Automated classification of supernovae (SNe) based on optical photometric light-curve information is essential in the upcoming era of wide-field time domain surveys, such as the Legacy Survey of Space and Time (LSST) conducted by the Rubin Observatory. Photometric classification can enable real-time identification of interesting events for extended multiwavelength follow-up, as well as archival population studies. Here we present the complete sample of 5243 "SN-like" light curves (in gP1rP1iP1zP1) from the Pan-STARRS1 Medium-Deep Survey (PS1-MDS). The PS1-MDS is similar to the planned LSST Wide-Fast-Deep survey in terms of cadence, filters, and depth, making this a useful training set for the community. Using this data set, we train a novel semisupervised machine learning algorithm to photometrically classify 2315 new SN-like light curves with host galaxy spectroscopic redshifts. Our algorithm consists of an RF supervised classification step and a novel unsupervised step in which we introduce a recurrent autoencoder neural network (RAENN). Our final pipeline, dubbed SuperRAENN, has an accuracy of 87% across five SN classes (Type Ia, Ibc, II, IIn, SLSN-I) and macro-averaged purity and completeness of 66% and 69%, respectively. We find the highest accuracy rates for SNe Ia and SLSNe and the lowest for SNe Ibc. Our complete spectroscopically and photometrically classified samples break down into 62.0% Type Ia (1839 objects), 19.8% Type II (553 objects), 4.8% Type IIn (136 objects), 11.7% Type Ibc (291 objects), and 1.6% Type I SLSNe (54 objects).