  We introduce the state-of-the-art deep learning Denoising Diffusion
Probabilistic Model (DDPM) as a method to infer the volume or number density of
giant molecular clouds (GMCs) from projected mass surface density maps. We
adopt magnetohydrodynamic simulations with different global magnetic field
strengths and large-scale dynamics, i.e., noncolliding and colliding GMCs. We
train a diffusion model on both mass surface density maps and their
corresponding mass-weighted number density maps from different viewing angles
for all the simulations. We compare the diffusion model performance with a more
traditional empirical two-component and three-component power-law fitting
method and with a more traditional neural network machine learning approach
(CASI-2D). We conclude that the diffusion model achieves an order of magnitude
improvement on the accuracy of predicting number density compared to that by
other methods. We apply the diffusion method to some example astronomical
column density maps of Taurus and the Infrared Dark Clouds (IRDCs) G28.37+0.07
and G35.39-0.33 to produce maps of their mean volume densities.
