Learning in the Transverse Subspace: A Minimal Representation for Divergence-Free Operator Learning

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.

We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier--Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our kernel method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields analytically and simultaneously preserve the aforementioned physical properties. We present universal approximation results and worst-case a priori convergence rates for our framework; empirically, the observed convergence rates exceed the pessimistic predictions across most benchmarks, motivating a formulation of more optimistic convergence rates. Another central contribution of this work is a novel computational framework revolving around streaming construction of kernel Gramians and recursive Schur-complement algorithms to solve the large block linear systems arising from property-preserving kernel methods. This kernel-based framework makes operator learning practical at scales up to $10{,}000$ training functions sampled at up to $10{,}000$ spatial locations. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative $\ell_2$ errors upon generalization and trains up to five orders of magnitude faster compared to neural operators. Moreover, while our method enforces incompressibility analytically, neural operators exhibit large divergence errors. Our results show that our method provides an accurate and efficient surrogate for incompressible flows.

Recovering divergence-free velocity fields from sparse and corrupted observations requires reconciling physical consistency, expressive representations, and robustness to missing data. We introduce DFK-Former, a set-to-field transformer that reconstructs divergence-free velocity fields from sparse, noisy, and irregularly sampled 3D observations. The model is inherently invariant to the number and ordering of input samples and predicts the parameters of a divergence-free kernel representation, producing continuous, mesh-free velocity fields that satisfy incompressibility exactly by construction rather than approximately through soft constraints or projection operators. Unlike existing divergence-free kernel approaches, which estimate kernel coefficients via a separate optimization for each field, DFK-Former predicts these coefficients directly in a single forward pass. This eliminates costly iterative fitting while providing a learned prior that enables accurate reconstruction even when observations are insufficient to uniquely determine the field. To enhance representational power, we introduce adaptive and anisotropic divergence-free kernels that preserve exact incompressibility guarantees while capturing more complex flow structures. Extensive experiments across varying levels of sparsity, noise, and occlusion, including transfer to real Direct Numerical Simulation (DNS) turbulence data, demonstrate substantial gains in both accuracy and efficiency. DFK-Former reduces endpoint error by up to 4.6x compared to per-field divergence-free-kernel optimization, while reconstructing each field in 7.5 ms instead of 9.8 s, yielding a speedup of more than three orders of magnitude. These results establish direct, feed-forward prediction of divergence-free kernel parameters as a practical and effective paradigm for physically consistent velocity-field reconstruction from sparse observations.