Beyond Blur: A Fluid Perspective on Generative Diffusion Models

We propose a novel PDE-driven corruption process for generative image synthesis based on advection-diffusion processes which generalizes existing PDE-based approaches. Our forward pass formulates image corruption via a physically motivated PDE that couples directional advection with isotropic diffusion and Gaussian noise, controlled by dimensionless numbers (Péclet, Fourier). We implement this PDE numerically through a GPU-accelerated custom Lattice Boltzmann solver for fast evaluation. To induce realistic ``turbulence,'' we generate stochastic velocity fields that introduce coherent motion and capture introduce multi-scale mixing. A diffusion model then learns to invert the advection-diffusion operator, reconstructing fine details from coarsely transported images and thus constituting a novel generative diffusion model. We discuss how previews methods emerge as specific cases (zero velocity or zero blur) of our operator, demonstrating that our advection-diffusion framework generalizes prior PDE-based diffusion techniques. This work bridges fluid dynamics, dimensionless PDE theory, and deep generative modeling, offering a fresh perspective on physically informed image corruption processes for diffusion-based synthesis.