Generative deep learning methods have become powerful tools for modeling and predicting complex data distributions. While they produce perceptually convincing samples in imaging tasks, many scientific applications in climate sciences require outputs to satisfy strict mathematical constraints, such as conservation laws or dynamical equations. Enforcing such constraints at sampling time is therefore critical for physically consistent predictions. In this talk, we present a mathematical framework for constrained sampling, based on the variational formulation of Langevin dynamics and duality. Building on this foundation, we introduce a sampling algorithm that progressively enforces constraints via variable splitting. We present preliminary experimental results on physically constrained generative modeling tasks, including energy- and mass-conserving diffusion models for data assimilation.