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Iterative Refinement Diffusion for Super-Resolved Data Assimilation of Multiscale Physical Systems

Mrigank Dhingra, Ramchandran Muthukumar, Rebecca Willett, Omer San

arXiv:2608.14744v1cs.LGphysics.flu-dyn

TL;DR

Recovering fine-scale physical states from coarse observations is ill-conditioned, while one-shot super-resolution may miss constraints from temporal dynamics. Iterative Refinement addresses this with hierarchical forecast–analysis updates and performs best on multiscale Kraichnan turbulence, while remaining close to one-shot diffusion on Burgers.

  • Problem

    Recovering high-resolution states from coarse observations is ill-conditioned because many fine-scale states can match the same measurement, particularly in turbulent or shock-dominated systems.

  • Method

    Iterative Refinement replaces one-shot reconstruction with multiresolution forecast–analysis updates using a shared mode-sliced Fourier Neural Operator and conditional diffusion corrector.

  • Results

    One-shot diffusion performs best on corrected Burgers, whereas iterative refinement leads across key reconstruction metrics on 2D Kraichnan turbulence.

  • Takeaways & Limitations

    Hierarchical forecast–analysis refinement is most beneficial for strongly multiscale, underdetermined reconstruction problems, while direct generative reconstruction can suffice for simpler settings.

  • Takeaways & Limitations

    The cascade can propagate errors from generated intermediate posteriors upward, especially during longer rollouts or in more chaotic regimes.

Abstract

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Recovering high-resolution states from sparse, low-resolution observations is a central challenge in scientific machine learning and data assimilation. Classical data assimilation exploits temporal information through forecast-analysis cycles, but often requires repeated access to expensive high-resolution forecast models. Generative super-resolution can recover unresolved structure from coarse observations, but is commonly used as a one-shot mapping that does not fully exploit constraints from past states. We introduce Iterative Refinement (IR), a learned data assimilation framework that combines these perspectives. Instead of performing a single coarse-to-fine reconstruction, IR decomposes the task into resolution-wise forecast-analysis operations across a multiresolution hierarchy. At each stage, a shared neural operator with resolution-dependent spectral mode slicing provides a dynamical prior, while a shared conditional diffusion corrector uses the current coarser-resolution state to produce a refined posterior at the next finer resolution. We evaluate IR on one-dimensional stochastically forced Burgers dynamics and two-dimensional Kraichnan turbulence. On the challenging 256x256 Kraichnan benchmark, IR achieves an RMSE of 0.184 and an SSIM of 0.836, outperforming spectral upsampling, one-shot diffusion super-resolution, enhanced deep super-resolution, and an autoregressive forecaster. On the more constrained Burgers testbed, IR remains competitive with one-shot diffusion, which achieves the lowest RMSE. These results show that one-shot generative reconstruction can be effective for simpler settings, while hierarchical forecast-analysis refinement becomes advantageous in strongly multiscale and underdetermined regimes. Overall, IR combines temporal priors, generative correction, and multiresolution reconstruction for learned data assimilation in complex physical systems.

1. Introduction

Super-resolved data assimilation must recover ambiguous unresolved structure while maintaining temporal and closed-loop consistency. The proposed iterative-refinement framework combines learned dynamical priors, generative correction, and progressive multiscale reconstruction, with previews showing improved fine-scale spectral recovery and an intermediate accuracy–cost regime.

  • Iterative refinement: Iterative refinement performs learned forecast–analysis steps across a multiscale hierarchy, using learned forecasts as dynamical priors and generative models for observation-based correction.Each step operates over a modest refinement ratio, progressively recovering fine-scale information.
  • Challenges: Coarse observations make high-resolution reconstruction severely ill-conditioned because many fine-resolution states can match the same measurements.The ambiguity is particularly prevalent in turbulent or shock-dominated systems.
  • Challenges: Temporal consistency requires dynamically coherent state sequences rather than isolated plausible snapshots, which one-shot generative models may violate through jitter and artifacts.Without explicit temporal conditioning, reconstructions can produce physically implausible transitions.
  • Empirical previews: Iterative refinement most closely follows the ground-truth spectrum, whereas spectral upsampling loses unresolved energy, EDSR decays too rapidly, and one-shot diffusion recovers only part of the high-wavenumber tail.This preview concerns 2D Kraichnan turbulence and unresolved modes beyond the coarse input’s Nyquist limit.
  • Empirical previews: Iterative refinement occupies an intermediate accuracy–cost regime between a highly accurate solver EnKF requiring repeated full-solver forecasts and a cheaper, less accurate learned EnKF.The solver EnKF uses oracle access to high-resolution Kraichnan dynamics, while the learned EnKF uses an FNO forecast.
  • Positioning: The approach combines a learned dynamical prior at each refinement level, a shared diffusion correction mechanism, and a hierarchical resolution cascade.These components distinguish it from prior diffusion-based data-assimilation approaches.

2. Problem Formulation

The paper formulates sequential super-resolution as data assimilation: reconstructing a temporally coherent fine-scale trajectory from coarse observations under unknown dynamics and an ill-posed observation operator. A multiresolution hierarchy decomposes this task into local conditional refinements that combine observational consistency, temporal dynamics, and generative correction.

  • 2. Problem Formulation: Fine-scale flow reconstruction from a single coarse observation is ill-conditioned because shocks, filaments, and vortices are only partially observed.Fine-scale structure also emerges through continuous fluid evolution along trajectories.
  • 2.1. Sequential Super-Resolution as Data Assimilation: Sequential super-resolution estimates a temporally coherent sequence of fine states consistent with coarse observations and underlying dynamics, rather than reconstructing states independently.This places the problem at the intersection of super-resolution and data assimilation.
  • 2.1. Sequential Super-Resolution as Data Assimilation: The inverse mapping from coarse observations to fine states is non-unique, so prior, dynamical, or statistical structure is required to regularize reconstruction.Temporal dependence across successive states provides additional information for filtering.
  • 2.2. Notation and Resolution Hierarchy: The state is represented across resolution levels r ∈ {0, 1, . . . , R}, with r = 0 coarsest and r = R the target finest resolution.Fixed downsampling and upsampling operators connect adjacent levels in the hierarchy.
  • 2.2. Notation and Resolution Hierarchy: The hierarchy decomposes direct coarse-to-fine recovery into sequential local refinement problems, each increasing resolution by a modest factor.This multiresolution viewpoint is central to the iterative refinement approach.
  • 2.3. Observation Model and Reconstruction Objective: The reconstruction objective is a finest-resolution trajectory that remains observationally consistent and dynamically coherent, corresponding ideally to a filtering distribution conditioned on observations through the current time.The formulation requires matching coarse observations while evolving plausibly in time.
  • 2.3. Observation Model and Reconstruction Objective: The multiresolution formulation factors a difficult full-scale filtering distribution into local conditional models estimating intermediate resolutions.This factorization avoids directly characterizing a distribution coupling the full range of scales.
  • 2.3. Observation Model and Reconstruction Objective: Each refinement conditions the finer state on the current coarser state and previous finer state, motivating learned forecast-analysis stages with dynamical priors and conditional generative corrections.This multilevel conditional structure is more tractable than one-shot mapping from a coarse observation to the finest state.

3. The Iterative Refinement Framework

The Iterative Refinement framework reconstructs high-resolution states through sequential forecast–analysis updates across time and resolution. A shared mode-sliced FNO supplies dynamical priors, while a shared resolution-conditioned diffusion corrector progressively assimilates coarse information.

  • Framework overview: IR applies a learned forecast–analysis update at each physical time step and resolution level instead of a single coarse-to-fine mapping.Updates propagate horizontally through time and vertically from coarse to fine within each time step.
  • Forecast–analysis update: Each refinement stage combines a prior forecast from the previous posterior with current coarse-level conditioning to produce the finer-resolution posterior.This mirrors classical data assimilation while learning both forecasting and correction components from data.
  • Shared models across scales: A shared FNO uses resolution-dependent spectral mode slicing, while a shared diffusion corrector adapts denoising through the resolution index.These mechanisms provide resolution awareness without independently trained networks at every scale.
  • Hierarchical propagation: Posterior outputs are propagated upward as conditioning for the next finer stage, distinguishing IR from cascades repeatedly conditioned only on the original coarse observation.Corrected information therefore moves progressively through the hierarchy within a single physical time step.
  • Temporal reconstruction: The resulting closed-loop cascade advances corrected states forward with the FNO and upward with diffusion analysis updates, assimilating new coarse observations while maintaining temporal coherence.At the initial time step, non-observed resolutions are initialized by spectrally upsampling the first coarse observation.
  • Inference sampling: At inference, posterior samples are generated by iteratively denoising an initial Gaussian field with a DDIM-style sampler whose stochasticity parameter η controls reverse-process noise.The paper reports sensitivity to η and the number of DDIM reverse steps in Appendix D.

4. Training Strategy

Training uses a two-stage pipeline: a shared mode-sliced FNO learns one-step dynamics across target resolutions, then its teacher-forced forecasts train a shared, resolution-conditioned diffusion corrector on aligned refinement tuples. One-step temporal alignment and stage-aware batching ensure the corrector learns the deployed forecast-error distribution across the multiresolution hierarchy.

  • Stage 2: Diffusion-corrector training: After FNO training, its weights are frozen and teacher-forced forecasts are generated from ground-truth states at every target resolution.These forecasts are paired with aligned coarse conditioning states and ground-truth fine-resolution targets for diffusion training.
  • Stage 2: Diffusion-corrector training: The diffusion corrector is trained jointly across all refinement stages as one resolution-conditioned model using forecast priors, lifted coarse inputs, diffusion timesteps, and target-resolution indices.Training tuples are aggregated across trajectories, time indices, and resolution transitions, while the corrector uses the standard DDPM epsilon-prediction objective.
  • Stage 1: Shared FNO training: The shared FNO is jointly trained across all non-observed target resolutions with a mean-squared one-step prediction loss.Each resolution selects a leading Fourier-mode slice from the same FNO backbone; the coarsest observed level is excluded from FNO training.
  • Temporal alignment: The diffusion dataset applies a one-step shift, pairing forecasts from time t with ground-truth targets at time t + 1 and aligned conditioning states.This alignment trains the corrector on the actual one-step prior and its characteristic forecast-error distribution rather than mismatched inputs.
  • Stage-aware batching: Resolution-aware batching groups examples by refinement stage so each minibatch has a consistent spatial shape while training cycles across stages in a balanced manner.This avoids padding or resizing and reduces domination by the most abundant or cheapest stage.

5. Experimental Setup

The framework is evaluated on 1D stochastically forced Burgers dynamics and more challenging 2D Kraichnan turbulence, using spectrally constructed resolution hierarchies. Experiments compare iterative refinement with baselines designed to isolate hierarchical correction, temporal conditioning, and learned dynamics.

  • Testbeds: The evaluation spans Burgers dynamics as a controlled proof-of-concept and Kraichnan turbulence as a substantially more challenging multiscale benchmark.Burgers contains shock-like structures and temporally evolving fine-scale ambiguity, whereas Kraichnan contains coherent vortices and thin filaments across broad scales.
  • Burgers setup: Coarse observations are insufficient to uniquely determine Burgers shock geometry, making temporal information essential for consistent fine-scale reconstruction.The hierarchy uses the coarsest level as the observed input and the finest level as the reconstruction target.
  • Burgers setup: The Burgers dataset contains 50 trajectories split into 40 training, 5 validation, and 5 test trajectories, with 800 saved snapshots per trajectory.
  • Kraichnan setup: Kraichnan turbulence uses a 256 × 256 pseudo-spectral simulation, with 32 × 32 observations and 256 × 256 target states in the resolution hierarchy.Coarse snapshots only partially reveal vortices, filaments, and merger debris, while previous high-resolution states constrain fine-scale organization.
  • Resolution hierarchy: The multiresolution hierarchy is formed by directly truncating unresolved high-frequency Fourier modes from the finest trajectory for each target level.This direct spectral restriction is used in both 1D and 2D rather than repeated chained downsampling.
  • Baselines: Baseline comparisons include spectral upsampling without temporal information or learned dynamics, alongside methods isolating hierarchical correction, temporal conditioning, and learned dynamics.Spectral upsampling zero-pads coarse Fourier coefficients and transforms them back to the target resolution, adding no inferred content beyond the observed bandwidth.

6. Results

Results show that one-shot diffusion is strongest on the constrained 1D Burgers reconstruction, whereas iterative refinement is superior on the more underdetermined, multiscale 2D Kraichnan benchmark. Spectral, local-structure, and temporal-stability diagnostics support the benefits of hierarchical forecast–analysis refinement and repeated diffusion correction.

  • Benchmark results: 2.776 × 10−3 RMSE makes one-shot diffusion the best method on 1D Burgers, while iterative refinement achieves 4.355 × 10−3 RMSE.Both diffusion-based methods substantially outperform deterministic baselines on the aligned 5-trajectory, 700-step benchmark.
  • Benchmark results: 0.18401 RMSE and 0.835870 SSIM make iterative refinement the best method on 2D Kraichnan turbulence.The benchmark uses 3 trajectories over 200 time steps at 256 × 256 resolution, and iterative refinement outperforms spectral upsampling, one-shot diffusion, EDSR, and the FNO-only forecaster.
  • Benchmark results: Iterative refinement gains the most in 2D turbulence by decomposing the ill-posed 32 × 32 →256 × 256 recovery into smaller resolution-wise correction steps.Coarse observations do not uniquely determine thin vorticity filaments, vortex interfaces, and merger debris.
  • Ablation and stability: The FNO-only autoregressive baseline becomes unstable and eventually blows up on 1D Burgers, showing that repeated observation-driven diffusion analysis is needed for stable long-horizon reconstruction.The diffusion corrector suppresses forecast drift and maintains consistency with the observed trajectory.
  • Qualitative comparisons: Iterative refinement most closely reproduces vortex-core shape and intensity in the local zoom, while EDSR is smoother and one-shot diffusion shows local distortions.The comparison indicates improved recovery of coherent small-scale structure beyond aggregate RMSE.
  • Spectral and temporal diagnostics: Iterative refinement best matches the 2D ground-truth energy spectrum across all sampled times and maintains the lowest, most nearly constant RMSE profile throughout rollout.Its spectral advantage is strongest in the intermediate-to-high-wavenumber range, while EDSR is more stable than spectral upsampling but has a higher error floor.

7. Ablation Studies

The ablations show that deeper forecast–analysis cascades improve multiscale reconstruction, especially for underconstrained 2D turbulence, and that propagating corrected diffusion posteriors is essential for transferring useful information across stages. Replacing posterior propagation with forecasts or repeated coarse observations substantially degrades fine-resolution accuracy.

  • Cascade depth: The full 3-stage cascade outperforms shallower variants in aggregate RMSE, temporal stability, and spectral reconstruction on 1D Burgers.The 1-stage and 2-stage variants retain larger error floors, while the full cascade rapidly settles into a low-error regime.
  • Cascade depth: 0.1844 RMSE is achieved by the 3-stage cascade at 256 × 256, versus 0.2834 for 1-stage and 0.2641 for 2-stage Kraichnan refinement.At 128 × 128, the 3-stage cascade also outperforms the 2-stage alternative, 0.1859 versus 0.2830.
  • Cascade depth: The 3-stage cascade preserves the closest match to the ground-truth E(k) curve at intermediate and high wavenumbers beyond the 32 × 32 Nyquist limit.The 1-stage variant loses high-wavenumber energy most rapidly, while the 2-stage variant remains too dissipative.
  • Propagation choice: 0.0049 RMSE at N = 256 is obtained with posterior propagation, compared with 0.0271 for forecast propagation and 0.0502 for raw-observation propagation.At N = 512, the posterior variant remains near 0.0053, while forecast and raw-observation propagation remain around 0.0538 and 0.0536.
  • Propagation choice: 0.1859 RMSE at 128 × 128 is achieved with posterior propagation, versus 0.2390 for forecast propagation and 0.2831 for raw-observation propagation in Kraichnan turbulence.All variants perform similarly at the first refinement level because they condition on the same initial coarse observation; differences emerge at finer resolutions.

8. Summary Across Methods

Across benchmarks, one-shot diffusion is strongest on the constrained 1D Burgers problem, whereas iterative refinement leads on the more challenging, strongly multiscale 2D Kraichnan task. Ablations further show that propagating corrected posteriors is central to refinement performance.

  • In 1D Burgers, one-shot diffusion achieves the lowest RMSE and spectral error, with iterative refinement close behind.
  • Both stochastic methods substantially outperform deterministic EDSR and spectral upsampling on the corrected 1D Burgers benchmark.
  • In 2D Kraichnan turbulence, iterative refinement achieves the best RMSE, spectral RMSE, and SSIM among all methods while maintaining competitive temporal consistency.
  • On 2D Kraichnan, EDSR improves over spectral upsampling but remains less accurate than iterative refinement, while one-shot diffusion produces larger errors and variability.
  • Propagation-signal ablations identify the corrected posterior as the strongest signal, yielding the lowest aggregate RMSE, most stable temporal behavior, and smallest spectral error.
  • On Kraichnan, posterior propagation remains most accurate at finer resolutions and reaches the lowest final RMSE despite similar radial energy spectra to the forecast variant.

9. Discussion

The discussion frames iterative refinement as a learned multiscale filtering framework whose benefits are greatest for strongly multiscale, temporally coupled, and underdetermined inverse problems. It also identifies computational and modeling limitations, including error propagation, shared-corrector assumptions, sampler sensitivity, and teacher-forced training.

  • Comparative interpretation: Iterative refinement is not universally superior: one-shot diffusion performs best on corrected 1D Burgers, whereas hierarchical refinement is most advantageous for strongly multiscale, temporally coupled, and underdetermined problems.On 1D Burgers, one-shot diffusion achieves the lowest RMSE and spectral error, while iterative refinement remains competitive and outperforms deterministic EDSR and spectral upsampling.
  • Data assimilation interpretation: The shared FNO supplies a dynamical prior, while the diffusion corrector learns a stochastic analysis conditioned on the forecast and current coarser-resolution state.This replaces an explicit Kalman-style update with a data-learned analysis step.
  • Hierarchical mechanism: Intermediate corrected posteriors propagate information upward, progressively refining the estimate rather than independently synthesizing each resolution.Prior–posterior comparisons and propagation-signal ablations distinguish this mechanism from one-shot super-resolution.
  • Problem complexity: Iterative refinement helps 2D Kraichnan reconstruction by decomposing the 32 × 32 →256 × 256 recovery into smaller resolution-wise corrections.This improves spectral recovery, SSIM, and RMSE when thin filaments, vortex interfaces, and localized merger structures are not uniquely determined by coarse observations.
  • Limitations: The method is limited by upward error propagation, potentially suboptimal parameter sharing across scales, diffusion-sampler sensitivity, and residual train–test mismatch from teacher-forced priors.Later stages depend on generated posteriors, different transitions may have different statistics, sampler settings affect accuracy and cost, and inference-aware training could improve stability and long-horizon performance.
  • Efficiency and future directions: Iterative refinement costs more than spectral upsampling, deterministic EDSR, or one-shot diffusion, but this expense is most justified in difficult multiscale settings such as 2D turbulence.A promising extension is aggregate training with separately or partially specialized correctors for different resolution transitions.

10. Conclusion

Iterative refinement combines resolution-wise forecast–analysis updates, learned dynamical priors, and conditional diffusion correction for super-resolved data assimilation. Its benefits are strongest for challenging, multiscale inverse problems, while one-shot diffusion remains highly effective when coarse observations constrain fine-scale structure.

  • Method: Iterative refinement replaces one-shot coarse-to-fine reconstruction with sequential resolution-wise forecast–analysis updates across a multiresolution hierarchy.A shared Fourier Neural Operator supplies spectral-mode-sliced dynamical priors, while a shared conditional diffusion corrector produces analysis posteriors.
  • Benchmark findings: On 1D stochastic Burgers, one-shot diffusion achieves the lowest RMSE and spectral error, while iterative refinement remains close and outperforms EDSR and spectral upsampling.The result indicates that one-shot generative reconstruction can be sufficient when coarse observations strongly constrain missing fine-scale structure.
  • Benchmark findings: On challenging 32 × 32 →256 × 256 Kraichnan turbulence, iterative refinement achieves the best RMSE, spectral fidelity, and SSIM among learned super-resolution baselines.It also yields temporally stable reconstructions with improved recovery of vortex interfaces, thin filaments, and localized turbulent structures.
  • Ablation studies: Increasing cascade depth improves reconstruction, and propagating corrected diffusion posteriors between stages is essential for fine-scale recovery.Using uncorrected forecasts or repeatedly upsampled coarse observations substantially degrades performance.
  • Computational comparison: Compared with solver-based EnKF, iterative refinement avoids online full-solver forecasts, trading a potentially lower aggregate RMSE for learned super-resolved assimilation.EnKF requires repeatedly advancing an ensemble of full-resolution states when the high-resolution solver is available online.
  • Future work: Future work includes inference-aware training, spectral or physics-informed regularization, ensemble posterior sampling, broader physical settings, and aggregate or stage-specialized training.The proposed direction targets train–test mismatch, high-wavenumber fidelity, uncertainty quantification, and extensions to partially observed, noisy, three-dimensional, or more chaotic systems.

Declaration of Generative AI use · Appendix A. Additional Baseline Comparisons

The authors used generative AI only for minor language editing and retained responsibility for all technical content. Appendix A provides qualitative diagnostics supporting the paper’s baseline-comparison conclusions across Burgers and Kraichnan settings.

  • Declaration of Generative AI use: Claude Sonnet 5.0 was used solely to correct spelling and grammatical errors during manuscript preparation.
  • Declaration of Generative AI use: The tools did not generate scientific content, data, analyses, or interpretations.
  • Declaration of Generative AI use: The authors developed, reviewed, and approved the manuscript’s technical content, results, and conclusions.
  • Appendix A. Additional Baseline Comparisons: Appendix A adds qualitative diagnostics for the baseline comparisons discussed in Section 6.3.
  • Appendix A. Additional Baseline Comparisons: Autoregressive forecasting without analysis is unstable over long rollouts, while one-shot diffusion performs strongly on the simpler 1D Burgers benchmark.
  • Appendix A. Additional Baseline Comparisons: Iterative refinement provides the strongest reconstruction quality in the more underdetermined 2D Kraichnan setting.

Appendix A.1. Autoregressive FNO-Only Rollout Diagnostics · Appendix A.2. Additional 1D One-Shot Baseline Diagnostics · Appendix B. Additional Temporal Stability Diagnostics

The diagnostics show that FNO-only autoregressive rollouts become unstable, while diffusion posteriors preserve physically coherent high-resolution structure. On 1D Burgers, one-shot diffusion and iterative refinement closely recover shocks, whereas EDSR has larger localized errors.

  • Appendix A.1. Autoregressive FNO-Only Rollout Diagnostics: The FNO-only baseline tests whether the shared mode-sliced FNO dynamical prior alone can support stable high-resolution reconstruction.It removes the diffusion analysis step and uses autoregressive rollout.
  • Appendix A.1. Autoregressive FNO-Only Rollout Diagnostics: At t ≈54, the 1D Burgers FNO-only rollout blows up after initially tracking the large-scale shock structure.Afterward, the predicted field loses physical coherence and no longer represents a plausible Burgers trajectory.
  • Appendix A.1. Autoregressive FNO-Only Rollout Diagnostics: Over the full Burgers time horizon, the diffusion posterior remains well aligned with the ground-truth Hovmöller shock structure.This contrasts with the unstable autoregressive FNO-only trajectory.
  • Appendix A.1. Autoregressive FNO-Only Rollout Diagnostics: In 2D Kraichnan dynamics, FNO-only rollout accumulates error, produces spurious small-scale oscillations, and loses vortex–filament organization over time.The diffusion posterior preserves large-scale morphology and fine-scale coherence with substantially smaller residual error.
  • Appendix A.2. Additional 1D One-Shot Baseline Diagnostics: One-shot diffusion and iterative refinement both closely track the ground-truth shock structure on the 1D Burgers benchmark.Both methods reconstruct the finest-resolution state directly from the current coarse observation rather than propagating intermediate posteriors through a hierarchy.
  • Appendix A.2. Additional 1D One-Shot Baseline Diagnostics: EDSR improves over spectral upsampling but exhibits larger localized deviations near sharp gradients in 1D Burgers.The qualitative comparison is consistent with the quantitative ordering in Table 2, where one-shot diffusion slightly outperforms iterative refinement.
  • Appendix A.2. Additional 1D One-Shot Baseline Diagnostics: Across the full 1D Burgers rollout, one-shot diffusion and iterative refinement remain closest to the ground truth among the compared learned methods.The Hovmöller comparison includes EDSR, one-shot diffusion, iterative refinement, and spectral upsampling.

Appendix B.1. Temporal Stability on the 1D Burgers Benchmark

On the 1D Burgers benchmark, learned super-resolution methods remain temporally stable and outperform spectral upsampling, with one-shot diffusion generally strongest and iterative refinement a close second. Both methods track ground-truth temporal variation, whereas EDSR and spectral upsampling miss sharper changes or appear too smooth.

  • Temporal stability: All learned super-resolution methods remain bounded over the full rollout and substantially outperform spectral upsampling.The comparison uses RMSE over time at the finest resolution N = 512, averaged over test trajectories.
  • Temporal stability: One-shot diffusion attains the lowest RMSE across most of the rollout, while iterative refinement remains a close second.This result concerns the finest-resolution N = 512 temporal RMSE averaged over test trajectories.
  • Temporal variation: One-shot diffusion and iterative refinement closely track the ground-truth temporal variation, preserving the dominant temporal dynamics.The frame-to-frame displacement norm ∥u_t − u_{t−1}∥_2 is averaged over test trajectories.
  • Temporal variation: EDSR underestimates some sharper temporal changes, whereas spectral upsampling is systematically too smooth.These differences are observed in the frame-to-frame displacement norm comparison.

Appendix C. Forecast-Prior Diagnostics · Appendix D. Sensitivity to Diffusion Sampling Settings · Appendix D.1. Effect of Stochasticity

Forecast-prior diagnostics show that iterative refinement begins from a strong dynamical approximation and applies localized diffusion corrections that restore fine-scale structure. Sampling stochasticity has little effect on 1D Burgers but is important for robust performance on turbulent 2D Kraichnan dynamics.

  • Appendix C. Forecast-Prior Diagnostics: The forecast prior already approximates the next-step state strongly, while diffusion refines it through structured, localized corrections rather than reconstructing the full field.This forecast–analysis separation underlies the method’s accuracy and rollout stability.
  • Appendix C. Forecast-Prior Diagnostics: In 1D Burgers, the FNO prior captures the overall shock profile but deviates near steep gradients, where the diffusion posterior reduces discrepancies against ground truth.Corrections are especially concentrated around sharp transitions.
  • Appendix C. Forecast-Prior Diagnostics: In 2D Kraichnan, the FNO prior reproduces large-scale vortices and filaments, while the posterior adds small, spatially coherent corrections along filaments, shear layers, and vortex interfaces.The prior–posterior differences indicate localized, flow-aware analysis corrections.
  • Appendix D. Sensitivity to Diffusion Sampling Settings: The method’s inference quality is evaluated for sensitivity to DDIM stochasticity η and the number of reverse diffusion steps.These sampler hyperparameters determine practical reconstruction behavior during iterative diffusion sampling.
  • Appendix D.1. Effect of Stochasticity: In 1D Burgers, RMSE decreases slightly as η increases from 0 to 1, while temporal smoothness remains essentially unchanged.The weak sensitivity suggests a relatively concentrated conditional posterior in this simpler setting.
  • Appendix D.1. Effect of Stochasticity: In 2D Kraichnan, deterministic or nearly deterministic sampling severely degrades RMSE and temporal smoothness, whereas moderate stochasticity substantially improves performance.For η ≥0.5, both metrics stabilize and remain nearly flat.

Appendix D.2. Effect of DDIM Step Count · Appendix D.3. Implications · Appendix E. Hyperparameters and architecture details

DDIM step counts reveal a quality–cost trade-off with a clear plateau beyond moderate values, while Kraichnan performance depends more strongly on sampling stochasticity. The appendices also document the PDE, architecture, training, and parameter-count configurations used for the experiments.

  • Appendix D.2. Effect of DDIM Step Count: In 1D Burgers, RMSE improves sharply from 10 to 25 DDIM steps, while 50 and 100 steps add little gain and increase wall-clock time.The practical operating point lies near the onset of this plateau.
  • Appendix D.2. Effect of DDIM Step Count: The forecast prior captures Burgers’ global solution profile, while the diffusion posterior corrects local errors near shocks and steep gradients.The diagnostic characterizes the corrector as targeted refinement of the forecast prior.
  • Appendix D.2. Effect of DDIM Step Count: For 2D Kraichnan, the forecast captures large-scale flow organization, while localized posterior corrections concentrate around filaments and vortex interfaces.This indicates residual analysis rather than full re-synthesis.
  • Appendix D.2. Effect of DDIM Step Count: In 2D Kraichnan, mean RMSE changes only slightly from 25 to 200 DDIM steps, with substantially overlapping error bars and monotonically increasing inference cost.Additional denoising iterations therefore provide diminishing returns after a moderate step count.
  • Appendix D.3. Implications: Sampling stochasticity matters more than chain length: η is critical in 2D Kraichnan, whereas reverse-step count mainly controls the quality–cost trade-off and plateaus beyond moderate values.The method therefore does not require extremely long diffusion chains.
  • Appendix D.3. Implications: The 1D Burgers results vary only mildly with η, while 2D Kraichnan performs poorly under deterministic or weakly stochastic sampling and improves with moderate stochasticity (η ≥0.5).In the harder setting, moderate stochasticity yields lower RMSE and substantially smoother temporal behavior.
  • Appendix E. Hyperparameters and architecture details: Appendix E specifies PDE and dataset configurations, model architecture details, training hyperparameters, and learnable parameter counts for IR and competing baselines.The architecture uses a diffusion corrector G shared across resolution stages and an FNO forecaster trained per stage; experiments use one NVIDIA RTX 4090 (24 GB).
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