Equation vs. AI: Predict Density and Measure Width of molecular clouds by Multiscale Decomposition

Abstract: Interstellar medium widely exists in the universe at multi-scales. In this study, we introduce the {\it Multi-scale Decomposition Reconstruction} method, an equation-based model designed to derive width maps of interstellar medium structures and predict their volume density distribution in the plane of the sky from input column density data. This approach applies the {\it Constrained Diffusion Algorithm}, based on a simple yet common physical picture: as molecular clouds evolve to form stars, the density of interstellar medium increases while their scale decreases. Extensive testing on simulations confirms that this method accurately predicts volume density with minimal error. Notably, the equation-based model performs comparably or even more accurately than the AI-based DDPM model(Denoising Diffusion Probabilistic Models), which relies on numerous parameters and high computational resources. Unlike the "black-box" nature of AI, our equation-based model offers full transparency, making it easier to interpret, debug, and validate. Their simplicity, interpretability, and computational efficiency make them indispensable not only for understanding complex astrophysical phenomena but also for complementing and enhancing AI-based methods.