Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

• URL: http://arxiv.org/abs/2602.17773v1
• Date: Thu, 19 Feb 2026 19:10:27 GMT
• Title: Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds
• Authors: Noah Trupin, Rahul Ghosh, Aadi Jangid,
• Abstract summary: We present a generative modeling framework for physically feasible two-dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions.<n>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.
• Score: 1.1668521361095043
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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