Abstract: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.
| Comments: | Code coming soon |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2601.21033 [cs.LG] |
| (or arXiv:2601.21033v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2601.21033 arXiv-issued DOI via DataCite |
Submission history
From: Omer Rochman-Sharabi [view email]
[v1]
Wed, 28 Jan 2026 20:50:19 UTC (9,587 KB)
[v2]
Wed, 13 May 2026 16:41:20 UTC (8,073 KB)