Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting

Ramzi Idoughi; Radouan Boukharfane; Peter Wonka

cs.LGOct 12, 2026

Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting

Authors: Ramzi Idoughi, Radouan Boukharfane, Peter Wonka

Organizations: King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia · Mohammed VI Polytechnic University (UM6P), Benguerir, Morocco

Abstract

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.

Figures & tables

Figure 1: DFK-Former architecture . Sparse spatiotemporal velocity observations are encoded into tokens and jointly processed with learned kernel-query and register tokens using a Perceiver-style spatiotemporal transformer. Decoding heads predict the parameters of a divergence-free kernel representation, yielding a continuous time-varying incompressible velocity field by construction.

Method	RMSE/ $\EPE$ $\downarrow$	rel. $L^{2}$ $\downarrow$	MAE $\downarrow$	cosine $\uparrow$	angle ( ∘ ) $\downarrow$	$\lVert\nabla\!\cdot\!v\rVert_{2}$ $\downarrow$	ms/field $\downarrow$	div-free?
Wendland RBF (scalar)	0.598	0.736	0.280	0.663	37.9	$3.2$	26	no
SIREN ( $+$ soft penalty)	0.392	0.519	0.195	0.805	26.7	$1.4$	53600	no (soft)
Curl-SIREN	0.950	1.352	0.475	0.629	42.0	$\sim\!10^{-14}$	52700	by construction
Vector-potential INR	0.778	1.092	0.388	0.661	39.1	$\sim\!10^{-15}$	48000	by construction
DFK fitting (fixed kernels)	0.689	0.924	0.335	0.482	52.9	$\sim\!10^{-15}$	22	by construction
DFK fitting (adaptive kernels)	0.546	0.719	0.264	0.568	45.5	$\sim\!10^{-15}$	9821	by construction

Table 1: Main reconstruction benchmark. Reconstruction accuracy, divergence residual, and per-field inference cost of DFK-Former against optimization-based, operator-learning, and physics-constrained baselines. Best in bold , second-best underlined .

Regime	Temporal DFK-Former (ours)	Per-frame DFK	Shared-kernel DFK	ST-SIREN	ST-Curl-SIREN
clean&dense	0.114	0.100	0.083	0.442	0.781
sparse	0.109	0.847	1.093	0.388	0.590
occluded	0.098	0.951	1.511	0.413	0.766
Overall (median $\EPE$ )	0.114	0.767	0.626	0.428	0.842
Overall (mean $\EPE$ )	0.124	20.29 †	0.762	0.466	0.821
Overall (mean angular error)	4.8 ∘	40.0 ∘	25.4 ∘	21.3 ∘	37.1 ∘

Table 2: Temporal flow reconstruction on 60 time-resolved JHTDB flows. Median

\EPE

by observation regime, plus overall mean

\EPE

and angular error; lower is better. Best results are in bold , second-best underlined .

Figure 2: Temporal reconstruction in the hardest evaluation regime (Forced MHD (dynamic JHTDB), hard combo : sparse observations, noise, occlusions, and outliers). Top: ground truth and predictions from Temporal DFK-Former, per-frame DFK fitting, iterative DFK fitting, space-time Curl-SIREN, and space-time SIREN. Bottom: input observations and reconstruction errors. DFK-Former remains robust under severe degradation, recovering the dominant flow structures with substantially lower EPE and angular error than competing methods.

Table 5

Target structure	Isotropic	Anisotropic	improvement
Vortex tube (Lamb–Oseen)	0.0763	0.0656	$1.16\times$
Shear layer	0.0719	0.0592	$1.22\times$
Vortex ring	0.0835	0.0623	$1.34\times$

Table 5: Anisotropy vs. a parameter-matched isotropic control on structured analytic flows: our anisotropic

K{=}1024

model against a separately trained isotropic model at

K{=}1542

, matched to within four scalar parameters. rel

\EPE

, lower is better; improvement is the isotropic/anisotropic ratio.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Static model: backbone, heads and kernel geometry
Component	Setting
Token width $d_{\mathrm{model}}$	$256$
Backbone blocks	$6$ ; each block $=$ one masked cross-attention sublayer $+$ one self-attention sublayer
Attention heads / head dimension	$8$ / $32$
Feed-forward hidden width	$4\,d_{\mathrm{model}}=1024$ , GELU
Dropout	$0$

Appendix

Table 6: DFK-Former architecture, as resolved for the reported static run. The upper block is the static model; the lower block gives the temporal model’s additions and the two places where it departs from the static geometry.

K

is the kernel count and

\Delta

the mean kernel spacing.

Optimization and schedule
Setting	Value
Optimizer	AdamW, library-default $\beta$ and $\epsilon$ ; weight decay $0$
Peak learning rate	$3\times 10^{-4}$
Schedule	constant to step $60{,}000$ , cosine decay to step $145{,}000$ , held at $10^{-5}$ thereafter; no warm-up
Gradient clipping	global norm $1.0$
Batch size	$32$ fields per rank $\times\ 4$ ranks $=128$ fields per step

Appendix

Table 7: Training recipe of the reported static model, read from its fully resolved configuration. Ablation arms differ from it only in the single factor each ablation names (Section E ); the temporal model’s departures are in the last block. The auxiliary block acts on the placement and shape heads during training only and has no effect at inference. Per-stage data distributions are in Table 9 ; supervision point counts are in Table 8 .

Figure 3: Training-data pipeline (2D shown, identical principles in 3D). Top: a Gaussian random field splits exactly in Fourier space into the solenoidal target ${{\bm{v}}^{\ast}}$ and an irrotational part, sparse observations are sampled and corrupted, and the resulting measurements are used to reconstruct the clean divergence-free target. Panel divergences are measured spectrally. Bottom: examples from the CAF library.

Symbol	Meaning	Value / range
Domain and precision
$L$	periodic domain size, $[0,L)^{3}$	$1.0$
$G$	FFT generation grid	$64^{3}$
—	generation / training precision	float64 / float32
Spectral generator (Eq. 10 )
$k_{0}$	energy-containing wavenumber (cycles/domain)	${\mathcal{U}}[1.5,3.0]$

Appendix

Table 8: Data-generation parameters. Ranges are drawn independently per field;

{\mathcal{U}}

denotes a uniform draw (continuous, or over integers where the range is written

\{\cdot\}

). Corruption entries give the values outside the curriculum; per-stage overrides are in Table 9 .

Stage	Steps	Sources (spectral / coherent)	$\chi_{\max}$	$\nu_{\max}$	$M$	$p_{\mathrm{occ}}$
Core	$60{,}000$	$1.0$ / $-$	$0.5$	$0.05$	$[512,4096]$	$-$
Hard	$85{,}000$	$0.6$ / $0.4$	$1.5$	$0.20$	$[128,2048]$	$0.5$
Cooldown	$5{,}000$	$0.7$ / $0.3$	$1.0$	$0.15$	$[256,4096]$	$0.5$

Appendix

Table 9: Training curriculum. Only the data distribution and loss weights change between stages. “

-

” means the term is off.

\chi_{\max}

is the maximum irrotational contamination,

\nu_{\max}

the maximum relative noise,

p_{\mathrm{occ}}

the occlusion probability.

Set	Generator	Fields	Role
in-dist	spectral, training band-limit	$128$	matched-distribution accuracy
CAF	coherent-structure mixture	$64$	coherent structure (in training)
Taylor–Green	single analytic family	$64$	per-family accuracy
Lamb–Oseen	single analytic family	$64$	per-family accuracy
vortex ring	single analytic family	$64$	per-family accuracy
shear layer	single analytic family	$64$	per-family accuracy

Appendix

Table 10: Composition of the

582

-flow evaluation benchmark. Synthetic sets use seeds disjoint from training. The CAF and single-family sets draw on the training structure library (Table 9 ), so they measure in-distribution accuracy; held-out families are evaluated in Table 21 . JHTDB sets are real DNS ( Li et al., 2008 ) , resampled to

64^{3}

and used for evaluation only.

Figure 4: The mean-flow gap. Kernels alone cannot represent net transport because their sum is zero-mean. Adding a mean-flow term substantially reduces approximation error.

Radius / spacing	channel	transitional b.l.	isotropic turb.	forced MHD
$1.5$	0.106	0.165	0.483	0.739
$2.5$	0.037	0.056	0.173	0.261
$3.5$	0.012	0.019	0.050	0.075
$4.5$	0.010	0.016	0.040	0.058
$6.0$	0.009	0.014	0.035	0.051

Appendix

Table 11: Best achievable fit (relEPE) of the anisotropic DFK span to clean real-DNS fields ( Li et al., 2008 ) at fixed

K=8192

, as a function of the support radius expressed in kernel spacings. Overlap is the dominant design variable: the floor falls monotonically as the kernels overlap more.

		same- $K$		same-memory
$K$	isotropic	aniso, equal-volume	aniso, extend-only	aniso, extend-only
$256$	0.173	0.251	0.139	0.173
$512$	0.137	0.247	0.092	0.122
$1024$	0.128	0.282	0.055	0.076
$2048$	0.124	0.356	0.038	0.045
$4096$	0.215	0.592	0.049	0.038

Appendix

Table 12: Anisotropic vs isotropic fit (relEPE) on a vortex-tube target: identical centers, radius scale

3.5\,\Delta

, Piola construction, aspect ratio

4

8000

fit samples,

4000

scoring points, ridge

10^{-4}

. Same-memory reduces the anisotropic count to

\lfloor 7K/11\rfloor

. The equal-volume column shows that preserving support volume is worse than not using anisotropy at all.

Factor	Change	$\Delta$ EPE	Budget
Set encoder	attention $\to$ DeepSets pooling	$+49.1\%$	$3$ k steps
Kernel count	$K{:}~512\to 2048$ (no $1024$ point)	$-12.0\%$	$3$ k steps
Adaptive placement	fixed $\to$ predicted centers/radii	$-7.4\%$	$3$ k steps
Width	$d_{\mathrm{model}}{:}~128\to 384$	$-4.8\%$	$30$ k steps
Fourier position enc.	off $\to$ on	$-2.4\%$	$30$ k steps
Orthogonality loss	$\lambda_{\mathrm{ortho}}{:}~0.1\to 0$	$+0.04\%$	$30$ k steps

Appendix

Table 13: Controlled architecture matrix: single-factor ablations on a shared reduced-scale base (

K{=}512

d_{\mathrm{model}}{=}128

4

blocks, Core stage only), scored on the

128

-field Gaussian Random Fields set. These are ablation signal runs, not paper-scale numbers; small effects are quoted at the

10\times

step budget (Section E.2 ). Negative

\Delta

is an improvement.

Figure 5: Axis-isolated robustness sweeps Reconstruction error (EPE; mean

\pm

95% bootstrap CI;

72

fields per point) as a function of corruption strength along a single axis, with all other factors fixed to nominal settings. DFK-Former (blue) is the strongest, exactly divergence-free feed-forward method. The dashed pink curve is an ablated variant of DFK-Former, while the grey dotted line denotes the oracle representation floor obtained by least-squares fitting the basis to the clean target, i.e., the best achievable error within the representation.

Method	$M{=}64$	noise $0.2$	occl. $0.8$	$k_{\mathrm{peak}}[12,18]$	$\lVert\nabla\!\cdot\!v\rVert_{2}$
DFK-Former (ours)	0.714	0.477	0.713	0.687	$2\text{--}7\times 10^{-7}$
FNO	0.742	0.501	0.749	0.760	$0.40\text{--}0.67$
DeepONet	0.816	0.723	0.856	0.862	$\approx 0.2$
DFK fitting (fixed kernels) ‡	0.804	0.636	0.805	0.842	$\sim\!10^{-16}$
DFK fitting (adaptive kernels)	1.628	2.167	2.425	2.361	$\sim\!10^{-15}$
Wendland RBF (scalar)	0.795	0.657	0.803	0.756	$1.5\text{--}3.1$

Appendix

Table 14: Multi-method sweep at each axis’s hard endpoint (

M{=}64

;

\sigma_{\mathrm{rel}}{=}0.2

M{=}512

; occluded fraction

0.8

; target

k_{\mathrm{peak}}\in[12,18]

). EPE, lower is better; the last column is residual divergence. The row before the floor is our own ablation, not a baseline: on the identical backbone it matches our accuracy and leaks divergence.

Axis	Range (easy $\to$ hard)	model EPE	margin vs fit	margin vs aniso. fit
Sample count $M$	$2048\to 64$	$0.315\to 0.713$	$36.3\%\to 11.2\%$	$\to 15.8\%$
Noise $\nu$	$0\to 0.4$	$0.326\to 0.390$	$36.8\%\to 26.5\%$	$\to 27.7\%$
Occluded fraction	$0\to 0.5$	$0.306\to 0.492$	$36.2\%\to 26.0\%$	$\to 29.5\%$
Target $k_{\min}$ – $k_{\max}$	$[3,5]\to[12,18]$	$0.441\to 0.544$	$29.3\%\to 32.0\%$	$\to 38.0\%$

Appendix

Table 15: Single-axis corruption sweep: model EPE against per-field DFK fitting (fixed kernels) at its per-point best ridge,

500

fields

\times

3

seeds per point,

95\%

bootstrap confidence intervals non-overlapping at every point. The margin is

1-\mathrm{EPE}_{\mathrm{model}}/\mathrm{EPE}_{\mathrm{fit}}

; the last column repeats it at the hard endpoint against a fit over the model’s own anisotropic basis.

	Synthetic (band-limited)							Real DNS (zero-shot)
Method	GRF	CAF	Taylor–Gr.	Lamb–Os.	vtx. ring	shear	ABC	iso.	MHD	chan.	tr. BL
Wendland RBF (scalar)	0.634	0.615	0.607	0.627	0.560	0.622	0.618	0.566	0.553	0.601	0.572
SIREN ( $+$ soft penalty)	0.550	0.537	0.560	0.435	0.444	0.438	0.418	0.256	0.310	0.183	0.180
Curl-SIREN	0.998	1.177	1.167	1.102	1.028	1.108	1.005	0.673	0.729	0.769	0.699
Vector-potential INR	0.819	1.029	0.969	0.943	0.942	0.989	0.821	0.469	0.470	0.623	0.485
DFK fitting (fixed kernels)	0.545	0.643	0.482	0.715	0.692	0.743	0.548	0.752	0.732	0.889	0.842

Appendix

Table 16: Per-set breakdown of the main benchmark (

\EPE

, lower is better). Bold marks the column minimum. GRF is the frozen Gaussian Random Fields set and CAF the coherent-structure mixture (Table 10 ). No method has trained on JHTDB, so those four columns are zero-shot throughout.

Method	$\lVert\nabla\!\cdot\!{\color[rgb]{0.1211,0.6055,0.3516}{\bm{v}}_{\Theta}}\rVert_{\infty}$ $\downarrow$	div-free
Wendland RBF (scalar)	$1.9{\times}10^{1}$	no
SIREN ( $+$ soft penalty)	$7.9{\times}10^{0}$	no (soft)
Curl-SIREN	$5.4{\times}10^{-14}$	by construction
Vector-potential INR	$2.7{\times}10^{-14}$	by construction
DFK fitting (fixed kernels)	$2.8{\times}10^{-14}$	by construction
DFK fitting (adaptive kernels)	$4.3{\times}10^{-14}$	by construction

Appendix

Table 17: Divergence quality , macro-averaged over the eleven static sets; the

L_{2}

column is in Table 1 . Our

\sim\!10^{-7}

is the float32 autograd diagnostic’s floor, not a property of the field: the DFK construction is solenoidal for any coefficients, centers and radii (Section A ).

	Three turbulence sets			Ten-family battery
Method	rel. KE $\downarrow$	rel. $\Omega$ $\downarrow$	band $E(k)$ $\downarrow$	rel. $\Omega$ $\downarrow$	band $E(k)$ $\downarrow$
SIREN ( $+$ soft penalty)	0.077	0.296	0.175	–	–
Curl-SIREN	1.388	4.962	1.576	–	–
DFK fitting (fixed kernels)	0.148	0.486	0.230	–	–
FNO	0.202	0.696	0.258	2.660	0.211
DeepONet	0.388	0.875	0.435	–	–

Appendix

Table 18: Physical-consistency battery , macro-averaged over the three turbulence sets (Gaussian Random Fields, JHTDB isotropic, JHTDB MHD;

128{+}18{+}18

fields; spectral curl on the node-aligned

64^{3}

grid). The last two columns repeat enstrophy and

E(k)

over the ten-family battery, available for the three methods run over all families.

Figure 6: Energy-spectrum fidelity. Shell-averaged

E(k)

(log–log) for ground truth and every method. Both axes are cropped to the band the ground truth carries: the Gaussian Random Fields target is band-limited (Section C.3 ), so an uncropped axis would compress every curve into the top of the panel. DFK-Former tracks the ground truth across the resolved band.

Method	ms/field $\downarrow$	scenes/s $\uparrow$	peak mem (MB) $\downarrow$	GFLOPs $\downarrow$	# params (M)
SIREN / Curl-SIREN (per-field fit)	53000	0.019	204	9500	0.20
DFK fitting (fixed kernels)	22	45	1285	81	0
DFK fitting (adaptive kernels)	9821	0.10	623	786	0
DFK-Former (ours)	7.5	134	68	41	10.2

Appendix

Table 19: Efficiency battery Runtime, throughput, peak memory, and computational cost for reconstructing a single scene (

M{\approx}2

Q{=}256

, batch size

1

). Lower is better except for throughput.

Restricted training — sparsity arm $M\!\in\![256,1024]$ , noise arm $\sigma_{\rm rel}\!\leq\!10\%$ , every stage, retrained from step $0$
	Sparsity axis					Noise axis ( $\sigma_{\rm rel}$ )
Method	$1\%$ ( $M{=}64$ ) †	$2\%$ ( $M{=}128$ ) †	$5\%$	$20\%$	$50\%$	$0\%$	$5\%$	$10\%$	$20\%^{\dagger}$
FNO	0.753	0.665	0.580	0.407	0.362	0.495	0.496	0.497	0.501
DeepONet	0.812	0.782	0.766	0.756	0.750	0.757	0.755	0.748	0.724
DFK-Former (ours)	0.718	0.637	0.551	0.380	0.323	0.461	0.464	0.471	0.492
Full-range training ( $M\!\in\![128,2048]$ , $\sigma_{\rm rel}\!\leq\!20\%$ ) — unrestricted reference

Appendix

Table 20: Generalization to unseen sparsity and noise levels. Restricted models are trained only on moderate sparsity (

M\!\in\![256,1024]

) or noise (

\sigma_{\rm rel}\!\leq\!10\%

) and evaluated both within and beyond that range. Columns marked by † correspond to out-of-distribution settings not observed during training. DFK-Former remains competitive under both sparsity and noise extrapolation, with only modest degradation relative to its full-range counterpart. FNO and DeepONet use identical restricted-training protocols. Results report

\EPE

averaged over

24

fields and

3

random seeds. Lower is better.

	Set	restricted	full library	ratio
Excluded families	Shear layer	0.425	0.054	7.95
	Taylor–Green	0.301	0.053	5.71
	Vortex ring	0.277	0.062	4.50
	Lamb–Oseen	0.249	0.056	4.48
Included families	ABC (the one kept family)	0.074	0.036	2.04
Included families	Gaussian Random Fields	0.369	0.343	1.08

Appendix

Table 21: Flow-family generalization under training-set restriction.

\EPE

on the eleven evaluation sets. The restricted model is trained using only the ABC coherent-flow family, while keeping all other training components, sampling weights, corruption ranges, and spectral-flow data identical to the full setup. The ratio reports restricted/full . Large ratios on withheld families indicate strong transfer from training on diverse coherent structures, whereas ratios near one indicate limited dependence on flow-family diversity. The JHTDB datasets are zero-shot for both models.

Corruption regime	Temporal DFK-Former	Per-frame DFK	Shared-kernel DFK	ST-SIREN	ST-Curl-SIREN
clean&dense	0.114 (0.114)	0.100 (0.102)	0.083 (0.093)	0.442 (0.432)	0.781 (0.727)
contaminated	0.085 (0.122)	0.217 (0.291)	0.165 (0.191)	0.365 (0.448)	0.712 (0.781)
noisy	0.088 (0.097)	0.398 (1.874)	0.268 (0.281)	0.440 (0.482)	0.939 (0.926)
sparse	0.109 (0.129)	0.847 (0.828)	1.093 (1.067)	0.388 (0.453)	0.590 (0.617)
occluded	0.098 (0.110)	0.951 (107.1)	1.511 (1.519)	0.413 (0.451)	0.766 (0.819)
hard combo	0.138 (0.173)	0.916 (11.52)	1.420 (1.419)	0.497 (0.528)	1.029 (1.057)

Appendix

Table 22: Time-varying reconstruction of unobserved real DNS frames , all six corruption regimes (dynamic JHTDB:

60

flows,

180

test observations). Each cell is median (mean) endpoint error at times the model never sees. Per-frame DFK’s mean is inflated by

15/180

numerical failures.

Dataset	$n_{\mathrm{obs}}$	$\EPE$	angle ( ∘ )	$\lVert\nabla\!\cdot\!{\color[rgb]{0.1211,0.6055,0.3516}{\bm{v}}_{\Theta}}\rVert_{\mathrm{rms}}$	Per-frame DFK	ratio
JHTDB isotropic	48	0.141	6.8	$5.6{\times}10^{-7}$	0.752	5.3 $\times$
JHTDB MHD	48	0.196	9.5	$6.3{\times}10^{-7}$	0.790	4.0 $\times$
JHTDB channel	42	0.057	2.7	$2.1{\times}10^{-7}$	0.805	14.2 $\times$
JHTDB transitional BL	42	0.073	3.4	$2.3{\times}10^{-7}$	0.753	10.4 $\times$

Appendix

Table 23: Temporal reconstruction by flow type. Median metrics on the dynamic JHTDB benchmark (

180

test observations). Temporal DFK-Former consistently outperforms per-frame DFK fitting, with the largest gains on wall-bounded flows. The ratio column reports the relative improvement over a per-frame DFK baseline using the same kernel basis.

Corruption regime	Per-frame DFK	$F{=}2$ moving iso.	$F{=}7$ moving iso.	$F{=}7$ shared aniso.	$F{=}2$ shared aniso.	$+$ adaptive placement
clean&dense	0.100	0.115	0.174	0.133	0.114	0.117
contaminated	0.217	0.092	0.172	0.105	0.085	0.100
noisy	0.398	0.093	0.173	0.102	0.089	0.098
hard combo	0.916	0.146	0.222	0.152	0.138	0.152
sparse	0.847	0.118	0.194	0.125	0.109	0.117
occluded	0.951	0.103	0.186	0.117	0.098	0.108

Appendix

Table 24: Ablation of temporal parameterizations on real DNS sequences. Median EPE on the dynamic JHTDB benchmark (

60

flows,

180

test observations). All temporal variants share the same

K{=}1024

kernel budget, training data, and supervision. The best-performing design uses a shared anisotropic kernel geometry with low-order temporal dynamics (

F{=}2

), outperforming both moving-kernel variants and per-frame DFK fitting. Adding adaptive kernel placement provides no benefit on real DNS data. Lower is better.

Figure 7: The well-posed case the model does not win (Forced MHD (dynamic JHTDB), clean&dense ): the DFK-span solves reach the fixed-basis optimum — shared-kernel DFK edging the learned prior out ( $0.207$ ) and per-frame DFK matching it ( $0.230$ each). The space-time INRs remain far behind ( $0.751$ , $1.048$ ). The learned prior earns its keep in the ill-posed regimes, not everywhere.

Figure 8: Temporal reconstruction on a representative real-DNS example (Forced MHD (dynamic JHTDB), sparse regime). Top: ground truth and predictions from Temporal DFK-Former, per-frame DFK fitting, iterative DFK fitting, space-time Curl-SIREN, and space-time SIREN. Bottom: sparse observations and reconstruction errors. DFK-Former best preserves the flow structure and achieves the lowest EPE while remaining exactly divergence-free.

Figure 9: The reconstruction as a function of time (Forced MHD (dynamic JHTDB), sparse regime). Columns are three real-DNS times the model never observes — frame midpoints; rows are ground truth, our model, and two baselines. The emitted field is continuous in ${\color[rgb]{0,0,0}t}$ and exactly divergence-free at every ${\color[rgb]{0,0,0}t}$ , not only where it was supervised.

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Mar 25, 2026

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

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.

Xigui Li, Hongwei Zhang, Ruoxi Jiang +6

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

Neural Operators Computational Fluid Dynamics Physics-Informed Generative Modeling Physics-Informed Neural Operators

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Sep 27, 2026

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

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.

Yifei Sun

PDE Operator Learning Operator Learning Computational Fluid Dynamics Neural Operators

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