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Supervising the Path to Fine Scales: GalerkinFlow for Scientific-Field and Image Super-Resolution

Zikang Zhan

arXiv:2608.16546v1cs.CVcs.CEcs.LG

TL;DR

Super-resolution commonly supervises only the final fine output, leaving the reconstruction path between coarse and fine states underconstrained. GalerkinFlow supervises residuals along a deterministic path while anchoring coarse-endpoint inference, achieving the strongest reported distortion-oriented results across scientific fields and competitive image fidelity.

  • Problem

    Most super-resolution methods impose only a single coarse-to-fine endpoint constraint, leaving how partially restored states should evolve relatively underexplored.

  • Method

    GalerkinFlow samples deterministic intermediate states, trains their endpoint-directed residuals, and combines path supervision with coarse-anchor reconstruction and finite-difference consistency.

  • Results

    GalerkinFlow achieves the lowest Rel-L2, MSE, and MAE among evaluated equation-agnostic baselines on Navier–Stokes and Darcy Flow at 2× and 4×, while attaining high image fidelity.

  • Takeaways & Limitations

    Paired coarse–fine data can provide supervision along a reconstruction path without governing equations, physical coefficients, or solver metadata.

  • Takeaways & Limitations

    The current benchmarks do not demonstrate arbitrary-scale reconstruction with one shared model, and accuracy decreases when a single checkpoint is queried across uniform target scales.

Abstract

from arXiv · show

Most super-resolution models learn from paired data by supervising only the final high-resolution output. This provides little control over how the prediction should evolve between the downsampled observation and its fine target. We introduce GalerkinFlow, an equation-agnostic framework that turns each coarse--fine pair into supervision along an entire reconstruction path. At a random sample of intermediate states on the reconstruction path, the model predicts the coarse-to-fine residual velocity and uses coarse-anchor point to define a pseudo-endpoint. We show that the reconstruction loss of this pseudo-endpoint is exactly related to the intermediate velocity loss through a known time-dependent weight. Consequently, every intermediate state contributes supervision toward the same fine target, rather than serving only as an internal step toward an endpoint loss. Because intermediate states already reveal part of the missing fine-scale structure, we additionally supervise the coarse endpoint used during one-step inference. A finite-difference objective further constrains local spatial variation. GalerkinFlow combines convolutional features with scale-conditioned Galerkin operator mixing and requires no governing equation or physical metadata. It achieves the lowest raw-space errors among the evaluated equation-agnostic baselines on Navier--Stokes and Darcy Flow, while remaining competitive on DIV2K.

Introduction

GalerkinFlow reframes super-resolution as learning a progress-conditioned residual field along the continuum between a coarse anchor and fine target, rather than enforcing only a terminal constraint. It combines pathwise and coarse-endpoint supervision without governing equations, and improves equation-agnostic scientific baselines while remaining competitive on DIV2K.

  • Equation-agnostic setting: The method targets equation-agnostic reconstruction using only a downsampled field, paired fine target, and requested scale, without governing equations or physical metadata.Training excludes differential operators, coefficients, forcing terms, boundary conditions, temporal derivatives, and PDE residuals.
  • Motivation: Most deterministic super-resolution methods learn a single maximum corrective mapping from the coarse anchor to the fine target, leaving progressive refinement behavior underexplored.This endpoint-only formulation can fit the inference state while failing to constrain how partially restored fields should evolve.
  • Pathwise supervision: GalerkinFlow treats each coarse-fine pair as a deterministic path whose intermediate states provide exact residual targets to the fine endpoint.The model samples partially restored states and learns a progress-conditioned residual field rather than supervision only at one endpoint.
  • Training objectives: GalerkinFlow adds explicit t = 0 reconstruction and finite-difference losses to prevent shortcutting from leaked fine information and to constrain spatial variation.Intermediate states can reveal partial fine displacement unavailable during one-step inference from the coarse endpoint.
  • Empirical results: On Navier-Stokes and Darcy Flow, GalerkinFlow improves upon all evaluated equation-agnostic baselines across reported raw-space errors at 2× and 4× scales, while remaining strong on DIV2K.The evaluated scientific baselines include U-Net, FNO, U-NO, and SRNO; DIV2K results show strong PSNR and SSIM but mixed LPIPS.

Related Work

Prior super-resolution work spans direct coarse-to-fine predictors, equation-agnostic scientific-field reconstruction, neural operators, and trajectory-based generative methods. GalerkinFlow is distinguished by deterministic supervision from the observed coarse field, without governing equations, teacher models, or distribution-matching losses.

  • Direct Super-Resolution: Direct super-resolution methods learn coarse-to-fine maps using encoder–decoders, residual dense blocks, coordinate-query latent codes, or shifted-window transformers.Examples include U-Net, RDN, LIIF, and SwinIR.
  • Scientific-Field Super-Resolution: Equation-agnostic scientific-field methods reconstruct turbulent fields without embedding a governing equation in the network or loss, but do not specify behavior on partially restored fields.This limitation motivates supervision along intermediate reconstruction states.
  • Neural Operators: Neural operators learn function-space mappings across compatible discretizations, with FNO using Fourier kernels, U-NO using multiresolution structure, and SRNO combining image encoding with Galerkin-type attention.These approaches provide operator-based alternatives for scientific and image super-resolution.
  • Flow-Based Super-Resolution: Flow matching and related image methods supervise sampled trajectories or shared endpoints, including FlowSR, CTMSR, and RFMSR.These methods combine trajectory constraints with consistency, endpoint, distribution, or single-step objectives.
  • GalerkinFlow: GalerkinFlow is deterministic: it starts from the observed coarse field, uses paired endpoint residuals as vector-field targets, and avoids teacher models and distribution-matching losses.Its main experiments use one Euler step from a lifted coarse anchor.
  • Derivative-Aware Supervision: GalerkinFlow also compares adjacent output differences with paired fine-target differences, using observed samples to encourage shared local variation across image edges and scientific fields.This is a finite-difference analogue of derivative-aware Sobolev training.

Method

GalerkinFlow supervises reconstruction along a deterministic coarse-to-fine path rather than only at the final endpoint, using an equation-agnostic architecture that combines convolutional features with Galerkin operator mixing. It additionally aligns one-step endpoint inference and neighboring spatial variations with the paired fine target.

  • Equation-agnostic formulation: GalerkinFlow requires no differential operator, physical coefficient, forcing, boundary condition, or temporal derivative as input or loss supervision.
  • Architecture: The model combines RDN convolutional features with scale- and time-conditioned Galerkin operator mixing over target-grid locations.Inputs include the current state, positional encodings, time embedding, encoded coarse-anchor features, and scale conditions.
  • Path supervision: Training samples t uniformly from [0, 1) on the straight path xt = (1 − t)x0 + ty and supervises the residual vector field from each intermediate state to the fine endpoint.The same observed pair is reused at many partially restored states rather than creating independent labels.
  • Endpoint supervision: The residual-field objective controls the one-step endpoint predicted from every sampled state, while training also runs the endpoint solver to align inference with the paired target.This addresses the mismatch between local residual supervision during training and coarse-anchor-only endpoint prediction at test time.
  • Spatial-variation supervision: A finite-difference endpoint penalty supplements pointwise reconstruction by penalizing disagreement between neighboring horizontal and vertical changes.The term is computed on the reconstructed endpoint and is not a PDE residual.
  • Model interpretation: Because source, target, and intermediate states are paired and deterministic, GalerkinFlow is a reconstruction model rather than a calibrated conditional generator, and its path is not a PDE trajectory.

Experimental Setup

Experiments evaluate fixed-scale 2× and 4× super-resolution on controlled PDE fields and RGB image benchmarks, using separate scale-specific learned checkpoints and standardized preprocessing. Performance is assessed with raw-space scalar-field errors and established image-quality metrics.

  • PDE benchmark: PDE experiments use fixed-scale specialists at 2× and 4×, separating reconstruction accuracy from unseen-scale generalization.All experiments run on Nvidia GH200 GPUs.
  • PDE benchmark: Navier–Stokes uses 64 × 64 frames, with 32×32 and 16×16 coarse inputs at 2× and 4×, respectively, and evaluation on 1,000 held-out frames.Training draws frames from the first 4,000 trajectories, while coarse fields are bicubically lifted to the target grid.
  • PDE benchmark: Darcy Flow uses nonsquare 64×128 target fields from PDEBench, evaluating all 1,000 test samples with 32 × 64 or 16 × 32 coarse inputs at 2× or 4×.The provided train and test splits are used.
  • Baselines: PDE baselines are bicubic interpolation, U-Net, FNO, U-NO, and SRNO, while RGB baselines are bicubic interpolation, LIIF, SRNO, RDN, and SwinIR.Each learned PDE method has a separately trained checkpoint for the same dataset and scale; RGB checkpoints are trained on DIV2K.
  • Evaluation: Scalar fields use relative L2 as the primary metric alongside MSE and MAE, while RGB images use PSNR, SSIM, and LPIPS with higher PSNR and SSIM but lower LPIPS preferred.Metrics are accumulated after mapping predictions back from training normalization; image evaluation uses one 128 × 128 RGB crop per image and bicubic inputs at 2× or 4×.

Results and Analysis

GalerkinFlow achieves the strongest PDE results across datasets, scales, and raw-space error metrics, while remaining highly competitive on DIV2K and external image benchmarks. Its image results are strongest on distortion metrics, with more mixed perceptual performance against image-specialized baselines.

  • PDE super-resolution: GalerkinFlow records the lowest raw relative L2, MSE, and MAE on every PDE dataset and scale.Relative-L2 reductions versus SRNO are 94.6% on Navier–Stokes 2×, 98.3% on Navier–Stokes 4×, 78.3% on Darcy Flow 2×, and 83.9% on Darcy Flow 4×.
  • PDE super-resolution: GalerkinFlow maintains very small PDE errors at both 2× and 4×, whereas baseline behavior varies substantially across scales.Bicubic and SRNO degrade from 2× to 4×; FNO improves over bicubic but trails SRNO, U-NO is unstable, and U-Net remains far from GalerkinFlow.
  • DIV2K: 4.03 dB and 5.34 dB are GalerkinFlow’s PSNR improvements over the best non-GalerkinFlow method on DIV2K at 2× and 4×, respectively.GalerkinFlow also obtains the highest PSNR and SSIM at both scales; it has the lowest LPIPS at 4×, while SwinIR is better at 2×.
  • External image transfer: 0.27 dB to 1.94 dB is the range of GalerkinFlow’s PSNR gains over the best non-GalerkinFlow baseline across external image datasets and scales.GalerkinFlow has the highest PSNR in every dataset-scale setting, while SSIM is strongest in most settings and LPIPS is less uniformly favorable.
  • External image transfer: Image-specialized baselines, especially SwinIR, remain stronger on several external validation cases despite GalerkinFlow’s strong overall transfer results.GalerkinFlow is best on Set5 4×, but SSIM has exceptions on Set14 4× and Urban100 2×, and LPIPS is not uniformly favorable.

Ablation Study

The ablation study finds that endpoint reconstruction and gradient consistency progressively improve Navier–Stokes reconstruction over vector-field supervision alone. Results also distinguish strong PDE raw-space gains from less consistent RGB perceptual-feature alignment, while all reported evaluations use one-step reconstruction.

  • Loss ablation: Adding endpoint reconstruction reduces Rel-L2 by 51.8%, MSE by 78.7%, and MAE by 52.1% relative to vector-field supervision alone.This supports anchoring training to the coarse state encountered by the ODE solver at inference.
  • Loss ablation: Adding gradient consistency further reduces Rel-L2 by 68.5%, MSE by 88.8%, and MAE by 60.4% over the no-gradient variant.The passage identifies this as the full objective configuration, though its truncated text does not report the resulting absolute errors.
  • Evaluation protocol: All tables report one-step reconstruction from the lifted coarse anchor, not additional inference iterations through intermediate path states.More Euler intervals can evaluate the same learned field, but they change the compute budget and create an accuracy–cost tradeoff.
  • Domain-specific findings: On PDE fields, the CNN–Galerkin residual-field model with the coarse-anchored objective achieves large gains across every error metric in the scalar raw space used for training.The PDE evaluation uses the same raw-space representation used to train the checkpoints.
  • Domain-specific findings: On RGB images, the design provides strong PSNR and SSIM, while LPIPS shows less consistently improved perceptual-feature alignment than pointwise fidelity.This split motivates future work on lighter perceptual or backbone designs, as indicated by the passage’s truncated conclusion.

Conclusion

GalerkinFlow extracts pathwise supervision from paired coarse–fine samples by training residual fields at intermediate states, augmenting this with coarse-anchor endpoint reconstruction and finite-difference constraints. It shows strong scientific super-resolution performance while remaining limited by scale-specific evaluation and incomplete validation of arbitrary-scale and loss-weight generality.

  • Method: GalerkinFlow treats each coarse–fine pair as a deterministic restoration path and trains scale-conditioned residual fields toward the remaining displacement to the fine endpoint.The objective is supplemented by ODE endpoint reconstruction from the inference-time coarse anchor and finite-difference constraints.
  • Results: GalerkinFlow achieves the lowest Rel-L2, MSE, and MAE among evaluated equation-agnostic baselines on Navier–Stokes and Darcy Flow at 2× and 4×.On Navier–Stokes at 2×, endpoint reconstruction improves over vector-field supervision alone, while gradient consistency further improves all three errors.
  • Supervision principle: The supervision principle converts one paired endpoint into a continuum of endpoint-directed residual targets along a task-relevant path.Explicit coarse-anchor loss constrains behavior before the endpoint and preserves alignment with one-step inference.
  • Limitations: Current benchmarks do not establish arbitrary-scale reconstruction with one shared model because they use dataset-specific and mostly scale-specific checkpoints.One checkpoint can query arbitrary uniform target scales, but accuracy decreases.
  • Limitations: The loss ablation covers only Navier–Stokes at 2×, so it does not establish universal loss weights.The constructed path is also not a physical trajectory.
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