Equation versus AI: Predicting Density and Measuring Width of Molecular Clouds by Multiscale Decomposition

Abstract The interstellar medium is ubiquitous throughout the universe across multiple scales. In this study, we introduce the multiscale decomposition reconstruction method, an equation-based model designed to derive width maps of interstellar medium structures and predict their volume density distribution on the plane of the sky from input column density data. This approach applies the constrained diffusion algorithm , based on a simple yet common physical picture: as molecular clouds evolve to form stars, the density of the interstellar medium increases while the scale decreases. Validation on simulations confirms that this method accurately predicts volume density with minimal error. Notably, the equation-based model performs comparably to or even more accurately than the AI-based denoising diffusion probabilistic models that rely on vast parameters and substantial computational resources. Unlike the “black-box” nature of AI, our equation-based model offers full transparency, making it easier to interpret, debug, and validate. The simplicity, interpretability, and computational efficiency make it powerful not only for understanding complex astrophysical phenomena but also for complementing and enhancing AI-based methods.