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

Ramzi Idoughi, Radouan Boukharfane, Peter Wonka

arXiv:2610.16645v1cs.LGcs.AIcs.CE

TL;DR

Recovering physically plausible velocity fields from sparse, noisy, and incomplete observations is challenging. DFK-Former predicts divergence-free kernel parameters from observations, and the evaluation reports gains across sparsity, noise, and occlusion, including real DNS data.

  • Problem

    Sparse, incomplete, and noisy measurements make recovery of dense, physically plausible velocity fields challenging.

    Source: §1 · p. 1

  • Method

    DFK-Former predicts kernel parameters from observations to produce velocity fields that are exactly divergence-free by construction.

    Source: §1 · p. 2, §6 · p. 9

  • Results

    The evaluation demonstrates gains across sparsity, noise, and occlusion, including on real DNS data.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Direct prediction combines the accuracy and physical guarantees of kernel representations with the efficiency and robustness of learned flow priors.

    Source: §6 · p. 10

    Training uses synthetic data, and broader evaluation on experimental measurements and larger-scale studies is needed to characterize robustness and scaling.

    Source: §6 · p. 10

Abstract

from arXiv · show

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.

1 INTRODUCTION

The paper addresses reconstruction of physically plausible velocity fields from sparse, incomplete, and noisy measurements. DFK-Former predicts divergence-free kernel parameters from unordered observations, combining learned flow priors with exact incompressibility guarantees.

1 INTRODUCTION

  • Sparse, incomplete, and noisy measurements make recovery of dense, physically plausible velocity fields challenging across flow-measurement settings.Source: §1 · p. 1
  • Existing divergence-free kernels require per-instance optimization, while HDNet provides a learned prior but only approximate incompressibility and assumes dense structured 2D observations.Source: §1 · p. 2
  • DFK-Former maps unordered 3D positions, noisy velocities, and confidence scores to kernel parameters in a single forward pass, yielding an exactly divergence-free, mesh-free field.Source: §1 · p. 2
  • Anisotropic kernels improve reconstruction at matched model capacity, and temporal modeling preserves exact incompressibility at every instant.Source: §1 · p. 2
  • The evaluation demonstrates gains across sparsity, noise, and occlusion, including on real DNS data.Source: §1 · p. 2

2 RELATED WORK

Prior approaches reconstruct sparse flow measurements using varied modalities, representations, and learning methods, but often use per-instance optimization or only approximate physical constraints. DFK-Former predicts parameters of a continuous divergence-free representation from unordered observations in one forward pass.

Flow Reconstruction from Sparse Observations

  • Coordinate-based neural fields can reconstruct from sparse observations, but the cited approaches optimize separately for each scene and enforce physical constraints only approximately.Source: §2 · pp. 2–3

Divergence-Free Representations and Physical Constraints

  • DFKs analytically guarantee continuous-domain incompressibility, while prior DFK-based approaches fit each scene independently; DFK-Former predicts kernel parameters directly.Source: §2 · p. 3

Set-Based Geometric Inference and Operator Learning

  • Sparse measurements are unordered sets, motivating permutation-equivariant architectures; prior field-regression methods generally satisfy incompressibility only approximately.Source: §2 · p. 3
  • DFK-Former combines a permutation-equivariant transformer with a continuous divergence-free kernel representation, predicting its parameters in one forward pass.Source: §2 · p. 3

3 BACKGROUND: DIVERGENCE-FREE KERNEL FIELDS

Divergence-free kernels represent incompressible velocity fields as continuous combinations of matrix-valued basis functions. DFK-Former predicts the representation's parameters, so the field can be evaluated continuously while retaining exact incompressibility.

3 BACKGROUND: DIVERGENCE-FREE KERNEL FIELDS

  • Incompressible flow is represented by a velocity field whose divergence is zero, and DFKs guarantee this analytically rather than through penalties or projection.Source: §3 · p. 3
  • Each kernel is specified by a center x_k, positive support radius σ_k, and vector coefficient ω_k; the coefficient controls its contribution to the flow.Source: §3 · p. 3
  • The architecture encodes sparse spatiotemporal observations and predicts kernel-representation parameters to produce a continuous, time-varying incompressible field.Source: Figure 1 · p. 4
  • Because each kernel is divergence-free, any linear combination remains exactly divergence-free; the resulting continuous field can be evaluated at arbitrary query locations.Source: §3 · p. 4

4 METHOD

DFK-Former maps unordered, sparse observations directly to divergence-free kernel parameters, producing continuous fields without per-instance optimization. Its architecture extends this representation to adaptive and anisotropic kernels, temporal flows, and procedurally generated training conditions.

4 METHOD

  • DFK-Former maps sparse, noisy, unordered observations to kernel parameters whose evaluation yields a continuous, mesh-free, exactly divergence-free field in one forward pass.This replaces per-instance kernel fitting with a learned mapping from observations to the DFK representation.Source: §4 · p. 4

4.1 DFK-FORMER ARCHITECTURE

  • Kernel queries gather evidence through cross-attention and coordinate their explanations through self-attention, while prediction heads decode kernel coefficients and a global mean-flow term.The fixed-kernel formulation learns a data-driven prior for the ill-conditioned sparse, noisy, partial-observation regime.Source: §4.1 · p. 5
  • Adaptive placement predicts kernel displacements and support radii, increasing reconstruction capacity while retaining divergence-free guarantees for arbitrary locations and radii.The adaptive model is initialized to behave like the fixed-kernel formulation.Source: §4.1 · p. 5
  • Oriented ellipsoidal kernels use rotations and positive semi-axis lengths to capture directional structures while preserving exact incompressibility under invertible transformations.The paper uses an axisymmetric parameterization with transverse and longitudinal scales.Source: §4.1 · p. 5

4.2 TEMPORAL DFK-FORMER

  • Temporal DFK-Former preserves kernel identities across frames and decodes continuous coefficient trajectories and mean flow from spatial and temporal attention.A low-order Fourier parameterization provides a compact continuous-time representation and naturally encourages temporal smoothness.Source: §4.2 · pp. 5–6

4.3 PROCEDURAL TRAINING

  • Procedural training combines projected Gaussian random fields with analytic-flow mixtures, varies corruption and sampling conditions, and band-limits targets to the kernel basis capacity.Training uses sampled spatial query points and a corruption curriculum progressing to heavy noise, sparse sampling, and large occlusions.Source: §4.3 · p. 6
  • Temporal training generates flows from time-varying coefficients of divergence-free modes or advected divergence-free primitives, then supervises randomly sampled space-time queries.Observation counts, noise realizations, and occlusion masks are sampled independently for each frame.Source: §4.3 · pp. 6–7

5 RESULTS

DFK-Former leads the reported reconstruction benchmarks, including temporal flow evaluation. The ablations examine its output representation, attention, and kernel design choices.

5.2 MAIN BENCHMARK

  • DFK-Former achieves the best performance across reconstruction metrics, with approximately 10^-7 divergence and 7.5 ms inference per field.Table 1 compares reconstruction accuracy, divergence residual, and inference cost against optimization-based, operator-learning, and physics-constrained methods.Source: §5.2 · pp. 7–8, Table 1 · p. 7

5.3 TIME-VARYING FLOW

  • Temporal DFK-Former achieves the lowest overall median EPE (0.114), mean EPE (0.124), and mean angular error (4.8°) on 60 time-resolved JHTDB flows.Its median EPE is 0.767 for per-frame DFK fitting, 0.626 for shared-kernel fitting, and 0.428/0.842 for Space-Time SIREN/Curl-SIREN; gains reach 9.7× over per-frame fitting in sparse, occluded settings.Source: §5.3 · p. 8, Table 2 · p. 8
  • Per-frame DFK fitting’s mean error is inflated by 15 of 180 near-singular per-frame solves.This caveat qualifies the temporal comparison’s mean-error results.Source: §5.3 · p. 8

5.4 ABLATIONS

  • Direct velocity regression attains similar accuracy but substantial divergence (0.26); predicting DFK parameters preserves incompressibility without sacrificing accuracy.Projecting the regressed field into the DFK basis raises EPE to 0.165 and latency to 29 ms, compared with 0.120 EPE and 7.5 ms for DFK-Former.Source: §5.4 · p. 8

5.3 TIME-VARYING FLOW

  • In the hardest Forced MHD regime, temporal DFK-Former recovers dominant flow structures with substantially lower EPE and angular error than competing methods.The comparison combines sparse observations, noise, occlusions, and outliers; the figure shows predictions above and observations and reconstruction errors below.Source: Figure 2 · p. 9

5.4 ABLATIONS

  • Adaptive placement yields modest gains, while multi-scale radii reduce CAF EPE from 0.171 to 0.102 and improve results across datasets.The component ladder adds adaptive placement and multi-scale radii to the anisotropic base model; multi-scale radii provide the largest gains, particularly on CAF and real DNS data.Source: §5.4 · p. 9, Table 3 · p. 9
  • EPE improves from 0.281 at K = 32 to 0.120 at K = 1024, then worsens to 0.144 at K = 2048; runtime stays nearly constant across a 32× range.An independent K = 8192 run has similar error, and even K = 32 outperforms closed-form DFK fitting.Source: §5.4 · p. 9, Table 4 · p. 9
  • At matched parameter count, anisotropic kernels improve relEPE across analytic flows, from 1.16× on Lamb–Oseen vortices to 1.34× on vortex rings.The comparison uses anisotropic K = 1024 and isotropic K = 1542 models.Source: §5.4 · p. 9, Table 5 · p. 9

6 CONCLUSION AND LIMITATIONS

DFK-Former predicts divergence-free kernel parameters from sparse, noisy, irregular observations, yielding exact, mesh-free fields without per-instance inference optimization. The conclusion reports extensions to more complex flows and identifies limits in training data, model-capacity exploration, and evaluation scope.

6 CONCLUSION AND LIMITATIONS

  • DFK-Former combines learned flow priors with exact incompressibility and avoids per-instance optimization, while anisotropic kernels and temporal modeling extend it to more complex flows.The fields are mesh-free and continuously queryable by construction.Source: §6 · pp. 9–10
  • The study uses synthetic training data and only limited exploration of larger model and kernel capacities.Although the learned representation generalizes to DNS turbulence without retraining, broader evaluation on experimental measurements and larger-scale studies is needed to characterize robustness and scaling.Source: §6 · p. 10
  • The authors plan to release code, procedural data generators, configurations, and scripts; real JHTDB data require individual authorization and are not redistributed.Scripts and manifests support reproducing the JHTDB protocol for users with valid database access.Source: REPRODUCIBILITY STATEMENT · p. 10

A DIVERGENCE PRESERVATION UNDER THE PIOLA TRANSFORMATION

The appendix establishes that the anisotropic kernel is a contravariant Piola transform of a divergence-free reference kernel. This map preserves divergence-freedom for each kernel and for their superposition.

A DIVERGENCE PRESERVATION UNDER THE PIOLA TRANSFORMATION

  • The anisotropic kernel is constructed as a contravariant Piola transform of the isotropic reference kernel, a standard divergence-preserving map.The proposition assumes an invertible matrix M_k with positive determinant.Source: A DIVERGENCE PRESERVATION UNDER THE… · p. 14
  • Each transformed kernel is divergence-free in every column, as follows from the reference kernel's divergence-free columns and the Piola identity.The proof applies the map to an arbitrary column combination of the reference kernel.Source: A DIVERGENCE PRESERVATION UNDER THE… · p. 14
  • With M_k = R_k diag(a_k), any kernel superposition remains exactly divergence-free while its compact support is reshaped to an ellipsoid.This holds for arbitrary weights, centers, rotations, and strictly positive semi-axis lengths.Source: A DIVERGENCE PRESERVATION UNDER THE… · p. 14

B EXPERIMENTAL DETAILS

The appendix specifies the resolved static and temporal architectures, training recipe, optimization settings, losses, and hardware configuration. It also records the reported static run and the temporal model's departures.

B EXPERIMENTAL DETAILS

  • Table 6 resolves the static architecture and lists the temporal model's additions and departures from the static geometry.K denotes kernel count and ∆ mean kernel spacing.Source: Table 6 · p. 15, B EXPERIMENTAL DETAILS · p. 15
  • The static training recipe uses 150,000 steps, batches of 128 fields per step, float32 model precision, and EMA weights for reported evaluations.The schedule comprises Core, Hard, and cooldown stages; bf16 autocast is limited to the encoder forward pass.Source: B EXPERIMENTAL DETAILS · p. 16
  • The objective combines field, data, cosine, and orthogonality losses with training-only auxiliary regularizers, while the temporal model omits the auxiliary block.The run used four NVIDIA A100 80GB PCIe GPUs and lasted 28 h 02 m (Core), 31 h 20 m (Hard), and 2 h 20 m (cooldown).Source: B EXPERIMENTAL DETAILS · p. 16

C DATA GENERATION

The data engine generates divergence-free spectral and analytic targets, then creates sparse, corrupted observations for static and temporal reconstruction. It also documents capacity-matched training data, evaluation sets, verification, and a fitting protocol for kernel modifications.

C.1 CONVENTIONS: DOMAIN, PRECISION, AND RANDOMNESS

  • Training fields use deterministic online generation on a periodic unit cube, with roughly 1.9 × 10^7 unique fields across training.The pipeline figure depicts the same principles in 2D and 3D and includes CAF examples.Source: §C.1 · pp. 16–17, Figure 3 · p. 17

C.2 PRIMARY GENERATOR: BAND-LIMITED GAUSSIAN RANDOM FIELDS WITH AN EXACT LERAY SPLIT

  • A band-limited Gaussian random field is split by an exact Leray projector into a divergence-free target and an irrotational component.The target's spectral divergence RMS is approximately 10^-14 relative to field RMS.Source: §C.2 · p. 17

C.3 CAPACITY-MATCHED BAND-LIMITING

  • Training spectra are matched to kernel-basis capacity because a fixed compactly supported basis cannot represent structure much finer than about twice its mean kernel spacing.The synthetic benchmark therefore measures reconstruction within basis capacity; real-DNS benchmarks are not band-limited and are reported separately.Source: §C.3 · pp. 17–18

C.4 COHERENT STRUCTURES: THE CAF LIBRARY

  • The CAF library complements Gaussian fields with superpositions of up to four coherent analytic structures, including vortices, ABC flows, and vortex rings.Each structure family is divergence-free, and superposition preserves that property.Source: §C.4 · p. 18

C.5 OBSERVATION CORRUPTION

  • Observations are corrupted through sparse uniform, clustered, or surface sampling, occlusion, irrotational contamination, and Gaussian velocity noise while targets remain clean.Clustered sampling mimics patchy Lagrangian measurements, and surface sampling mimics a camera-visible sheet.Source: §C.5 · pp. 18–19

C.6 NORMALIZATION AND SUPERVISION; C.7 CURRICULUM

  • Per-example normalization uses only corrupted input statistics, and training supervises resampled query points with padding masks; a three-stage curriculum changes data distributions and loss weights.The Hard stage uses heavier corruption and noise, a smaller sample budget, occlusion, and coherent-structure mixing.Source: §C.6 · p. 19, §C.7 · p. 20

C.8 TIME-VARYING DATA

  • Temporal targets are divergence-free at every instant, using time-varying spectral modes or translated coherent structures rather than nonlinear self-advection.Frames are independently corrupted, and training queries include unobserved times.Source: §C.8 · p. 20

C.9 FROZEN EVALUATION SETS; C.10 VERIFICATION; D FROM THE ORIGINAL DFK REPRESENTATION TO OURS: A FITTING STUDY; D.1 PROTOCOL

  • Frozen evaluation sets contain 582 flows, with real DNS reported separately; verification finds divergence RMS near 10^-14 for spectral targets and residuals ≲10^-14 for analytic families.The fitting study compares representation modifications on clean fields using ridge least squares and held-out query points; fitted coefficients are optimal within each fixed basis.Source: §C.9 · p. 20, Table 10 · p. 21, §C.10 · p. 21

D.2 A CONSTANT MEAN-FLOW TERM

These appendix studies refine the divergence-free kernel representation for mean transport, support overlap, anisotropy, adaptive placement, and temporal parameterization. They identify design choices that improve fit while preserving the representation’s physical guarantees.

D.2 A CONSTANT MEAN-FLOW TERM

  • Adding a constant mean-velocity term reduces relEPE from 0.334 to 0.094 on a flow with uniform transport and a zero-mean vortical field.The term is essential for non-zero-mean flows and incurs no cost on zero-mean fields.Source: §D.2 · p. 22, Figure 4 · p. 22

D.3 SUPPORT RADIUS: THE KERNELS MUST OVERLAP HEAVILY

  • At fixed K=8192, increasing support overlap consistently lowers real-DNS approximation floors; supports smaller than mean kernel spacing poorly capture smooth large-scale structures.The radius is parameterized as σk = scale · ∆, with the scale tuned directly.Source: §D.3 · p. 22, Table 11 · p. 22
  • Excessive overlap can reduce basis rank and conditioning; a {5.0, 4.0, 3.0} radius mixture balances large-scale coverage with reduced redundancy.The mixture proportions are {0.3, 0.5, 0.2}.Source: §D.3 · p. 22

D.4 ANISOTROPIC KERNELS

  • On a vortex-tube target, anisotropic kernels improve accuracy at equal kernel count and remain advantageous at equal memory; extend-only scaling and cylindrical anisotropy offer the preferred trade-offs.Orientation alignment helps when coherent directional structure is present, while aligned and random orientations perform similarly on isotropic spectral flows.Source: §D.4 · pp. 22–23, Table 12 · p. 23

D.5 WHAT FITTING ALONE DOES not JUSTIFY: ADAPTIVE PLACEMENT

  • Optimizing centers and radii helps flows needing precise localization but can worsen fits when a fixed basis already works well, and raises the per-kernel parameter count from three to seven.The mixed outcomes support learning adaptivity from data rather than optimizing it independently for each field.Source: §D.5 · p. 23

D.6 KERNEL NORMALIZATION WITH PER-KERNEL RADII

  • For multi-scale and adaptive radii, the σk^-2 normalization keeps kernels with different support sizes at comparable physical scales; omitting it changes their relative amplitudes.The factor follows from the chain rule for the normalized coordinate r = ∥z∥/σk.Source: §D.6 · p. 23

E.1 TEMPORAL FOURIER ORDER

  • On 60 real-DNS flows, F=2 beats F=7 in median EPE (0.114 vs. 0.128) and mean angular error (4.8° vs. 5.7°), but F=7 performs better on synthetic data.The comparison varies temporal Fourier order alone; the result supports matching capacity to temporal complexity, not universally preferring smaller F.Source: §E.1 · p. 24

E.2 CONTROLLED ARCHITECTURE MATRIX

The architecture ablations identify the set encoder and kernel representation as major performance contributors. Corruption sweeps find DFK-Former to be the strongest feed-forward method that preserves incompressibility exactly across the tested operating conditions.

E.2 CONTROLLED ARCHITECTURE MATRIX

  • The set encoder and kernel representation dominate the controlled ablations: replacing attention with DeepSets pooling degrades accuracy, while more kernels and adaptive placement provide gains.Width and Fourier positional encoding yield modest improvements with longer training; depth and orthogonality-loss weight have minor impact (≤1.5%).Source: §E.2 · p. 24, Table 13 · p. 24

E.3 SINGLE-AXIS CORRUPTION SWEEP

  • Across sample-count, noise, occlusion, and spectral-content sweeps, DFK-Former is the strongest feed-forward method that exactly preserves incompressibility.Its advantage over fixed-kernel DFK fitting is consistent across corruption axes; the margin narrows under severe corruption.Source: §E.3 · p. 24, Figure 5 · p. 25, Table 14 · p. 25

F.1 PER-SET BREAKDOWN OF THE MAIN BENCHMARK

The main benchmark, physical-consistency tests, and efficiency results favor DFK-Former across diverse flow sets. Generalization is stronger across corruption levels than across withheld flow families, where prior coverage matters.

F.1 PER-SET BREAKDOWN OF THE MAIN BENCHMARK

  • DFK-Former achieves the lowest EPE on all eleven main-benchmark evaluation sets, with no competing method winning any column.Its average EPE is 0.343 on GRF and 0.153 across JHTDB subsets; no method trained on JHTDB, making those results zero-shot.Source: §F.1 · p. 25, Table 16 · p. 26

F.2 ENFORCING EXACT DIVERGENCE-FREENESS IN OPERATOR-LEARNING BASELINES

  • Leray projection makes operator-learning baselines exactly divergence-free but leaves them behind DFK-Former by 2.10× (FNO) and 3.57× (DeepONet) in macro EPE.Projection improves EPE on 8/11 datasets for FNO and 7/11 for DeepONet, but a periodic-boundary discontinuity in the shear-layer case drives most aggregate degradation.Source: §F.2 · pp. 26–27

F.3 PHYSICAL CONSISTENCY: DIVERGENCE, ENERGY, ENSTROPHY AND SPECTRUM

  • DFK-Former preserves divergence to numerical precision and achieves the lowest band-integrated spectrum error on the turbulence and ten-family evaluations, alongside the lowest overall enstrophy and spectral errors.Exact incompressibility alone is insufficient for physical fidelity: Curl-SIREN is divergence-free yet has large kinetic-energy, enstrophy, and spectral errors.Source: §F.3 · pp. 27–28, Table 17 · p. 27, Table 18 · p. 28

F.4 EFFICIENCY

  • DFK-Former reconstructs a scene in 7.5 ms at 134 scenes/s using 68 MB; its 41 GFLOPs are about half the 81 GFLOPs of fixed-kernel DFK fitting.The forward pass replaces per-field optimization with learned prediction, shifting that cost to training.Source: §F.4 · pp. 28–29, Table 19 · p. 29

F.5 GENERALIZATION

  • Corruption extrapolation causes modest degradation, but withholding coherent-flow families increases error by 4.5–8.0× on withheld families and 2.13× overall.Wall-bounded DNS shows the largest degradation, while homogeneous turbulence remains comparatively stable; the restricted model stays exactly divergence-free at the same inference cost.Source: §F.5 · pp. 29–30, Table 20 · p. 29, Table 21 · p. 30

F.6 TIME-VARYING FLOW: THE FULL BREAKDOWN

Temporal DFK-Former performs best when real-DNS observations leave the flow under-constrained, while optimization-based DFK fitting can match or edge it in clean, dense settings. The temporal model produces continuous, exactly divergence-free fields at unobserved times.

F.6 TIME-VARYING FLOW: THE FULL BREAKDOWN

  • Temporal DFK-Former has the lowest error in every under-constrained dynamic-JHTDB setting, improving over per-frame DFK fitting by 2.6× to 9.7×; clean&dense is the exception.In clean&dense conditions, observations are informative enough for optimization-based reconstruction to approach the fixed-basis optimum.Source: §F.6 · p. 30, Table 22 · p. 30, Figure 7 · p. 32
  • Gains over per-frame fitting reach 14.2× on channel flow and 10.4× on the transitional boundary layer, with smaller but substantial gains on homogeneous turbulence.All predicted fields remain exactly divergence-free to numerical precision.Source: §F.6 · p. 31, Table 23 · p. 31
  • On real-DNS sequences, shared anisotropic geometry with F=2 outperforms moving-kernel variants and per-frame fitting; adaptive placement adds no benefit.All temporal variants share K=1024, training data, and supervision.Source: Table 24 · p. 31
  • In Forced MHD clean&dense data, optimization-based DFK fitting reaches the lowest errors and the learned model is comparable; in sparse data, Temporal DFK-Former preserves flow structures as fitting errors grow.Per-frame DFK median error rises from 0.100 to 0.847 between clean&dense and sparse, while Temporal DFK-Former changes from 0.114 to 0.109.Source: §F.6 · pp. 31–32, Figure 8 · p. 32
  • Space-time SIREN can look plausible in sparse examples but has divergence residuals several orders of magnitude larger than kernel-based methods.The comparison illustrates optimization fitting’s strength under dense constraints and the learned temporal prior’s benefit when observations under-constrain reconstruction.Source: §F.6 · p. 32
  • The temporal model emits continuous fields at unobserved intermediate times while remaining exactly divergence-free.Figure 9 shows three real-DNS frame midpoints the model never observes.Source: Figure 9 · p. 33, §F.6 · p. 33

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