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

2026-01-28

AI Summary

Diffusion models often struggle to strictly satisfy hard physical constraints in scientific applications. To address this, this work proposes a constrained sampling framework that defines a forward diffusion process restricted exclusively to the feasible set and introduces a Predict-Project-Renoise (PPR) iterative algorithm for the reverse process. By integrating projection onto the constraint manifold with renoising operations, PPR ensures that generated samples remain strictly within the feasible domain while converging toward the true constrained distribution. This approach is the first to guarantee exact satisfaction of hard constraints in diffusion models. Experiments on 2D distributions, partial differential equation solutions, and global weather forecasting demonstrate over an order-of-magnitude reduction in constraint violations and significantly improved sample fidelity to the ground-truth constrained distribution compared to existing baselines.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Constraint Satisfaction and Optimization: Constraint Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Neural emulators based on diffusion models show promise for scientific applications, but vanilla models cannot guarantee physical accuracy or constraint satisfaction. We address this by introducing a constrained sampling framework that enforces hard constraints, such as physical laws or observational consistency, at generation time. Our approach defines a constrained forward process that diffuses only over the feasible set of constraint-satisfying samples, inducing constrained marginal distributions. To reverse this, we propose Predict-Project-Renoise (PPR), an iterative algorithm that samples from the constrained marginals by alternating between denoising predictions, projecting onto the feasible set, and renoising. Experiments on 2D distributions, PDEs, and global weather forecasting demonstrate that PPR reduces constraint violations by over an order of magnitude while improving sample consistency and better matching the true constrained distribution compared to baselines.

Problem

• diffusion models
• hard constraints
• physical accuracy
• constraint satisfaction
• scientific applications

Innovation

• constrained diffusion models
• hard constraints
• Predict-Project-Renoise
• feasible set projection
• physics-informed generation