With the observations of an unprecedented number of oscillating subgiant stars expected from NASA's TESS mission, the asteroseismic characterization of subgiant stars will be a vital task for stellar population studies and for testing our theories of stellar evolution. To determine the fundamental properties of a large sample of subgiant stars efficiently, we developed a deep learning method that estimates distributions of fundamental parameters like age and mass over a wide range of input physics by learning from a grid of stellar models varied in eight physical parameters. We applied our method to four Kepler subgiant stars and compare our results with previously determined estimates. Our results show good agreement with previous estimates for three of them (KIC 11026764, KIC 10920273, KIC 11395018). With the ability to explore a vast range of stellar parameters, we determine that the remaining star, KIC 10005473, is likely to have an age 1 Gyr younger than its previously determined estimate. Our method also estimates the efficiency of overshooting, undershooting, and microscopic diffusion processes, from which we determined that the parameters governing such processes are generally poorly constrained in subgiant models. We further demonstrate our method's utility for ensemble asteroseismology by characterizing a sample of 30 Kepler subgiant stars, where we find a majority of our age, mass, and radius estimates agree within uncertainties from more computationally expensive grid-based modelling techniques.