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Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

Noah Trupin, Rahul Ghosh, Aadi Jangid

TL;DR

This work derives 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.

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.

Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

TL;DR

This work derives 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.

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.
Paper Structure (38 sections, 21 equations, 2 figures, 3 tables)

This paper contains 38 sections, 21 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our approach.
  3. Related Works
  4. Physics-informed diffusion and generative PDE surrogates.
  5. Constraint-enforcing generative sampling.
  6. Diffusion models for incompressible and geometry-conditioned flows.
  7. Flow-field representations for robotics and physics-informed planning.
  8. Methods
  9. Forward Diffusion on Vector Fields
  10. Geometry-aware training data
  11. Boundary-Conditioned Score Network
  12. Physics-Informed Objective
  13. Projection-Constrained Reverse Diffusion
  14. Periodic Domains
  15. Obstacle-Bounded Domains
  16. ...and 23 more sections

Figures (2)

  • Figure 1: Periodic energy spectra. Comparison of the learned velocity-field energy spectra $E(k)$ under the vanilla diffusion model (V) and the full physically-constrained model (TCP). Both models recover physically plausible $k$-decay, while TCP exhibits slightly improved mid–high frequency attenuation, consistent with enforcing incompressibility and boundary correctness.
  • Figure 2: Aggregate divergence, velocity error, and boundary violation metrics. Summary of quantitative performance across three dataset splits (Periodic, Obstacle, Obstacle–OOD) and four model variants: vanilla diffusion (V), training-constrained diffusion (TC), projection-only (P), and the fully constrained method (TCP). TCP and P achieve the lowest divergence and boundary error, while TCP provides consistently strong generalization across obstacles.