Supervising the Path to Fine Scales: GalerkinFlow for Scientific-Field and Image Super-Resolution
2026-08-17 • Computer Vision and Pattern Recognition
Computer Vision and Pattern RecognitionComputational Engineering, Finance, and ScienceMachine Learning
AI summaryⓘ
The authors introduce GalerkinFlow, a new method for improving super-resolution models by supervising not just the final high-resolution image but also intermediate steps in the reconstruction process. Instead of only focusing on the endpoint, their framework guides the model along a whole path from low to high resolution, helping it learn better details gradually. GalerkinFlow uses a technique that predicts changes at different stages and links these to the final target, improving accuracy without needing knowledge of underlying physical equations. They tested their method on fluid flow problems and an image dataset, showing it outperforms other equation-agnostic approaches.
super-resolutionreconstruction pathGalerkinFlowcoarse-to-fine residualNavier-StokesDarcy Flowfinite-differenceconvolutional featuresscale-conditioned mixing
Authors
Zikang Zhan
Abstract
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.