Constrained Synthesis with Projected Diffusion Models (2402.03559v3)

Abstract: This paper introduces an approach to endow generative diffusion processes the ability to satisfy and certify compliance with constraints and physical principles. The proposed method recast the traditional sampling process of generative diffusion models as a constrained optimization problem, steering the generated data distribution to remain within a specified region to ensure adherence to the given constraints. These capabilities are validated on applications featuring both convex and challenging, non-convex, constraints as well as ordinary differential equations, in domains spanning from synthesizing new materials with precise morphometric properties, generating physics-informed motion, optimizing paths in planning scenarios, and human motion synthesis.

• The paper presents PGDM, which integrates constraint satisfaction into the diffusion sampling process for high-quality data synthesis.
• It utilizes iterative projection to enforce complex, non-convex constraints without compromising the fidelity of generated samples.
• Empirical evaluations in areas like physics-informed video, motion planning, and material synthesis demonstrate state-of-the-art performance and practical applicability.

Introduction

Generative diffusion models have garnered significant attention for their ability to create high-fidelity data from complex distributions. While they perform exceptionally well in image synthesis and other applications, their direct use in scenarios with specific, strict requirements remains a formidable challenge. Relying on standard methods, such as conditional diffusion models or post-processing techniques, often leads to outputs that may look plausible but do not strictly adhere to the required constraints.

The approach introduced in this paper, Projected Generative Diffusion Models (PGDM), presents a solution to this problem. PGDM reframes the traditional diffusion sampling strategy as a constrained optimization challenge, whereby adhering to constraints or physical laws is as critical as the generation quality. The proposed methodology uses iterative projections across the diffusion process, demonstrating the ability to generate samples that satisfy complex non-convex constraints and physical principles.

Diffusion Models and PGDM

Generative models, such as diffusion models, progress by systematically introducing noise into data and reversing this process for sample synthesis. Traditional diffusion models struggle to ensure generated content meets precise specifications, often generating samples that, while similar to real-world data, fail to comply with stringent criteria.

PGDM addresses these limitations by integrating a projection operator into the iterative sampling process, ensuring each generated sample falls within a feasible solution space, defined by the imposed constraints. This is achieved without compromising the model's goal of generating samples resembling the true data distribution, striking a balance between fidelity and constraint compliance. Noteworthy is PGDM's demonstrated capability to achieve state-of-the-art FID scores while strictly adhering to constraints.

Constraint-Aware Diffusion and Applications

PGDM's utility is underscored through rigorous empirical evaluations across domains that demand stringent compliance with constraints. These include synthesizing physics-informed video sequences consistent with differential equations, generating optimized motion planning trajectories that circumvent obstacles, and fabricating materials with specific morphometric properties. The empirical evidence from these domains underscores PGDM's capacity to generate high-quality, constraint-abiding content—a capability both theoretically supported and practically demonstrated through the approach's versatility in various complex scenarios.

Implications and Considerations

PGDM's introduction presents to the AI community a generative model architecture capable of honoring specific constraints and physical principles without sacrificing generation quality. As innovations in LLMs continue, methodologies like PGDM open pathways to deploy AI models in science and engineering, where data generation must often meet exacting standards.

A consideration in deploying PGDM involves the computational overhead incurred by iterative projections—a factor that may require trade-offs between performance and computational efficiency. Additionally, forward processing of constraints may appear an obvious extension, but evidence suggests this could actually decrease performance, further emphasizing PGDM's current practical design.

Conclusion

Projected Generative Diffusion Models stand out by seamlessly integrating constraint satisfaction into the generative sampling process, producing results that have immediate implications for applied research and industry applications requiring precision. PGDM heralds a significant step forward, enabling diffusion models to expand beyond traditional domains into fields where strict adherence to constraints is non-negotiable. This innovation paves the way for future research endeavors aimed at refining constraint representation and optimization in large-scale, multifaceted generative modeling tasks.