Topology-Inspired Fine-Tuning for Wind-Field Super-Resolution

Scientific super-resolution (SR) models are typically optimized with pointwise reconstruction losses, yet scientific interpretation often hinges on extrema, connected regions, and multiscale scalar structure. We characterize this tension with pretrained CNN and GAN wind-field SR models. Pointwise metrics, bottleneck persistence-diagram (PD) distance, and merge-tree (MT) distance favor the CNN, whereas order-2 Wasserstein PD distances favor the sharper GAN, so even different distances on the same topological descriptor reward different properties. Motivated by this disagreement, we fine-tune the pretrained CNN with differentiable topology-inspired auxiliary losses on scalar speed, gradient magnitude, soft high-speed level sets, and local maxima, and with fixed-pair supervision, which constrains predicted values and persistence at ground-truth persistence-pair endpoints located offline with the Topology ToolKit. A loss study of 16 fine-tuning configurations, each compared with a matched standard fine-tuning control that uses only the original reconstruction loss, on a fixed 168-field benchmark (with topology evaluated on a fixed 160 by 160 subregion of each field), shows that, among the scalar-field losses, gradient supervision provides the strongest PD improvement but does not improve mean MT distance, whereas fixed-pair supervision gives the largest and most consistent joint improvement in PD and MT distances. The final objective combines these two signals. Relative to the pretrained CNN, it lowers mean bottleneck, $W_{2,\infty}$, and $W_{2,2}$ distances by 31%, 25%, and 27% (and by 32-35% relative to the control, which on its own worsens all three), lowers all three on 150 of 168 fields, and gains 1.30 dB PSNR, about half the gain of standard fine-tuning. Mean MT distance also improves, but on only 91 of those 150 fields, indicating that PD and MT capture complementary structure.