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High-fidelity Three-Dimensional Flow Field Reconstruction Driven by Turbulent Fluid Dynamics

Hanlin Kong, Jianan Wei, Yuntian Chen, Yujia Wang, Wenguan Wang

arXiv:2610.18045v1physics.flu-dyncs.AIcs.LG

TL;DR

High-fidelity 3D turbulent flow fields are costly to obtain, and reconstruction methods still struggle with multiscale volumetric structure and coherent temporal evolution. The paper combines a multiscale neural operator, a state space model, physical residual constraints, and MoE adaptation; across turbulent systems and atmospheric reanalysis, it reports lower errors and better preservation of structure and temporal coherence than baselines.

  • Problem

    Reconstructing time-resolved 3D turbulence requires recovering multiscale volumetric structure and coherent dynamics, while data-driven methods remain challenged by these coupled requirements.

    Source: §1 · p. 2

  • Method

    A multiscale neural operator and state space model jointly reconstruct spatial structure and temporal evolution, with Navier–Stokes residuals and MoE adaptation for unknown coarse-graining.

    Source: §1 · p. 2, §3 · p. 19

  • Results

    Across FIT, MHDT, and HBDT under tested filters, nRMSE reductions reach approximately 30% relative to the best-performing baseline, alongside better-preserved multiscale structure and temporal coherence across evaluations.

    Source: §1 · p. 3, §3 · pp. 19–20

  • Takeaways & Limitations

    The evaluations support reconstruction of high-fidelity 3D flows from coarse-grained inputs across canonical turbulence systems and atmospheric reanalysis data.

    Source: §1 · p. 3

    The framework produces one deterministic reconstruction per LR sequence and does not quantify variability among compatible HR realizations or establish recovery of the full conditional distribution.

    Source: §3 · p. 20

Abstract

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High-fidelity three-dimensional turbulent flow fields are essential for scientific and engineering analyses, yet obtaining them through direct numerical simulation or high-resolution measurements remains expensive. Super-resolution reconstruction from coarse-grained data therefore offers a promising alternative, but remains fundamentally challenging because fine-scale structures are intrinsically three-dimensional and dynamically coupled across scales. Existing data-driven approaches still struggle to recover multiscale volumetric structures and their coherent temporal evolution. To address these limitations, we propose a turbulence-aware framework for time-resolved three-dimensional flow reconstruction. The framework couples a multiscale neural operator, designed to recover volumetric structures, with a state space model to promote temporal consistency in the reconstructed dynamics. In addition, Navier-Stokes residuals are incorporated as soft constraints to regularize the reconstruction toward physically plausible dynamics. A mixture-of-experts mechanism is also introduced to support unsupervised adaptation to unknown coarse-graining. Across evaluations on canonical turbulent systems and atmospheric reanalysis data, the proposed framework reduces reconstruction errors relative to learning-based and classical baselines, while better preserving multiscale spatial structure and coherent temporal evolution. These results validate the effectiveness of the proposed framework in reconstructing high-fidelity three-dimensional flow fields from coarse-grained data.

1 Introduction

High-fidelity, time-resolved 3D flow data are costly to obtain, while existing reconstruction approaches face challenges recovering volumetric multiscale structure and coherent dynamics. The paper proposes a physics-regularized framework and evaluates it across turbulent flows and atmospheric reanalysis data.

1 Introduction

  • High-fidelity volumetric flow data are important for science and engineering, but DNS is computationally prohibitive and less costly alternatives sacrifice fidelity through simplifying assumptions.Consequently, time-resolved volumetric data are often available only at limited spatial resolution.Source: §1 · p. 1
  • Three challenges motivate the work: recovering coupled 3D multiscale structures and temporal evolution, representing dynamical rather than frame-wise physical consistency, and handling coarse-graining that differs at inference.Prior frame-wise constraints can yield spatially plausible reconstructions that deviate from consistent dynamical trajectories.Source: §1 · p. 2
  • The framework combines a multiscale neural operator for volumetric structure, a state space model for temporal evolution, Navier–Stokes residual constraints, and MoE adaptation to unknown coarse-graining.The MoE combines pretrained experts, with residuals guiding adaptation without HR supervision.Source: §1 · pp. 2–3
  • Across canonical turbulent systems and ERA5, the framework improves reconstruction accuracy over learning-based and classical baselines while better preserving multiscale structure and dynamically consistent evolution.The evaluations include accuracy, spatial-structure diagnostics, temporal consistency, and turbulence statistics.Source: §1 · p. 3
  • The benchmark and evaluation span forced isotropic, magnetohydrodynamic, and buoyancy-driven turbulence, multiple coarse-graining settings and upscaling factors, decaying turbulence, and atmospheric reanalysis.TurbSR3D provides standardized splits; decaying isotropic turbulence tests non-stationary conditions and independently evolved LES inputs paired with DNS references.Source: §1 · pp. 2–3

2 Results

The Results section evaluates the framework’s reconstruction of turbulent flows and ERA5 data, including its multiscale spatial modeling, temporal state-space memory, Navier–Stokes residual regularization, and expert mechanism for unknown coarse-graining. It also examines MoE adaptation to unseen coarse-graining and SGS-modeled LES inputs, reporting reconstruction and physical-diagnostic improvements without high-resolution supervision.

2.1 Problem set-up

  • The LR-to-HR framework uses latent state-space temporal memory, multiscale Fourier and convolutional refinement, Navier–Stokes residual regularization, and expert gating across coarse-graining hypotheses.The state-space model evolves latent features over time; the multiscale neural operator combines long-range frequency-domain interactions with local convolutional refinement.Source: Figure 1 · p. 4, §2.1 · p. 5

2.2 Data preparation

  • Evaluations cover FIT, MHDT, HBDT, DIT, and ERA5, using box/Favre, Gaussian, and spectral-cutoff training degradations plus tricubic, Lanczos, and noisy uniform-subsampling tests.FIT and MHDT use 4×/8× upscaling, while HBDT uses 8×/16×; supervised settings use 6:2:2 training, validation, and test splits.Source: §2.2 · pp. 5–7

2.3 Accurate reconstruction of turbulent flows

  • The method achieves the lowest nRMSE in every TurbSR3D setting, improving FIT nRMSE by 29.9% (4×, box), 28.7% (4×, Gaussian), and 12.6% (4×, spectral) over second-best.At 8× FIT, the nRMSE improvements are 17.0%, 19.4%, and 5.9%; HBDT reductions are 14.1%–27.3%. Across all 18 settings, the method also achieves the highest SSIM and PSNR and lowest fRMSE.Source: Table 1 · p. 6, §2.3 · p. 7

2.4 Recovering multiscale spatial structure from coarse-grained data

  • Across FIT, MHDT, and HBDT, reconstructions better preserve fine-scale structures, high-wavenumber energy, intermittent PDF tails, and velocity-gradient invariant topology.The method recovers more continuous vortex filaments, sharper current sheets, and better-organized divergence features; its spectra, PDFs, and invariant contours more closely match reference data.Source: §2.4 · pp. 7, 9, 11, Figure 2 · p. 8, Figure 3 · p. 10

2.5 Capturing the temporal evolution of turbulent flows

  • Across FIT, MHDT, and HBDT, the method achieves the lowest tRMSE, reducing temporal-increment errors 23%–31% versus the best-performing baseline.On FIT, the full model reduces nRMSE and tRMSE versus the no-state-space variant by 14.36% and 16.05% at 4×, and 12.87% and 6.52% at 8×. On decaying DIT, physical-loss constraints improve nRMSE, tRMSE, and dissipation-ratio agreement with DNS throughout the plotted interval.Source: §2.5 · pp. 11, 12, 14, Figure 5 · p. 13, Figure 6 · p. 14

2.6 Atmospheric downscaling on ERA5

  • On ERA5 at both downscaling factors, the method achieves the lowest MSE and tRMSE, better preserves 850-hPa spatial fields and spectra, and leads on cross-level wind-shear metrics.It attains the highest vCoh and lowest vRMSE for the 850–500 hPa bulk wind-vector difference at both factors.Source: Figure 7 · p. 15, §2.6 · p. 16

2.7 Generalizing to unknown coarse-graining

  • For unknown coarse-graining, the study tests prescribed filtering, resampling, and measurement-noise changes alongside independently evolved LES inputs, where degradation includes evolving modeling and numerical errors.Because models trained under one filtering strategy degrade when inference coarse-graining differs, the proposed mechanism uses frozen experts for distinct coarse-graining hypotheses and a gating network to combine predictions.Source: §2.7 · p. 16
  • Across tricubic, Lanczos-filtered, and noisy uniform-subsampling inputs, the MoE achieves lower numerical error than any single expert.The noisy setting uses 5% additive Gaussian noise.Source: Figure 8 · p. 17, §2.7 · p. 18
  • For tricubic inputs, the MoE reduces high-wavenumber energy excess, better matches reference spectral decay and DNS Q–R topology, and preserves vortical-structure continuity.The individual experts show scale-dependent spectral bias, including premature decay at higher wavenumbers.Source: Figure 8 · p. 17, §2.7 · p. 18
  • Adaptation error drops sharply when gating-network training increases from 5 to 10 LR sequences, with only small improvements thereafter.Box and spectral reconstruction errors are also weakly correlated, supporting their complementary use.Source: Figure 8 · p. 17, §2.7 · p. 18
  • With only 10 LES patches and no HR supervision, the MoE has the lowest reconstruction nRMSE throughout LES evolution and better recovers DNS fine-scale vortices.Its normalized mean dissipation stays close to unity, unlike the Box and Gaussian experts, which overestimate it, and the Spectral expert, which underestimates it.Source: §2.7 · p. 18, Figure 9 · p. 19
  • The MoE achieves the lowest relative flux-field error at every examined filter scale, indicating improved recovery of multiscale energy-transfer magnitude and spatial distribution.Together with the LES results, this supports data-efficient generalization to unseen coarse-graining and SGS-modeled inputs without HR supervision.Source: §2.7 · p. 19, Figure 9 · p. 19

3 Discussion

The framework combines spatial, temporal, and physics-informed modeling, and reports lower reconstruction errors alongside improved recovery of turbulence structure and temporal coherence. Its scope remains bounded by deterministic outputs and challenges in generalizing across regimes and settings.

3 Discussion

  • The framework combines a multiscale neural operator and state space model with Navier–Stokes residual regularization and MoE adaptation to unknown coarse-graining using LR data alone.Source: §3 · p. 19
  • Across FIT, MHDT, and HBDT under box/Favre, Gaussian, and spectral-cutoff filtering, nRMSE reductions reach approximately 30% versus the best-performing baseline.The framework also better recovers multiscale energy distributions and turbulence statistics, and improves temporal coherence relative to baselines.Source: §3 · pp. 19–20
  • The deterministic framework returns one reconstruction per LR sequence, so agreement with reference fields and selected statistics does not establish the full conditional distribution.Representing compatible HR variability with generative approaches may add training and sampling costs.Source: §3 · p. 20
  • Generalization across physically distinct turbulent regimes remains challenging, while broader training coverage is difficult because high-fidelity three-dimensional datasets are expensive to obtain.Source: §3 · p. 20
  • Extensions to unstructured or curvilinear meshes and settings with partly unknown governing dynamics remain challenges; training multiple experts also raises one-time training costs.Source: §3 · p. 20

4 Methods

The framework combines temporal state-space modeling, multiscale spatial operators, physical soft constraints, and unsupervised adaptation to reconstruct time-resolved volumetric turbulence. It evaluates these components across turbulence and atmospheric datasets using accuracy, structural, temporal, and spectral measures.

4.1 State space model

  • A state space model evolves local latent states over time on the 3D Eulerian grid, with input-conditioned updates adapting to changing flow patterns.The resulting temporally evolved features feed the multiscale neural operator.Source: §4.1 · p. 21

4.2 Multiscale neural operator

  • Hybrid blocks couple global spectral interactions with local convolutional refinement to capture large-scale structures and fine turbulent details.The spectral operator uses an adaptive Fourier neural operator; the depthwise convolution supplies a local spatial inductive bias.Source: §4.2 · p. 22

4.3 Physics-informed neural optimization

  • Paired LR–HR voxel-wise L1 supervision is combined with physical residual penalties to encourage consistency with governing equations.For FIT and MHDT, the momentum residual is high-pass filtered to emphasize finer scales; HBDT and DIT use unfiltered momentum penalties.Source: §4.3 · pp. 22–23

4.4 Unsupervised generalization with mixture-of-experts

  • An unsupervised mixture-of-experts adapts to unknown coarse-graining by combining frozen experts through a learned gate optimized with LR inputs and physics residuals.The gating network requires no HR supervision during inference-time adaptation.Source: §4.4 · p. 24

4.5 Evaluation metrics

  • Evaluation combines pointwise accuracy, structural similarity, temporal consistency, and multiscale turbulence statistics.Metrics include nRMSE, PSNR, 3D SSIM, frame-increment tRMSE, Fourier-space RMSE, energy spectra, and velocity-gradient diagnostics.Source: §4.5 · pp. 24–25

4.6 Datasets

  • TurbSR3D contains over 3,500 samples spanning three turbulence regimes and 4×, 8×, and 16× coarse-graining; DIT and ERA5 provide additional evaluation settings.DIT tests non-stationary conditions and independently evolved LES inputs against DNS references; ERA5 covers ten January 2022 days with 4× and 8× horizontal downsampling.Source: §4.6 · p. 25

4.7 Baselines

  • The framework is compared with tricubic and bicubic interpolation and deep-learning baselines including RRDB, EDSR, RCAN, FNO, PhySR, SwinIR, and DiffSR.The baseline set includes image super-resolution, operator-learning, physics-informed, transformer, and diffusion approaches.Source: §4.7 · p. 25

1 Implementation and methodology details

TurbSR3D experiments use a shared training protocol, with physics-loss settings selected through sensitivity tests on DIT and FIT.

1.1 Benchmark experimental settings of TurbSR3D

  • TurbSR3D models train for 1,000 epochs at 4× and 500 epochs at 8×/16×, using AdamW, L1 data and physical losses, and λ_phy = 10^-3.DIT uses λ_phy = 10^-2; both settings use β = 0.3.Source: §1.1 · p. 33

1.2 Sensitivity to physical-loss weights

  • With β = 0.3, λ_phy = 0.01 for DIT and 0.001 for FIT give the best tested nRMSE, SSIM, and PSNR; increasing λ_phy to 0.1 worsens all three metrics.At those respective λ_phy values, β = 0.3 outperforms tested β values of 0.1 and 1 across all three metrics.Source: §1.2 · p. 33, Table 1 · p. 33, Table 2 · p. 34

1.3 Benchmark experimental settings of ERA5

ERA5 training uses a separate protocol and omits explicit governing-equation terms from the loss.

1.3 Benchmark experimental settings of ERA5

  • ERA5 models train for 100 epochs on one GPU, and the loss uses only L1 data fidelity because explicit governing equations are not incorporated.The stated rationale is ERA5’s coarse spatial resolution of approximately 30 km.Source: §1.3 · p. 34

1.4 Details of state space model

The state space model discretizes continuous-time dynamics and uses a spatially varying gate to modulate local temporal updates.

1.4 Details of state space model

  • A continuous-time state space system is discretized with a zero-order hold, and a spatially varying selection gate modulates local temporal updates from the instantaneous flow state.The gate is computed from a locally transformed latent representation; a point-wise projection then aligns feature channels.Source: §1.4 · pp. 34–35

1.5 Methods and training configurations of TurbSR3D

The section specifies the benchmark baselines and TurbSR3D configuration. Across the reported test cases, the framework has the lowest average parameter count while achieving superior performance.

Methods and training configurations of TurbSR3D

  • Learnable baselines are generally retrained under a unified protocol, and the section details their architectures alongside the proposed model’s configuration.TurbSR3D uses one state-space-model layer and six multiscale neural-operator blocks, with a four-block diagonalization to reduce parameter count and improve computational efficiency.Source: §1.5 · p. 35
  • The decoder upsamples volumetric features through staged 3D convolutions and pixel shuffle, doubling resolution along all three Cartesian coordinates at each stage.It uses two, three, or four stages for 4×, 8×, or 16× reconstruction, respectively; a final learned projection maps features to physical variables.Source: §1.5 · p. 36
  • The proposed framework achieves superior performance with the fewest average parameters across all test cases, averaging 3.87 M.The average covers six configurations across FIT, MHDT, and HBDT.Source: §1.5 · p. 36, Figure 1 · p. 36

1.6 Scale-selective momentum regularization

The section describes a high-pass momentum-residual penalty and compares alternative formulations. On FIT at 4×, high-pass L1 gives the lowest reported high-wavenumber energy-spectrum error among the three formulations.

Scale-selective momentum regularization

  • The high-pass operator subtracts a Gaussian-smoothed component from the momentum residual, suppressing its direct forcing contribution to the scale-selective constraint.The filtered forcing RMS is at most 0.1989% of the common pre-filter reference.Source: §1.6 · pp. 36–37, Table 3 · p. 37
  • The implemented momentum term uses linear-growth L1 rather than squared L2, limiting the influence of large local residuals that may include numerical-derivative errors.The authors motivate this choice by turbulent intermittency in velocity gradients and small-scale velocity increments.Source: §1.6 · p. 37
  • The benchmarks use state-dependent low-mode energy maintenance for FIT and deterministic Taylor–Green forcing for MHDT, where L1 serves as a soft dynamical constraint.The passage contrasts these settings with the quadratic residual term in the Onsager–Machlup action for additive Gaussian white-in-time forcing.Source: §1.6 · p. 37
  • 0.1736 is the relative L1 error for high-pass L1, lower than 0.1920 for high-pass squared L2 and 0.2125 for unfiltered L1 on FIT at 4×.All three formulations closely follow the reference spectrum at low and intermediate wavenumbers.Source: §1.6 · p. 37, Figure 2 · p. 38

1.7 Hard incompressibility constraint on the physical output

The section presents hard constraint projection as an optional way to enforce discrete continuity on the reconstructed physical velocity. In the DIT comparison, it greatly reduces divergence RMS with a small nRMSE increase.

Hard incompressibility constraint on the physical output

  • HCP projects the reconstructed physical velocity onto the discrete divergence-free subspace, making the smallest discrete L2 correction needed to satisfy continuity.The constraint couples the three velocity components through discrete spatial derivatives; the projection adds no learnable parameters or penalty weight.Source: §1.7 · pp. 38–39
  • HCP is an optional extension evaluated only on DIT at 4× with spectral-cutoff inputs, while the soft momentum term continues to regularize dynamics during training.The reconstruction loss for the HCP-trained model is evaluated on the projected velocity.Source: §1.7 · p. 39
  • 1.52 × 10−6 is the divergence RMS with HCP, down from 1.41 × 10−2 without HCP, while nRMSE increases from 0.0773 to 0.0779.The energy spectra remain close to the reference at low and intermediate wavenumbers; the HCP model is closer over part of the high-wavenumber range.Source: §1.7 · p. 39, Figure 3 · p. 39, Table 4 · p. 40

1.8 Methods and training configurations of ERA5

The ERA5 configuration adapts the reconstruction setup to the dataset’s characteristics, using slice-wise bicubic and SwinIR baselines alongside FNO and the proposed model.

Methods and training configurations of ERA5

  • For ERA5, bicubic and SwinIR reconstruct 2D slices independently, while FNO retains horizontal, vertical, and temporal spectral modes.The section specifies two hidden layers and width 64 for FNO, with 4 horizontal spatial modes, 2 vertical spatial modes, and 2 temporal modes.Source: §1.8 · p. 40
  • The ERA5 TurbSR3D configuration uses four multiscale neural-operator blocks with channel width 224 and a decoder shared across time.Upsampling is performed at each pressure level using 2D pixel shuffle; the 4× and 8× configurations use two and three stages while retaining all eight levels.Source: §1.8 · p. 40

1.9 Supplementary definitions of evaluation metrics

The evaluation metrics measure reconstruction fidelity at multiple levels, from pointwise error and local structure to Fourier-scale error and energy distribution.

Peak signal-to-noise ratio (PSNR)

  • PSNR evaluates reconstruction quality logarithmically from mean squared error.Its definition also uses the data range of the ground-truth field.Source: §1.9 · p. 40

Structural similarity (SSIM)

  • 3D SSIM assesses local structural fidelity using local means, variances, covariance, and stability constants.Source: §1.9 · p. 40

Fourier-space RMSE (fRMSE)

  • Fourier-space RMSE measures scale-dependent errors between the predicted and ground-truth 3D spectra.The calculation uses the 3D discrete Fourier transform over spatial dimensions and frequency indices.Source: §1.9 · pp. 40–41

Energy spectra

  • Energy spectra are computed by summing Fourier-mode energies within spherical wavenumber shells and dividing by shell width.FIT uses the kinetic energy spectrum, MHDT the total kinetic-plus-magnetic spectrum, and HBDT a density-weighted kinetic spectrum.Source: §1.9 · p. 41

HBDT spectrum

  • For HBDT, the density-weighted kinetic spectrum uses mass density in its normalization.With this normalization, summing across Fourier shells gives P.Source: §1.9 · p. 41

2 Datasets and preprocessing

The dataset and preprocessing pipeline combines three canonical turbulent systems, additional decaying-turbulence and LES–DNS data, and atmospheric fields, with multiple coarse-graining and evaluation settings.

2.1 JHTDB data sources and target variables; 2.2 System-specific configurations and parameters

  • TurbSR3D draws on FIT, MHDT, and HBDT, with system-specific governing equations and reconstructed variables.The configurations use periodic domains unless stated otherwise and specify HR grid spacing and system parameters.Source: §2.1 · p. 42, §2.2 · p. 43, Table 5 · p. 43

2.3 TurbSR3D corpus statistics and splits; 2.6 JHTDB acquisition notes

  • The corpus contains 3,600+ high-quality spatiotemporal HR samples with non-overlapping train, validation, and test splits; JHTDB data are acquired through block-wise sampling with retries and SciServer.Supplementary Table 9 reports sample counts and storage footprints.Source: §2.3 · p. 43, Table 9 · p. 44, §2.6 · p. 45

2.4 Coarse-graining and LR generation

  • LR inputs are generated by filtering and subsampling, with box/Favre and Gaussian filtering among the coarse-graining options.The generic process can also include additive measurement noise.Source: §2.4 · pp. 43–44
  • Coarse-graining settings also include spectral cutoff, tricubic interpolation, Lanczos filtering, and noisy uniform downsampling; upscaling factors are typically 4, 8, or 16.Source: §2.4 · p. 44

2.5 Normalization and windowing

  • Variables are standardized with training-set z-scores, and TurbSR3D uses non-overlapping 8-frame windows over 64³ volumes with 60%/20%/20% splits.Zero padding preserves spatial feature-map dimensions in convolutional layers.Source: §2.5 · p. 45

2.7 DNS of decaying homogeneous isotropic turbulence

  • The DIT dataset comprises 201 velocity–pressure frames over t = 0.200–0.600, and its nominal dissipation-scale resolution is reported as kmaxη = 1.716–1.917.Grid convergence was not assessed.Source: §2.7 · p. 45

2.8 LES simulation and LES–DNS pairing

  • For LES–DNS pairing, LES starts from DNS at t0 = 0.522 on a 64³ grid, evolves independently for 160 steps, and is paired with DNS at eight matching output times.The LES uses a Smagorinsky closure and retains the DNS viscosity and numerical scheme.Source: §2.8 · p. 46

2.9 ERA5 data access and experimental protocol

  • The ERA5 downscaling experiment maps eight-frame, eight-pressure-level fields to 128 × 128 fields with temperature and horizontal wind channels.The pressure levels range from 1000 to 300 hPa.Source: §2.9 · p. 46

3 Additional details on experiments

This section presents additional evaluations of reconstruction performance, robustness, ablations, flow structures, and atmospheric shear downscaling. It examines cross-level shear reconstruction in ERA5, sensitivity to training randomness, and generalization across unseen coarse-graining methods.

3.1 Accurate reconstruction of turbulent flows

  • Across 18 settings, the framework achieves the highest PSNR and SSIM and the lowest nRMSE and fRMSE among compared methods.The settings span three turbulent systems, three coarse-graining strategies, and two upscaling factors per system; learning-based methods were independently trained on the same LR–HR pairs without cross-dataset fine-tuning or weight sharing.Source: §3.1 · p. 47, Table 10 · p. 48, Table 11 · p. 49

3.2 Noise robustness

  • At σξ = 0.10, the model’s nRMSE is 0.0587, 31.0% lower than PhySR, and its increase from σξ = 0 to 0.10 is 0.0242.It achieves the lowest nRMSE at every tested noise level; mean nRMSE rises from 0.0345 to 0.0587, while other methods’ increases range from 0.0302 to 0.0621.Source: Figure 4 · p. 51, Table 13 · p. 51, §3.2 · pp. 51–52

3.3 Ablation study of the multiscale neural operator

  • Removing the global Fourier operator most increases low-range errors, whereas removing the local convolutional operator most increases mid- and high-range errors.Without Os, low-range velocity and vorticity errors rise by 69.9% and 66.1%, versus 14.9% and 10.1% without Oc; without Oc, mid- and high-range increases are 23.9%/18.9% and 10.3%/10.1%, versus 11.0%/13.5% and 1.3%/0.8% without Os. Both removals affect the full spectrum, while the full model is best at every evaluated wavenumber. At matched capacity, replacing the proposed spatial design with an MS-ESRGAN-derived block raises overall nRMSE from 0.0615 to 0.0739 (20.1%); the proposed design also has lower shell-wise velocity nRMSE in all three ranges and reduces high-wavenumber energy underestimation.Source: Figure 5 · p. 52, Table 14 · p. 52, §3.3 · pp. 52–53

3.4 Ablation and sensitivity study of the state space model

  • At 4× and 8×, adding the SSM reduces nRMSE by 14.36% and 12.87%, and tRMSE by 16.05% and 6.52%, respectively, versus the SSM-free model.The full model has the lowest nRMSE and tRMSE among five evaluated models at both factors; visualizations show it closely matches reference structures and preserves smoother temporal evolution than the SSM-free model and baselines.Source: Table 16 · p. 54, §3.4 · p. 54, Figure 7 · p. 54
  • At matched parameter counts, the SSM has the lowest nRMSE, tRMSE, and fRMSE and the highest SSIM and PSNR among the temporal modules tested.Compared with ConvLSTM and the causal Transformer, respectively, it reduces nRMSE by 4.95% and 7.44%, tRMSE by 5.10% and 6.28%, and fRMSE by 4.87% and 7.32%, while increasing PSNR by 0.49 and 0.73 dB.Source: §3.4 · pp. 54–55, Table 17 · p. 55
  • On FIT and MHDT, the default SSM dimensions (df, ds) = (64, 16) offer a practical accuracy–cost compromise; reducing df to 32 cuts SSM GFLOPs by 53.2% with an approximately 5% accuracy penalty.Halving ds increases nRMSE by 1.4% on FIT and 1.7% on MHDT, with larger relative tRMSE increases; expanding either capacity yields modest accuracy gains at higher computational cost.Source: Table 18 · p. 55, §3.4 · pp. 55–56

3.5 Recovering multiscale spatial structure from coarse-grained data

  • In FIT’s 8× box-filtered comparison, RRDB’s displaced positive-Q∗ contours and shortened lower-right extension indicate distorted rotation–strain configurations and deficient recovery of strongly strain-dominated states.For FIT at 8× with spectral-cutoff coarse-graining, removing Oc or Lphy shifts the right flank of rotation-dominated contours toward larger R∗, while removing Os contracts the negative-R∗ side; all removals degrade the strain-dominated branch, and the full model retains some outer-contour deviations.Source: Figure 11 · p. 60, Figure 12 · p. 60, §3.5 · pp. 61–62

3.6 Atmospheric downscaling on ERA5

  • For ERA5 atmospheric downscaling, the section defines vRMSE and vCoh using wind-vector differences between 850 and 500 hPa to assess cross-level shear.The shear magnitude is a bulk-difference proxy, and the two metrics evaluate amplitude accuracy and spatial organization of strong cross-level shear.Source: §3.6 · p. 64
  • ERA5 cross-level diagnostics assess both the spatial organization and amplitude of the 850–500 hPa bulk-shear field.Vertical coherence is Pearson correlation restricted to the reference’s top 10% strongest-shear points (q = 0.9); metrics are computed per time step and averaged over the evaluation set.Source: §3.6 · p. 64, Table 19 · p. 64

3.7 Training-seed uncertainty

  • Across three fixed-protocol seeds, the method achieved the best mean for every reported metric on FIT at 4× and ERA5 at 4× and 8×.On FIT, nRMSE was 0.0292 ±0.0001 versus RCAN’s 0.0410 ±0.0006, and tRMSE was 0.0042 ±0.0000 versus SwinIR’s 0.0053 ±0.0000. On ERA5, MSE was 0.0762 ±0.0011 at 4× and 0.4451 ±0.0007 at 8×, versus SwinIR’s 0.0947 ±0.0005 and 0.5333 ±0.0006; standard deviations were small relative to between-method differences.Source: §3.7 · pp. 64–65, Table 20 · p. 65, Table 21 · p. 65

3.8 Generalizing to unknown coarse graining

  • The main configuration uses a two-expert box+spectral MoE, whose near-zero error correlation under tricubic downsampling indicates complementary error modes.The Gaussian expert’s error correlation with both was approximately 0.5, suggesting partially redundant errors; the results support expert diversity rather than simply adding experts. The quantitative evaluation covers tricubic downsampling, Lanczos filtering, and uniform subsampling with 5% additive Gaussian noise.Source: §3.8 · pp. 65–66, Table 22 · p. 66
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