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

Yifei Sun

cs.LGSep 27, 2026

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

Authors: Yifei Sun

Abstract

Divergence-free vector fields are fundamental state variables in incompressible flows and many PDE systems. Redundant parameterizations, including Neural Conservation Law (NCL) potentials, map multiple auxiliary representations to the same physical field. Our experiments show that this redundancy can reduce static representation-fitting error by enlarging the set of equivalent solutions, but the resulting many-to-one mapping does not provide a unique state for operator learning. We introduce a minimal representation that encodes a real $D$ -component divergence-free vector field on a $D$ -dimensional domain as a real $(D-1)$ -component field on the same domain. Exploiting the transverse structure imposed by incompressibility in Fourier space, we use a Householder orthogonal transformation to construct the reduced coordinates directly. For periodic and closed impermeable fields, the transform is invertible, isometric, and angle-preserving. For open nonperiodic flows, Fourier extension constructs a compatible periodic field, and a minimum-energy rule selects a unique reduced representation. Neural operators then learn temporal evolution entirely in this reduced space. At inference, the predicted $(D-1)$ -component field is decoded directly into a physical divergence-free $D$ -component field, without predicting an ambient field or applying post-hoc projection. Experiments on static fitting and temporal prediction reveal a task-dependent trade-off: redundancy facilitates static optimization, whereas unique invertible coordinates provide a well-defined state for temporal dynamics. By removing unconstrained longitudinal or null directions from the learned state space, the proposed formulation achieves lower prediction error and greater robustness while preserving divergence freedom by construction.

Figures & tables

Figure 1: Mode-wise function evolution in the minimal transverse representation. For every nonzero wavenumber

k_{i}

, both the input mode

\widehat{u}_{t}(k_{i})

and the decoded next mode

\widehat{u}_{t+1}(k_{i})

lie in

k_{i}^{\perp}

. The learned map

\mathcal{G}_{D-1}

evolves only the

D-1

intrinsic transverse coordinates.

Figure 2: Machine-precision reconstruction with the proposed $D\!\to\!D-1\!\to\!D$ encoder and decoder under two boundary conditions. (a) A closed impermeable three-dimensional no-slip cube: the physical magnitude, two real reduced channels, reconstruction, and pointwise error are shown for a representative slice. The two reduced panels retain the nonzero values outside the black physical boundary. (b) An open nonperiodic two-dimensional channel with a $20\%$ Fourier extension ( $g=1.2$ ) and density $r=2$ : the original speed augmented by its Fourier continuation, decoded $U_{x}$ , decoded $U_{y}$ , and physical-domain pointwise error are shown. The cyan line marks the physical/virtual boundary. In both settings, encoding to $D-1$ real channels and decoding back to $D$ components reconstructs the physical field to machine precision.

$\mathsf{I}$	$\varnothing$	$\displaystyle A_{0}v=u$	$\displaystyle v_{0}=A_{0}^{-1}u=A_{0}^{}u=F_{D-1}^{-1}E^{}F_{D}u$
$\mathsf{II}$	$\displaystyle\min_{v}\;\frac{1}{2}\\|v\\|_{2}^{2}$	$\displaystyle A_{E}v=u$	$\displaystyle v_{E}^{\star}=A_{E}^{\dagger}u=A_{E}^{}(A_{E}A_{E}^{})^{-1}u$
$\mathsf{III}$	$\varnothing$	$\displaystyle A_{K}v=u$	$\displaystyle v_{K}=A_{K}^{-1}u$

Table 3

Model	Val. 10-step mean	Holdout 10-step mean	Holdout step-10
A—Raw	0.1343289	0.1350950	0.1847896
B—Leray	0.1332429	0.1338184	0.1917674
C—Potential	0.1005770	0.1012601	0.1465121
D—Latent2 (ours)	0.0993804	0.0996544	0.1421250

Table 1: 2D MHD physical relative-

L_{2}

error. Lower is better.

Figure 3: 2D MHD closed-loop prediction on a selected held-out trajectory at the final $+10$ step. The top and middle rows compare velocity and magnetic-field magnitudes, and the lower rows show absolute errors on common color scales. This instance is selected for its large Latent2 advantage over A/B/C; Table 1 remains the aggregate evaluation.

Model	$h=1$	$h=5$	$h=10$	$h=20$
A—Raw	0.074044	0.297128	0.643126	1.060287
B—Leray	0.075585	0.297055	0.617995	1.010129
C—Potential	0.078994	0.308620	0.626087	1.079068
D—Latent2 (ours)	0.082842	0.308565	0.592485	1.000849

Table 2: 3D NS physical relative-

L_{2}

error at different rollout horizons. Lower is better.

Figure 4: 3D Navier–Stokes final autoregressive $+10$ frame for held-out trajectory g46. The panels show the solver velocity-speed slice, predictions from the four representations, and their voxelwise velocity-error magnitudes. This selected trajectory illustrates the long-horizon advantage; Table 2 reports the aggregate horizon-wise result.

Model	Clean step-10 velocity Rel- $L_{2}$
FNO-vorticity	0.039315
D—Latent2 (ours)	0.075430
B—Leray	0.436152
A—Raw	0.596742
C—Potential	1.003959

Table 3: 2D NS clean step-10 physical velocity error under the common reevaluation protocol.

Figure 5: The same 2D Navier–Stokes held-out case with the FNO-vorticity prediction included. At this selected instance, FNO-vorticity has the lowest physical velocity error, followed by Latent2; the aggregate clean-error comparison is reported in Table 3 .

Model	Absolute error increase at $h=10$
B—Leray	0.001056259
D—Latent2 (ours)	0.000028114

Table 4: 3D NS response to the same B-targeted observation perturbation.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Geometric comparison of three temporal-evolution representations for each nonzero wavenumber $k_{i}$ . Top (unconstrained): neither the input nor the output is required to be divergence-free or to lie in $k_{i}^{\perp}$ , so the learned evolution uses ambient $D$ -component modes. Middle (Project-and-Generate): the model uses an ambient $D$ -component representation with no constraint on the longitudinal direction; its raw output is likewise unconstrained and is returned to $k_{i}^{\perp}$ only by a subsequent projection. Bottom (ours): the input and output are represented directly by the $D-1$ transverse coordinates in $k_{i}^{\perp}$ . The neural operator never predicts the longitudinal component that projection would discard, and no projection is needed after prediction.

Retained $D-1$ payload / original data	$\\|RU_{\rm decoded}-u\\|_{2}/\\|u\\|_{2}$	Interpretation
$1.20$ (full payload)	$8.45\times 10^{-15}$	machine-precision crop reconstruction
$1.18$	$9.28\times 10^{-6}$	below $10^{-5}$ , but exceeds the original storage
$1.10$	$3.62\times 10^{-5}$	$10^{-5}$ – $10^{-4}$ accuracy range
$1.00$	$6.57\times 10^{-5}$	$10^{-5}$ – $10^{-4}$ accuracy range at equal storage
$0.90$	$9.46\times 10^{-5}$	$10^{-4}$ accuracy with $10\%$ fewer stored values
$0.50$	$3.01\times 10^{-4}$	stronger compression with reduced accuracy

Appendix

Table 5: Open-channel Fourier-extension storage–accuracy frontier. The extension uses

g=1.2

and axial density

r=2

before coefficient pruning. Every nonzero retained mode uses only the intrinsic

D-1

transverse coordinates; no longitudinal residual is stored.

Representation	rank / nullity	best validation loss	optimizer steps	loss decades / s
Minimal Fourier $a\in\mathbb{R}^{3}$ (ours)	$3/0$	$2.734\!\times\!10^{-2}\pm 2.619\!\times\!10^{-3}$	$16{,}106\pm 80$	$5.286\!\times\!10^{-3}$
Direct complex Fourier coefficients	$3/0$	$7.735\pm 1.404\!\times\!10^{-2}$	$16{,}205\pm 71$	$1.199\!\times\!10^{-3}$
Quaternion three-potential	$3/0$	$1.373\!\times\!10^{-3}\pm 1.977\!\times\!10^{-4}$	$28{,}306\pm 909$	$7.445\!\times\!10^{-3}$
NCL vector auxiliary field $B\in\mathbb{R}^{4}$	$3/1$	$2.516\!\times\!10^{-2}\pm 3.241\!\times\!10^{-3}$	$15{,}034\pm 54$	$5.400\!\times\!10^{-3}$
NCL antisymmetric potential $A\in\mathbb{R}^{6}$	$3/3$	$1.039\!\times\!10^{-3}\pm 6.298\!\times\!10^{-5}$	$28{,}351\pm 233$	$7.661\!\times\!10^{-3}$

Appendix

Table 6: Ten-initialization representation-learning record for the three-dimensional ball over time. Mean

\pm

standard deviation is reported for loss and optimizer steps. The rank and nullity are per nonzero

D=4

Fourier mode.

Potential coordinates	output coordinates	nullity	validation loss at $10$ k $\downarrow$	training s
Quaternion three-potential construction	$3$	$0$	$0.1164051$	$111.21$
Vector auxiliary field $B$	$4$	$1$	$0.1784628$	$245.97$
Direct antisymmetric potential $A$	$6$	$3$	$0.1038496$	$117.72$
Canonical skew potential	$8$	$5$	$0.1123825$	$128.40$
Unconstrained dense coordinates followed by a fixed map to $A$
Dense $z\mapsto A$ , $m=12$	$12$	$9$	$0.0974009$	$121.50$

Appendix

Table 7: Full controlled redundancy sweep from the original

10

k-update Ball experiment. “Nullity” is the generic output-coordinate nullity of the map to the three physical current components, i.e., output coordinates minus rank

3

. The deliberately overcomplete dense, multi-potential, and outer-product constructions mostly reach a lower loss than direct six-entry

A

at comparable training time.

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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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