trained solely using simulated images. It is anticipated that these results will allow for an enhancement of current optical communications technologies.