Projected Generative Diffusion Models for Constraint Satisfaction
Jacob K Christopher Stephen Baek Ferdinando Fioretto
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.