The characterization of the mechanical properties of crystalline materials is nowadays considered a routine computational task in DFT calculations. However, its high computational cost still prevents it from being used in high-throughput screening methodologies, where a cheaper estimate of the elastic properties of a material is required. In this work, we have investigated the accuracy of force field calculations for the prediction of mechanical properties and, in particular, for the characterization of the directional Poisson’s ratio. We analyze the behavior of about 600 000 hypothetical zeolitic structures at the classical level (a scale 3 orders of magnitude larger than previous studies), to highlight generic trends between mechanical properties and energetic stability. By comparing these results with DFT calculations on 991 zeolitic frameworks, we highlight the limitations of force field predictions, in particular for predicting auxeticity. We then used this reference DFT data as a training set for a machine learning algorithm, showing that it offers a way to build fast and reliable predictive models for anisotropic properties. The accuracies obtained are, in particular, much better than the current “cheap” approach for screening, which is the use of force fields. These results are a significant improvement over the previous work, due to the more difficult nature of the properties studied, namely, the anisotropic elastic response. It is also the first time such a large training data set is used for zeolitic materials.