Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling

2025-05-03

AI Summary

Diffusion models traditionally operate in continuous intensity spaces, making them ill-suited for scientific tasks—such as particle counting and materials unit modeling—that demand strict conservation of discrete quantities like mass. This work introduces the first diffusion framework grounded in continuous-time, discrete-state jump processes, ensuring **strict mass conservation** in both forward and reverse dynamics while supporting intensity-conditioned generation. Our method integrates discrete-space stochastic dynamics, mass-constrained parameterization, and conditional sampling mechanisms to construct a differentiable architecture for discrete image synthesis and inpainting. Experiments demonstrate high-fidelity generation, classifier-guided sampling, and inpainting on standard image benchmarks; moreover, on materials microstructure generation—a canonical conservation-law-driven task—our approach significantly outperforms continuous-space diffusion baselines. These results validate the framework’s effectiveness and generalizability for scientific modeling governed by physical conservation laws.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

Generative diffusion models have achieved remarkable success in producing high-quality images. However, because these models typically operate in continuous intensity spaces - diffusing independently per pixel and color channel - they are fundamentally ill-suited for applications where quantities such as particle counts or material units are inherently discrete and governed by strict conservation laws such as mass preservation, limiting their applicability in scientific workflows. To address this limitation, we propose Discrete Spatial Diffusion (DSD), a framework based on a continuous-time, discrete-state jump stochastic process that operates directly in discrete spatial domains while strictly preserving mass in both forward and reverse diffusion processes. By using spatial diffusion to achieve mass preservation, we introduce stochasticity naturally through a discrete formulation. We demonstrate the expressive flexibility of DSD by performing image synthesis, class conditioning, and image inpainting across widely-used image benchmarks, with the ability to condition on image intensity. Additionally, we highlight its applicability to domain-specific scientific data for materials microstructure, bridging the gap between diffusion models and mass-conditioned scientific applications.

Problem

• Existing diffusion models fail with discrete, mass-preserving scientific data
• Proposing Discrete Spatial Diffusion for strict mass conservation
• Enabling diffusion models for scientific applications with discrete quantities

Innovation

• Discrete-state jump process for mass preservation
• Continuous-time discrete spatial diffusion framework
• Stochasticity via discrete formulation in diffusion