Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds Research

Generating physically consistent incompressible velocity fields remains challenging for diffusion modes: standard DDPMs produce samples with nonzero divergence and boundary violations, while training-time physics penalties alone cannot guarantee validity during sampling. We present projection-guided diffusion, a task-specialized approach that integrates a Helmholtz–Hodge projection into every step of the reverse diffusion process. Our method combines (i) geometry-conditioned score estimation, (ii) a divergence-aware training objective, and (iii) per-step incompressible projection that corrects intermediate states rather than only the final sample. We evaluate vanilla, training-constrained, projection-only, and full projection-guided variants on periodic domains and obstacle-rich geometries. The full model consistently yields the lowest divergence, smallest boundary-normal velocities, and the most accurate flow reconstruction. Additional experiments on out-of-distribution geometries, including unseen obstacle sizes, shapes, and multi-obstacle configurations, show improved geometric generalization relative to baselines. Our results demonstrate that integrating domain-specific projections with score-based generative modeling substantially improves the physical feasibility and robustness of diffusion-generated incompressible flows, providing a practical foundation for physics-aware generative models in fluid simulation and scientific machine learning.