Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets $Σ$ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto $Σ$, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.

翻译：在满足物理、几何或安全需求（例如分子生成、机器人学）的科学与工程应用中，约束集合内的生成建模至关重要。我们提出了一个统一框架，用于在一般非凸可行集$Σ$上构建约束扩散模型，该模型在整个扩散过程中同时强制执行等式与不等式约束。该框架整合了过阻尼与欠阻尼动力学，用于前向与反向采样。一项核心算法创新是引入了一种计算高效的着陆机制，以替代昂贵且通常定义模糊的到$Σ$上的投影操作，从而无需迭代牛顿求解或投影失败即可确保可行性。通过利用欠阻尼动力学，我们加速了向先验分布的混合过程，有效缓解了约束扩散通常带来的高模拟成本。实验表明，该方法在保持样本质量的同时，降低了训练与推理过程中的函数评估次数与内存使用量。在包含等式与混合约束的基准测试中，我们的方法达到了与现有最优基线相当的样本质量，同时显著降低了计算成本，为非凸可行集上的扩散提供了一种实用且可扩展的解决方案。