  Generative models such as denoising diffusion models are quickly advancing
their ability to approximate highly complex data distributions. They are also
increasingly leveraged in scientific machine learning, where samples from the
implied data distribution are expected to adhere to specific governing
equations. We present a framework to inform denoising diffusion models of
underlying constraints on such generated samples during model training. Our
approach improves the alignment of the generated samples with the imposed
constraints and significantly outperforms existing methods without affecting
inference speed. Additionally, our findings suggest that incorporating such
constraints during training provides a natural regularization against
overfitting. Our framework is easy to implement and versatile in its
applicability for imposing equality and inequality constraints as well as
auxiliary optimization objectives.
