Machine learning allows for efficient extraction of physical properties from stellar spectra that have been obtained by large surveys. The viability of machine-learning approaches has been demonstrated for spectra covering a variety of wavelengths and spectral resolutions, but most often for main-sequence (MS) or evolved stars, where reliable synthetic spectra provide labels and data for training. Spectral models of young stellar objects (YSOs) and low-mass MS stars are less well-matched to their empirical counterparts, however, posing barriers to previous approaches to classify spectra of such stars. In this work, we generate labels for YSOs and low-mass MS stars through their photometry. We then use these labels to train a deep convolutional neural network to predict logg, Teff, and Fe/H for stars with Apache Point Observatory Galactic Evolution Experiment (APOGEE) spectra in the DR14 data set. This "APOGEE Net" has produced reliable predictions of logg for YSOs, with uncertainties of within 0.1dex and a good agreement with the structure indicated by pre-MS evolutionary tracks, and it correlates well with independently derived stellar radii. These values will be useful for studying pre-MS stellar populations to accurately diagnose membership and ages.