Project and Generate: Divergence-Free Neural Operators for Incompressible Flows

Xigui Li; Hongwei Zhang; Ruoxi Jiang; Deshu Chen; Chensen Lin; Limei Han; Yuan Qi; Xin Guo

cs.LGMar 25, 2026

Project and Generate: Divergence-Free Neural Operators for Incompressible Flows

Authors: Xigui Li, Hongwei Zhang, Ruoxi Jiang, Deshu Chen, Chensen Lin, Limei Han, Yuan Qi, Xin Guo, +1 more

Organizations: Artificial Intelligence Innovation and Incubation Institute, Fudan University, Shanghai, China · Shanghai Academy of Artificial Intelligence for Science, Shanghai, China

Abstract

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.

Figures & tables

Figure 1: Motivation for the Project & Generate framework illustrated using numerical solvers. (a) Visual comparison. Velocity streamlines (top, colored by error magnitude), divergence fields (middle), and recovered physical pressure (bottom). The Baseline approach (middle), which enforces only the momentum equation, exhibits severe violations of mass conservation with nonzero divergence and noisy pressure fields. A Leray-projected solution (right) enforces incompressibility via a spectral Leray projection, yielding a divergence-free velocity field ( $\nabla\cdot\boldsymbol{u}=0$ ) with machine precision and recovering a smooth, physically consistent pressure field. The result is visually indistinguishable from the Ground Truth numerical solution (left). (b) Computational paradigms. Left: Classical numerical solvers (CFD) enforce incompressibility through iterative Poisson solves, which require accessing the pressure variable and are computationally expensive. Middle: Unconstrained end-to-end learning approaches typically violate the continuity constraint. Right: We leverage Leray projection to decouple dynamics from constraint enforcement, enabling hard physical constraints to be imposed via a fast, differentiable spectral operator.

Figure 2: Schematic of the proposed Projective Framework. The grey surface represents the divergence-free subspace $\mathcal{V}$ (the physical manifold). (a) Projective Regression: The framework takes an input and uses a base neural operator $f_{\theta}$ to produce an intermediate prediction in the ambient space (shown hovering above the manifold). This unconstrained prediction is then strictly mapped onto $\mathcal{V}$ via the spectral Leray projection $\mathcal{P}$ . The composite system constitutes a Divergence-Free Neural Operator , ensuring the final output is physically consistent. (b) Projective Generative Model: The generation process is initialized with a sample from a divergence-free noise distribution (bottom left) constrained to $\mathcal{V}$ . During the probability flow evolution ( $\tau\in[0,1]$ ), the vector field predicted by $f_{\theta}$ is continuously projected via $\mathcal{P}$ (red arrows) back onto the manifold. This enforces hard constraints at every step of the flow matching process, ensuring the entire trajectory stays within the physically admissible subspace.

Panel of Figure 3 — Figure 3: Flow Field Visualization. Top Row: Snapshot of velocity components ( $u,v$ ). Bottom Row: Streamlines and Pressure field. The pressure is reconstructed from the predicted velocity field via the pressure Poisson equation, serving as a proxy for the dynamical consistency of the flow structure.

Regression Models
Model	Prediction ( $T_{15\text{-}49}$ )			Short-term ( $T_{50\text{-}100}$ )			Long-term ( $T_{101\text{-}300}$ )
Model	U-MSE	V-MSE	Div	U-MSE	V-MSE	Div	U-MSE	V-MSE	Div
Base	$4.0\pm 3.6$	$5.6\pm 5.4$	$625.0\pm 576.0$	$16.6\pm 17.5$	$17.5\pm 15.5$	$21.7\pm 18.6$	$4.3\text{e}3\pm 5.1\text{e}3$	$4.4\text{e}3\pm 5.2\text{e}3$	$1.5\text{e}5\pm 1.9\text{e}5$
Ours	$\mathbf{3.4\pm 3.0}$	$\mathbf{3.2\pm 2.9}$	$\mathbf{0.1\pm 0.1}$	$\mathbf{5.9\pm 4.3}$	$\mathbf{6.9\pm 5.4}$	$\mathbf{0.0\pm 0.0}$	$227.0\pm 335.0$	$210.0\pm 328.0$	$118.0\pm 160.0$
Generative Models
Base	$4.3\pm 3.6$	$4.6\pm 4.3$	$1.7\pm 1.6$	$1.0\pm 0.7$	$1.2\pm 1.4$	$0.6\pm 2.5$	Inf	Inf	Inf

Table 1: Quantitative Comparison on Re = 1000. We report Mean

\pm

Std over the test set. Bold indicates best performance. Gray values indicate model divergence. Prediction and Short-term metrics are scaled by

10^{-3}

and

10^{-2}

respectively.

Figure 4: Enstrophy Spectrum Analysis. Comparison of energy spectra. Our method accurately recovers the inertial range scalings, preserving the correct energy cascade across scales.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Category	Meaning
$\mathbf{y}\in\mathcal{V}$	data sample	divergence-free velocity field from the dataset
$\nu$	data distribution	probability distribution of data samples on $\mathcal{V}$
$\mu_{0}$	reference distribution	divergence-free Gaussian noise on $\mathcal{V}$
$\mathbf{u}_{0}\sim\mu_{0}$	noise sample	reference divergence-free random field
$\mathbf{u}_{\tau}^{\mathbf{y}}$	interpolated sample	noisy interpolation between $\mathbf{u}_{0}$ and data sample $\mathbf{y}$
$\mu_{\tau}^{\mathbf{y}}$	conditional distribution	distribution induced by $\mathbf{u}_{\tau}^{\mathbf{y}}$

Appendix

Table 2: Summary of notation. We distinguish between data samples, probability measures, and vector fields used in divergence-free flow matching.

Model	Stage	Total MSE	U-MSE	V-MSE	Div-MSE
Gen. (Base)	Prediction ( $T_{15\text{-}50}$ )	$0.0046\pm 0.0035$	$0.0044\pm 0.0033$	$0.0047\pm 0.0038$	$0.0075\pm 0.0120$
	Short-term Ext. ( $T_{51\text{-}150}$ )	$2.40\text{e}28\pm\text{NaN}$	$9.22\text{e}30\pm\text{NaN}$	$2.60\text{e}28\pm\text{NaN}$	$2.64\text{e}26\pm\text{NaN}$
	Long-term Ext. ( $T_{151\text{-}\text{End}}$ )	$0.0076\pm 0.0009$	$0.0079\pm 0.0009$	$0.0072\pm 0.0012$	$0.0024\pm 0.0046$
Gen. (Ours)	Prediction ( $T_{15\text{-}50}$ )	$\mathbf{0.0011\pm 0.0010}$	$\mathbf{0.0011\pm 0.0009}$	$\mathbf{0.0011\pm 0.0011}$	$\mathbf{9.3\text{e-}7\pm 1.2\text{e-}7}$
	Short-term Ext. ( $T_{51\text{-}150}$ )	$\mathbf{0.0073\pm 0.0024}$	$\mathbf{0.0071\pm 0.0025}$	$\mathbf{0.0074\pm 0.0023}$	$\mathbf{7.5\text{e-}7\pm 4.0\text{e-}8}$
	Long-term Ext. ( $T_{151\text{-}\text{End}}$ )	$\mathbf{0.0074\pm 0.0013}$	$\mathbf{0.0073\pm 0.0013}$	$\mathbf{0.0076\pm 0.0012}$	$\mathbf{7.4\text{e-}7\pm 2.7\text{e-}8}$

Appendix

Table 3: Quantitative Comparison on Re = 1000 (Generative Models). We report the Mean

\pm

Std computed across all windows and realizations (

20\times

). Bold indicates best performance. Note the astronomical error values for the Base model in Short-term Extrapolation, indicating model blow-up.

Panel of Figure 5 — Figure 5: Physical Field Reconstruction (Prediction). A unified view of the kinematic and topological state. Top: Velocity fields. Middle: Conservation metrics showing baseline failure. Bottom: Topology.

Panel of Figure 6 — Figure 6: Spectral Fidelity and Error Distributions (Prediction). Top: Enstrophy Spectrum showing partial alignment with ground truth. Rows 2-4: Squared error distributions for primitive and derived variables.

Panel of Figure 7 — Figure 7: Short-term Stability (Extrapolation). Snapshots at $T=100$ . Top: Velocity drift in baseline. Middle: Conservation failure. Bottom: Topological degradation.

Panel of Figure 8 — Figure 8: Stationarity and Error Growth (Short-term). Top: Enstrophy spectrum remains stable. Rows 2-4: Significant divergence errors appear in the baseline.

Panel of Figure 9 — Figure 9: Long-term Stability (Extrapolation). Snapshots at $T=300$ . Top: Complete decoherence in baseline. Middle: Continued physical validity in ours. Bottom: Stationarity of topological features.

Panel of Figure 10 — Figure 10: Long-term Stability (Extrapolation). Top: Enstrophy spectrum. Rows 2-4: Baseline errors saturate, while ours remain bounded.

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