# Predict-Project-Renoise: Sampling Diffusion Models under Hard Constraints

arXiv: https://arxiv.org/abs/2601.21033 · HTML: https://mlsanity.com/paper/2601.21033/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [L-C2ST: Local Diagnostics for Posterior Approximations in Simulation-Based Inference](https://mlsanity.com/paper/2306.03580/) (2023)
- [Learn to Guide Your Diffusion Model](https://mlsanity.com/paper/2510.00815/) (2025)
- [Constrained Generative Modeling with Manually Bridged Diffusion Models](https://mlsanity.com/paper/2502.20371/) (2025)
- [Constrained Posterior Sampling: Time Series Generation with Hard Constraints](https://mlsanity.com/paper/2410.12652/) (2024)
- [Generating Physical Dynamics under Priors](https://mlsanity.com/paper/2409.00730/) (2024)
- [Constrained Diffusion for Protein Design with Hard Structural Constraints](https://mlsanity.com/paper/2510.14989/) (2025)
- [Enforcing governing equation constraints in neural PDE solvers via training-free projections](https://mlsanity.com/paper/2511.17258/) (2025)

## Built on by (most cited first, 2 of 2)

- [FlatLands: Generative Floormap Completion From a Single Egocentric View](https://mlsanity.com/paper/2603.16016/) (2026)
- [Inference-Time Projection for Physically Valid Biomolecular Diffusion Models](https://mlsanity.com/paper/2610.07037/) (2026)

## Abstract (arXiv)

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 108-dimensional atmospheric model, PPR simultaneously achieves low constraint violations and high distributional fidelity, a combination that existing methods fail to deliver.