Satellites enable widespread, regional or global surveillance of volcanoes and can provide the first indication of volcanic unrest or eruption. Here we consider Interferometric Synthetic Aperture Radar (InSAR), which can be employed to detect surface deformation with a strong statistical link to eruption. The ability of machine learning to automatically identify signals of interest in these large InSAR datasets has already been demonstrated, but data-driven techniques, such as convolutional neutral networks (CNN) require balanced training datasets of positive and negative signals to effectively differentiate between real deformation and noise. As only a small proportion of volcanoes are deforming and atmospheric noise is ubiquitous, the use of machine learning for detecting volcanic unrest is more challenging. In this paper, we address this problem using synthetic interferograms to train the AlexNet. The synthetic interferograms are composed of 3 parts: 1) deformation patterns based on a Monte Carlo selection of parameters for analytic forward models, 2) stratified atmospheric effects derived from weather models and 3) turbulent atmospheric effects based on statistical simulations of correlated noise. The AlexNet architecture trained with synthetic data outperforms that trained using real interferograms alone, based on classification accuracy and positive predictive value (PPV). However, the models used to generate the synthetic signals are a simplification of the natural processes, so we retrain the CNN with a combined dataset consisting of synthetic models and selected real examples, achieving a final PPV of 82%. Although applying atmospheric corrections to the entire dataset is computationally expensive, it is relatively simple to apply them to the small subset of positive results. This further improves the detection performance without a significant increase in computational burden.