Projected Generative Diffusion Models for Constraint Satisfaction

Diffusion models are a class of generative models that function by progressively introducing noise into data and then methodically demonising it [17, 11]. They have revolutionized high-fidelity creation of complex data, and their applications have rapidly expanded beyond mere image synthesis, finding relevance in areas such as engineering [22, 24], automation [3, 13], chemistry [1, 12], and scientific research [2, 6]. Although diffusion models excel at generating content that is coherent and aligns closely with the original data distribution, their direct application in scenarios requiring stringent adherence to predefined criteria poses significant challenges. Particularly in domains where the generated data needs to not only resemble real-world examples but also rigorously comply with established specifications, physical laws, or engineering principles, conventional diffusion models are unable to ensure this level of precision. Given these limitations, one may consider an alternative approach: training a diffusion model on a data distribution that already aligns with these constraints.