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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