This paper introduces an approach to endow generative diffusion processes the ability to satisfy and certify compliance with constraints and physical principles. The proposed method recast the traditional sampling process of generative diffusion models as a constrained optimization problem, steering the generated data distribution to remain within a specified region to ensure adherence to the given constraints. These capabilities are validated on applications featuring both convex and challenging, non-convex, constraints as well as ordinary differential equations, in domains spanning from synthesizing new materials with precise morphometric properties, generating physics-informed motion, optimizing paths in planning scenarios, and human motion synthesis.

翻译：本文提出一种方法，使生成式扩散过程具备满足并验证约束条件与物理原理的能力。该方法将传统生成扩散模型的采样过程重构为约束优化问题，引导生成数据分布保持在特定区域内，以确保对给定约束的遵循。这些能力在包含凸约束、具有挑战性的非凸约束以及常微分方程的应用中得到验证，其应用领域涵盖：合成具有精确形态特征的新材料、生成物理信息驱动的运动、规划场景中的路径优化以及人体运动合成。