Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

TL;DR

This paper introduces a diffusion-based generative model for physically feasible incompressible flows that enforces exact divergence-free conditions using a projection method, improving accuracy and boundary consistency.

Contribution

It combines boundary-conditioned diffusion, physics-informed training, and a projection process to ensure exact incompressibility in flow generation, unifying soft and hard physical constraints.

Findings

Significantly improved divergence and spectral accuracy.

Enhanced boundary condition enforcement.

Better vorticity statistics compared to baselines.

Abstract

We present a generative modeling framework for synthesizing physically feasible two-dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions. Whereas existing diffusion-based flow generators either ignore physical constraints, impose soft penalties that do not guarantee feasibility, or specialize to fixed geometries, our approach integrates three complementary components: (1) a boundary-conditioned diffusion model operating on velocity fields; (2) a physics-informed training objective incorporating a divergence penalty; and (3) a projection-constrained reverse diffusion process that enforces exact incompressibility through a geometry-aware Helmholtz-Hodge operator. We derive the method as a discrete approximation to constrained Langevin sampling on the manifold of divergence-free vector fields, providing a connection between modern diffusion models and geometric constraint enforcement in incompressible flow spaces. Experiments on analytic Navier-Stokes data and obstacle-bounded flow configurations demonstrate significantly improved divergence, spectral accuracy, vorticity statistics, and boundary consistency relative to unconstrained, projection-only, and penalty-only baselines. Our formulation unifies soft and hard physical structure within diffusion models and provides a foundation for generative modeling of incompressible fields in robotics, graphics, and scientific computing.