This paper aims at predicting the behavior of untethered magnetic microrobotic swarm (MMS), which also include emergent behaviors, from raw grayscale time-series image data. A fast dimensionality reduction algorithm for compressing image data set is proposed. Then based on the algorithm, an interpretable machine learning method which is based on Proper Orthogonal Decomposition (POD) and polynomial regression is proposed for feature extraction of compressed image data set and for further dimensionality reduction. The proposed method is reduced-order and purely data-driven not assuming any knowledge of the physics and/or interacting dynamics of MMS, which are extremely challenging to ascertain for systems of this nature and dimension. Finally, experimental results on several kinds of synthetic MMs show the effectiveness of the proposed method with less prediction error while using more dimensionality reduction than existing approaches in the literature.