SIMBAD references

We introduce the state-of-the-art deep-learning denoising diffusion probabilistic model 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. 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 G28.37+0.07 and G35.39-0.33 to produce maps of their mean volume densities.