让扩散模型在硬约束下采样，同时满足物理规律与数据分布。
Predict-Project-Renoise: Sampling Diffusion Models under Hard Constraints
- 通过预测-投影-重噪声三步迭代，实现约束下的采样。
- 在二维分布和气象模型中，约束违反率低且分布保真度高。
- 适合需要严格遵守物理定律的科学仿真任务。
扩散模型无法强制执行硬约束，但物理科学应用要求精确满足守恒律、边界条件和观测一致性。本文提出一个校正核，其唯一平稳分布即为每噪声水平下的约束边缘分布，并通过迭代地穿过去噪器投影并利用前向核重噪声来近似该核。由此得到的预测-投影-重噪声（PPR）算法，使预训练扩散模型能在硬约束下进行采样。其三个组件缺一不可：通过去噪器投影使样本贴近数据流形，而重噪声与迭代则推动样本趋近于约束边缘分布。在二维分布、Kuramoto-Sivashinsky方程以及维数达$10^8$的全球天气预报大气模型上，PPR同时实现了低约束违反率与高分布保真度，这是现有方法无法兼顾的。
原文摘要 · Abstract (English)
Diffusion models cannot enforce hard constraints, yet applications in the physical sciences demand exact satisfaction of conservation laws, boundary conditions, and observational consistency. In this work, we identify a corrector kernel whose unique stationary distribution is the constrained marginal at each noise level, and approximate it by iteratively projecting through the denoiser and renoising via the forward kernel. The resulting Predict-Project-Renoise (PPR) algorithm enables sampling from pretrained diffusion models under hard constraints. Its three components are each necessary: projecting through the denoiser keeps samples close to the data manifold, while renoising and iterating drive samples toward the constrained marginal. On 2D distributions, the Kuramoto-Sivashinsky equation, and global weather forecasting with a $10^8$-dimensional atmospheric model, PPR simultaneously achieves low constraint violations and high distributional fidelity, a combination that existing methods fail to deliver.
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