This project develops generative modeling methods for scientific domains where physical or mathematical constraints must be respected. We focus on Schrödinger Bridge matching models, a class of diffusion-like generative models, and study how to incorporate domain constraints in a way that provides theoretical guarantees. In parallel, we adapt ideas from diffusion models to improve sampling of complex probability distributions arising in statistical mechanics, with the aim of obtaining more accurate and efficient approximations of stationary ergodic stochastic processes. More broadly, the project explores how modern generative-modeling frameworks can be extended beyond mainstream AI applications to mathematically grounded scientific settings. Main supervisor: Joakim Andén, Dept. of Mathematics, KTH