监督精细尺度的路径：用于科学场与图像超分辨率的GalerkinFlow

中文摘要 AI 辅助

大多数超分辨率模型仅通过监督最终的高分辨率输出从配对数据中学习，这几乎无法控制预测结果在降采样观测值与其精细目标之间的演化方式。我们提出了GalerkinFlow，这是一种与方程无关的框架，它将每一对粗-细样本转化为沿整个重建路径的监督信号。在重建路径上的随机中间状态样本中，模型会预测粗到细的残差速度，并使用粗锚点定义伪端点。我们证明，该伪端点的重建损失通过已知的时变权重与中间速度损失直接相关，因此每个中间状态都能为指向同一精细目标的监督做出贡献，而非仅作为通向端点损失的内部步骤。由于中间状态已揭示了部分缺失的精细尺度结构，我们还对单步推理中使用的粗端点进行监督。有限差分目标进一步约束了局部空间变化。GalerkinFlow结合了卷积特征与尺度条件Galerkin算子混合，无需控制方程或物理元数据。在Navier-Stokes和Darcy Flow上，它在评估的与方程无关的基线中实现了最低的原始空间误差，同时在DIV2K上仍具有竞争力。

英文摘要

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.

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8 pages, 2 figures