Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

Noah Trupin; Rahul Ghosh; Aadi Jangid

physics.flu-dynFeb 19, 2026

Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds

Authors: Noah Trupin, Rahul Ghosh, Aadi Jangid

Organizations: Department of Computer Science, Purdue University

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.

Figures & tables

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.

Method	$\\|\nabla\!\cdot u\\|_{L^{2}(\Omega)}\ \downarrow$	$\\|u-u^{\star}\\|_{L^{2}(\Omega)}\ \downarrow$	$\int_{\Omega}\|\nabla\!\cdot u\|\,\mathrm{d}x\ \downarrow$
V	0.0033 $\pm$ 1.8e-04	0.0206 $\pm$ 2.0e-03	0.0035 $\pm$ 1.9e-04
TC	0.0021 $\pm$ 1.1e-04	0.0151 $\pm$ 2.0e-03	0.0022 $\pm$ 1.2e-04
P	0.0005 $\pm$ 1.5e-05	0.0493 $\pm$ 7.9e-04	0.0005 $\pm$ 1.6e-05
TCP	0.0004 $\pm$ 1.9e-05	0.0480 $\pm$ 6.1e-04	0.0004 $\pm$ 2.1e-05

Table 1: Periodic incompressible flows. Mean

L^{2}

divergence, velocity error, and integrated divergence over the periodic test set.

Method	$\\|\nabla\!\cdot u\\|_{L^{2}(\Omega)}\ \downarrow$	$\\|u-u^{\star}\\|_{L^{2}(\Omega)}\ \downarrow$	$\int_{\Omega}\|\nabla\!\cdot u\|\,\mathrm{d}x\ \downarrow$	$\\|u\cdot n\\|_{L^{2}(\partial\Omega)}\ \downarrow$
V	0.0896 $\pm$ 2.99e-02	0.1366 $\pm$ 5.04e-02	0.0856 $\pm$ 3.02e-02	0.4245 $\pm$ 9.46e-02
TC	0.0929 $\pm$ 2.95e-02	0.1419 $\pm$ 5.16e-02	0.0884 $\pm$ 2.91e-02	0.4345 $\pm$ 9.65e-02
P	0.0445 $\pm$ 1.19e-02	0.0889 $\pm$ 2.66e-02	0.0365 $\pm$ 1.03e-02	0.3219 $\pm$ 5.17e-02
TCP	0.0432 $\pm$ 1.17e-02	0.0840 $\pm$ 2.68e-02	0.0352 $\pm$ 9.96e-03	0.3180 $\pm$ 5.10e-02

Table 2: Obstacle flows (in-distribution geometries). Mean divergence, velocity error, integrated divergence over the fluid region, and boundary violation

\|u\cdot n\|_{L^{2}(\partial\Omega)}

Method	$\\|\nabla\!\cdot u\\|_{L^{2}(\Omega)}\ \downarrow$	$\\|u-u^{\star}\\|_{L^{2}(\Omega)}\ \downarrow$	$\int_{\Omega}\|\nabla\!\cdot u\|\,\mathrm{d}x\ \downarrow$	$\\|u\cdot n\\|_{L^{2}(\partial\Omega)}\ \downarrow$
V	0.0869 $\pm$ 2.68e-02	0.1312 $\pm$ 4.54e-02	0.0830 $\pm$ 2.74e-02	0.4190 $\pm$ 8.80e-02
TC	0.0939 $\pm$ 3.10e-02	0.1451 $\pm$ 5.37e-02	0.0897 $\pm$ 3.10e-02	0.4400 $\pm$ 9.82e-02
P	0.0445 $\pm$ 1.22e-02	0.0885 $\pm$ 2.78e-02	0.0366 $\pm$ 1.04e-02	0.3217 $\pm$ 5.35e-02
TCP	0.0423 $\pm$ 1.15e-02	0.0818 $\pm$ 2.62e-02	0.0347 $\pm$ 9.95e-03	0.3153 $\pm$ 5.07e-02

Table 3: Generalization to held-out obstacle geometries. Same metrics as Table 2 . Models are evaluated on unseen obstacle masks.

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Learning-based models for fluid dynamics often operate in unconstrained function spaces, leading to physically inadmissible, unstable simulations. While penalty-based methods offer soft regularization, they provide no structural guarantees, resulting in spurious divergence and long-term collapse. In this work, we introduce a unified framework that enforces the incompressible continuity equation as a hard, intrinsic constraint for both deterministic and generative modeling. First, to project deterministic models onto the divergence-free subspace, we integrate a differentiable spectral Leray projection grounded in the Helmholtz-Hodge decomposition, which restricts the regression hypothesis space to physically admissible velocity fields. Second, to generate physically consistent distributions, we show that simply projecting model outputs is insufficient when the prior is incompatible. To address this, we construct a divergence-free Gaussian reference measure via a curl-based pushforward, ensuring the entire probability flow remains subspace-consistent by construction. Experiments on 2D Navier-Stokes equations demonstrate exact incompressibility up to discretization error and substantially improved stability and physical consistency.

Abstract

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