Bypassing the need for energy-based calculations entirely, machine learning offers a pathway to accelerate materials discovery by allowing large-scale high-throughput screening of hypothetical materials. In particular, machine learning techniques have been used in many fields of chemistry including the pharmaceutical and material science industries. It is especially powerful in predicting properties for which molecular simulations are complex, such as melting points, solubility and crystallisation. Recently, machine learning methods have also demonstrated their effectiveness for identifying porosity in organic molecular crystals, hydrogen storage performance and xenon/krypton separation in the Materials Genome database. In this work, we use machine learning methods to predict the elastic properties of pure silica zeolites. Using an accurate training set of elastic properties calculated by density functional theory (DFT) the machine learning approach is able to predict bulk and shear moduli of zeolites with impressive accuracy. By comparison to five state-of-the-art widely used interatomic potentials, we show that machine learning based purely on the structures gives better results than all classical potentials. Moreover, intrinsic feature selection performed by the machine learning algorithm has highlighted the influence of a zeolite's local, structural and porosity features on the resulting elastic moduli. Finally, to demonstrate the efficiency of this model, we have predicted the elastic response of 590 448 hypothetical zeolitic structures, with a computational cost of mere hundreds of CPU hours, which are primarily required for descriptor calculation. The properties of this massive database provide important evidence to the physical and chemical intuition researchers use to understand the mechanical stability of zeolites and related nanoporous materials.