Projected Generative Diffusion Models for Constraint Satisfaction

Generative diffusion models excel at robustly synthesizing coherent content from raw noise through a sequential process. However, their direct application in scenarios requiring outputs to adhere to specific, stringent criteria faces several severe challenges. This paper aims at overcome these challenges and introduces Projected Generative Diffusion Models (PGDM), an approach that recast traditional diffusion models sampling into a constrained-optimization problem. This enables the application of an iterative projections method to ensure that generated data faithfully adheres to specified constraints or physical principles. This paper provides theoretical support for the ability of PGDM to synthesize outputs from a feasible subdistribution under a restricted class of constraints while also providing large empirical evidence in the case of complex non-convex constraints and ordinary differential equations. These capabilities are demonstrated by physics-informed motion in video generation, trajectory optimization in path planning, and morphometric properties adherence in material science.