FlowReconBench: Accuracy, calibration and cost of learned and classical flow reconstruction from sparse sensors

Nicolas Tricard; Linzheng Wang; Jason Chen; Sili Deng

physics.flu-dynNov 4, 2026

FlowReconBench: Accuracy, calibration and cost of learned and classical flow reconstruction from sparse sensors

Authors: Nicolas Tricard, Linzheng Wang, Jason Chen, Sili Deng

Organizations: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Abstract

Reconstructing full flow fields from sparse sensors requires prior assumptions about unmeasured regions and physical variables. Yet it remains unclear when learned models outperform classical estimators and which aspects of their predictions can be trusted. FlowReconBench evaluates ten learned and five classical methods on three regimes: a laminar cylinder wake across Reynolds numbers, two-dimensional Kolmogorov turbulence and a large-scale three-dimensional (3D) isotropic turbulence. Methods share leakage-controlled splits and sensor draws and are scored on per-channel error, ensemble spread, calibration, spectra and cost. The leading method changes for each regime. Classical methods perform strongly in low-order and dense-sensing regimes. On the cylinder wake, linear stochastic estimation and gappy proper orthogonal decomposition lead, including for unobserved pressure. In 3D, inverse-distance weighting matches the best learned methods on observed channels. Data-driven methods become superior at decreasing sensor density or poor sensor observability. On Kolmogorov flow, continuous ranked probability score and error yield different method rankings: generative models offer broader but underdispersed ensembles, while deterministic models prioritize pointwise error. Generative models balance this tradeoff through their sampler budget, with a larger budget improving ensemble quality at higher computational cost. In 3D, scaling tests demonstrate that queryable methods are flat in memory, while full-grid construction methods run out of memory. Observing only 2 of 4 fields leaves the problem kinematically underconstrained, yet a latent model can reproduce structural similarities to the ground truth.

Figures & tables

Regime	Dataset	Points	Channels	Split	Primary challenge
2D laminar periodic	Cylinder wake (new)	$2.4{\times}10^{4}$	$u_{x},u_{y},p$	cross-Re	Low-rank observability
2D chaotic	Kolmogorov [ 65 ]	$6.6{\times}10^{4}$	$\omega$	cross-traj.	Multiscale reconstruction
3D chaotic	JHTDB isotropic [ 35 ]	$1.95{\times}10^{6}$	$U_{x},U_{y},U_{z},p$	spatial	Unobserved channels

Table 1: The three regimes. The split column names the leakage control (Sec. II.2 ).

Family	Cyl.	Kolm.	JHTDB
Classical (mean, NN, IDW,	✓	✓	✓
gappy POD, LSE)
Point-attention det. (Senseiver)	✓	✓	✓
Point-kernel det. (MLP-RBF)	✓	✓	n.r.
Fourier operator det. (Geo-FNO; FNO)	✓	✓	✓
Set-DeepONet and Set-DeepONet++	n.r.	n.r.	✓

Table 2: Where each method is deployed. Every method run on the cylinder uses its native mesh; methods that need a Cartesian grid are not run there (“grid”). “n.r.”: admitted but not run. det.: deterministic. Abbreviations (Sec. III ): NN, nearest neighbor; LSE, linear stochastic estimation; MLP-RBF, multilayer perceptron with a radial-basis-function kernel; FNO, Fourier neural operator; Geo-FNO, geometry-aware FNO; DeepONet, deep operator network; FM, flow matching; CoNFiLD, conditional neural field latent diffusion; S3GM, sparse-sensor-assisted score-based generative model; SiT, scalable interpolant transformer. a) Scored on the central

120^{3}

crop of the held-out cube (Sec. III ).

	Rel. $L_{2}$
Method	agg.	$\mathrm{Re}=80$	$\mathrm{Re}=250$	$u_{x}$	$u_{y}$	$p$	CRPS	Spr/Err	Cov. 90	s/field
SiT-point ( $N{=}8$ )	0.073	0.038	0.108	0.057	0.105	0.057	0.034	0.51	0.40	0.50
Senseiver	0.134	0.118	0.150	0.115	0.192	0.095	0.082	—	—	0.06
MLP-RBF	0.318	0.260	0.376	0.220	0.536	0.197	0.165	—	—	0.07
Geo-FNO	0.079	0.058	0.100	0.063	0.109	0.064	0.045	—	—	0.26
LSE	0.043	0.036	0.050	0.032	0.058	0.039	0.022	—	—	—

Table 3: Cylinder wake at the held-out

\mathrm{Re}\in\{80,250\}

on 25 test frames each using 1% sensing of

u_{x}

and

u_{y}

. Generative methods use

K{=}8

members and

N

sampler steps. Spr/Err: spread–error ratio. Interpolation returns the training mean on

p

. Boldface denotes the lowest aggregate and unobserved-pressure errors.

Figure 1: Cylinder wake held-out $\mathrm{Re}=250$ frames sensed in the volume (238 sensors per field on $u_{x}$ and $u_{y}$ ; $p$ unobserved) and at the surface (128 taps per field on all three fields). Columns display $u_{y}$ and $p$ for each sensing geometry; circles on the truth row mark the sensors. Panel numbers are single-member relative $L_{2}$ , which exceeds the ensemble-mean errors of Tables 3 and 4 . Gappy POD is accurate from the volume sensors and fails from the taps; IDW returns the training mean on the unobserved pressure.

	Rel. $L_{2}$ at $n$ taps/field
Method	32	64	128	360	CRPS	Spr/Err
SiT-point	0.103	0.092	0.091	0.091	0.040	0.25
Senseiver	0.130	0.106	0.101	0.099	0.063	—
Geo-FNO	0.236	0.182	0.173	0.157	0.086	—
MLP-RBF	0.370	0.322	0.303	0.290	0.156	—
Ridge gappy POD	0.270	0.265	0.262	0.260	0.124	—

Table 4: Cylinder surface-to-field task: relative

L_{2}

of the full wake from taps on the body, held-out

\mathrm{Re}\in\{80,250\}

, 50 test frames. Columns give taps per field on the near-wall ring;

u_{x}

u_{y}

and

p

are read at the taps. CRPS and the spread–error ratio (Spr/Err) are at the full ring. The gappy POD rank, and the ridge strength of ridge gappy POD and LSE, are chosen per budget on training data; ridge gappy POD keeps

r{=}10

. Boldface denotes the lowest error with the full ring of taps.

	Rel. $L_{2}$
Method	mean	sample	CRPS	Spr/Err	Cov. 90	$E_{\rm s}$	s/field	GB
SiT-point ( $N{=}32$ )	0.293	0.310	0.146	0.37	0.42	0.27	5.00	0.22
Latent FM ( $N{=}4$ )	0.463	0.523	0.240	0.57	0.55	0.30	0.11	0.16
S3GM ( $N{=}1000$ )	0.273	0.363	0.110	0.95	0.75	0.93	6.80	7.30
Senseiver	0.465	0.465	0.329	—	—	0.023	0.07	0.81
MLP-RBF	0.500	0.500	0.337	—	—	0.13	0.11	1.36

Table 5: Kolmogorov flow reconstruction. Sampler budgets, step counts

N

and the S3GM guidance coefficient are tuning-half choices. The conditional methods train at 65–6,554 sensors per frame.

E_{\rm s}

: test-frame median of the single-member energy at

k=32

–128 relative to the truth. Inference time and peak memory per field on one H100. per./non-per.: periodic or non-periodic neighbor search. Boldface denotes the lowest value in each of the relative

L_{2}

and CRPS columns.

Figure 2: Kolmogorov spectral fidelity at 655 sensors (1%): one held-out frame, one member per method. (a) Kinetic energy spectra. Large scales (

k=1

–8) and small scales (

k=32

–128) are shaded. A curve below the truth at high

k

is too smooth; a curve above it carries spurious fine-scale noise. The

k^{-3}

line is shown for reference. (b) Small-scale band energy relative to the truth on a log axis centered at 1.

Figure 3: Kolmogorov $256^{2}$ , the held-out frame of Fig. 2 : truth and one member per method, with the same 655 sensors (dots on the truth panel) in every panel and one color scale. Panel numbers are single-member relative $L_{2}$ . Deterministic methods are smooth, most generative members carry small-scale texture, and gappy POD is over-smoothed.

Method	$U_{x}$	$U_{y}^{*}$	$U_{z}$	$p^{*}$	agg.	CRPS	Spr/Err	Cov. 90	$E_{\rm d}$
Latent flow matching	0.249	0.710	0.260	0.643	0.465	0.239	0.69	0.57	0.056
SiT-point ( $N{=}32$ )	0.175	0.957	0.182	0.904	0.555	0.323	0.46	0.40	11.7
FNO3D-FM ( $N{=}4$ )	0.190	0.921	0.193	1.054	0.589	0.272	0.98	0.63	0.012
CoNFiLD (single snapshot, $N{=}1000$ ) a)	0.500	1.024	0.543	1.034	0.775	0.378	0.77	0.62	—
S3GM (normalized guidance) b)	0.257	0.995	0.252	0.993	0.624	0.347	0.52	0.46	—
Gen4Turb c)	0.212	0.916	0.325	0.876	0.582	0.307	0.70	0.54	—

Table 6: 3D isotropic turbulence, spatially disjoint held-out cube (

1.95\times 10^{6}

points). Per-channel relative

L_{2}

of the ensemble mean over the 25 test-half snapshots; agg. averages the four channels. Asterisks mark the unobserved channels, where an error near 1 equals that of the training mean. Checkpoints and sampler settings are tuning-half choices.

E_{\rm d}

: test-snapshot median of the single-member energy at

k=32

–50 relative to the DNS on the observed channels. Each 3D method is trained once. Boldface denotes the best relative

L_{2}

or CRPS among learned methods. a) Published prior, outside the parameter budget; the 32-snapshot checkpoints conditioned on sensors in the target snapshot only, with the guidance scale (0.1) chosen on the tuning half. b) Guidance fitted on the tuning half, because the published coefficients diverge at this size; final checkpoint of a

\sigma_{\max}{=}20

prior trained before the frozen protocol. c) Scored on the central

120^{3}

crop (88% of the points and sensors) and not paired. Further CoNFiLD, SiT-point and S3GM tests are in SM Table S3.

Figure 4: 3D isotropic turbulence, held-out cube, frame 152 (a test-half frame). Each panel shows the $125^{3}$ cube with one octant cut away to expose the $x$ , $y$ and $z$ midplanes. The other visible surfaces are outer faces of the cube. $U_{x}$ and $U_{z}$ are observed at 19,531 identical sensors each (circles: sensors on the midplanes); $U_{y}$ and $p$ are unobserved. Gen4Turb shows its central $120^{3}$ crop. Panel numbers are single-member relative $L_{2}$ over the whole cube (the crop for Gen4Turb). Among the methods shown, only Latent FM is below 0.7 on an unobserved channel, and it identifies the low-pressure core. IDW returns the training mean on the unobserved channels.

Method	Params	GPU-h	Peak GB	s/field
Latent FM	6.68M	20.1	8.4	0.019
SiT-point ( $N{=}32$ )	6.57M	19.4	26.8	149
FNO3D-FM ( $N{=}4$ )	6.73M	27.4	48.6	0.097
CoNFiLD a)	118.9M	19.2	10.4	36 (d)
Gen4Turb	6.16M	16.6	47.4	1.65
S3GM	6.35M	18.5	26.3	73.8

Table 7: Cost of the 3D fleet on one H100-SXM (80 GB): parameters, training GPU-hours, peak training memory, and inference time per full

125^{3}

field. (d): derived from evaluation wall time, an upper bound that includes data loading and metric computation. It marks only slow samplers, so the orders of magnitude between families are unaffected. a) Published CoNFiLD prior, outside the parameter budget. Optimizer steps and time per step are in SM Table S4. Two-stage methods (Latent FM, CoNFiLD) report the sum of both stages and the larger peak. CPU: run on the central processing unit. RSS: peak resident set size in host memory.

		Local (interpolation)				Global (gappy POD)
Regime	Sensors	$\bar{d}/\ell$	$\hat{e}_{\mathrm{NN}}$	NN	IDW	$r$	$E_{r}$	Floor	$\kappa(H\Phi_{r})$	Gappy POD
Cylinder, volume	238 per field	0.95–1.3	0.55	0.569	0.568	20	0.997	0.048	3.2	0.057
Cylinder, surface	128 taps per field	13–19	2.10	2.146	2.131	5	0.929	0.163	$1.0{\times}10^{2}$	0.594
						20	0.997	0.048	$7.7{\times}10^{3}$	0.763
Kolmogorov	655	0.40	0.58	0.580	0.530	160	0.832	0.513	3.7	0.607
JHTDB, $U_{x}$ and $U_{z}$	19,531 per channel	$<0.05$	0.19	0.216	0.179	80	0.955 a)	0.648	11.9	0.953

Table 8: Classical diagnostics at the canonical sensor counts (test frames). Local columns average the observed channels:

\bar{d}/\ell

is the mean nearest-sensor distance over the decorrelation length (cylinder: wake region

x>D

|y|<4D

\hat{e}_{\mathrm{NN}}

the nearest-neighbor error predicted from the training structure function and the sensor count (location-resolved on the cylinder mesh), and NN and IDW the measured errors. Global columns aggregate all channels: POD rank

r

, training energy fraction

E_{r}

, relative

L_{2}

of projecting the test frames onto

\Phi_{r}

(the floor of any coefficient fit), median condition number of

H\Phi_{r}

over the sensor draws, and the measured gappy POD error. Kolmogorov local values use the periodic neighbor search. a) In-sample over 150 correlated training frames.

Figure 5: Kolmogorov reconstruction error versus sensors per field (vorticity, 50 test frames, identical draws for every method at every density, all densities inside the training ranges). Colored: learned methods; gray: classical anchors, with the gappy POD rank and the LSE penalty chosen per density. S3GM scales its guidance weight from the value tuned at 655 sensors with the inverse square root of the sensor fraction and diverges at 65 and 164 sensors. A 3D sweep of the classical anchors is in SM Fig. S1.

Figure 6: Spectral content of single samples on the held-out 3D cube (frame of Fig. 4 , one sensor draw, sampler settings of Table 6 , Hann window). (a) Energy spectra of the total velocity with the inertial (

k=8

–31) and dissipation (

k=32

–50) bands shaded. (b) Band energy relative to the DNS and (c) band-limited correlation with the DNS, pooled over the observed channels. SiT-point members carry grid-scale speckle, which shows as excess dissipation-band energy.

Figure 7: Measured peak GPU memory against output resolution (one H100, 80 GB; log–log) for (a) full-field reconstruction and (b) one training step. Crosses mark measured out-of-memory failures. Grid models run at batch size 1; the latent-FM curve is its autoencoder alone, a lower bound on sampling. The point model (Senseiver) evaluates fixed-size query chunks and stays flat to

1024^{3}

Computational Fluid Dynamics

Computational fluid dynamics numerically models fluid motion, turbulence, heat and mass transport, and fluid–structure interactions.

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Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

Qian Zhang, George Em Karniadakis

Division of Applied Mathematics, Brown University

PDE Operator Learning Operator Learning Computational Fluid Dynamics

Open paper AskPDF arXiv

Reconstructing flow fields from sparse measurements is a fundamental problem in fluid mechanics with broad implications for modeling, control, and design. In this work, we propose a novel operator learning framework that leverages the architecture of language models to perform flow reconstruction in a mesh-free manner. We reformulate flow field reconstruction as a sequence-to-sequence learning task, where sparse measurements are treated as context and unobserved locations as queries. Our model learns to reconstruct the full flow field from sparse inputs, effectively capturing spatial correlations and long-range dependencies. We evaluate the proposed approach on four benchmark datasets: (1) two-dimensional vortex street simulations, (2) daily average temperature data across the contiguous United States, (3) three-dimensional blood flow simulations based on dissipative particle dynamics, and (4) three-dimensional turbulent jet flow measurements obtained via particle tracking velocimetry. Across all cases, our method demonstrates competitive reconstruction accuracy, even with highly incomplete data (less than 10% observed), and achieves efficient performance. The results highlight the potential of language models as robust and scalable tools for scientific data reconstruction, and suggest a promising direction toward the development of foundation models for scientific and engineering applications.

Abstract

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