Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

2026-02-19

AI Summary

This work addresses the limitation of existing diffusion models in generating incompressible flow fields, which often neglect physical constraints or enforce them only through soft penalties. To overcome this, we propose a diffusion-based generative framework that integrates hard geometric constraints with soft physical regularization. Our approach combines boundary-condition-guided diffusion, a physics-informed loss incorporating divergence penalties, and a projection-constrained reverse sampling scheme based on a geometry-aware Helmholtz–Hodge decomposition. Furthermore, we bridge the generative model with the intrinsic geometry of incompressible flows via constrained Langevin dynamics on manifolds. Experiments demonstrate that our method significantly outperforms baseline approaches on both analytical Navier–Stokes solutions and complex obstacle scenarios, achieving substantial improvements in divergence error, spectral accuracy, vorticity statistics, and boundary consistency.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Sampling/Simulation-based Search
• Natural Language Processing: Generation

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Responsible Web: Consent frameworks and practices on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• incompressible flow
• generative modeling
• physical constraints
• manifold
• boundary conditions

Innovation

• projection-constrained diffusion
• incompressible flow generation
• Helmholtz-Hodge decomposition
• physics-informed diffusion models
• manifold-constrained sampling