We use machine learning to predict shape persistence and cavity size in porous organic cages. The majority of hypothetical organic cages suffer from a lack of shape persistence and as a result lack intrinsic porosity, rendering them unsuitable for many applications. We have created the largest computational database of these molecules to date, numbering 63,472 cages, formed through a range of reaction chemistries and in multiple topologies. We study our database and identify features which lead to the formation of shape persistent cages. We find that the imine condensation of trialdehydes and diamines in a [4+6] reaction is the most likely to result in shape persistent cages, whereas thiol reactions are most likely to give collapsed cages. Using this database, we develop machine learning models capable of predicting shape persistence with an accuracy of up to 93%, reducing the time taken to predict this property to milliseconds, and removing the need for specialist software. In addition, we develop machine learning models for two other key properties of these molecules, cavity size and symmetry. We provide open-source implementations of our models, together with the accompanying data sets, and an online tool giving users access to our models to easily obtain predictions for a hypothetical cage prior to a synthesis attempt.