cs.LGOct 15, 2026

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

Authors: Arjun Dadhwal, Brian Summa

Organizations: Tulane University

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Abstract

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, W2,∞W_{2,\infty}, and W2,2W_{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.

Figures & tables

DescriptorRetainsDoes not explicitly retain
PDPaired birth/death values and feature persistenceFull component ancestry and branch hierarchy; spatial location of features
MTCritical values, component ancestry, and merge hierarchyOriginal spatial embedding of the complete field
Table 1: Complementary information retained by the two topological descriptors used in this study.
Figure 1: Conceptual relationship among scalar threshold sweeps, persistence diagrams, and merge trees. The scalar field produces critical-point events as the threshold changes. A persistence diagram keeps paired birth/death values and persistence, whereas a merge tree additionally preserves ancestry and merge order.
Figure 2: Overview of the data-generation, fine-tuning, and evaluation pipeline. Wind Toolkit fields are streamed through HSDS, converted to two-channel vector fields, and organized into PhIRE-compatible paired-resolution fields (MR →\rightarrow HR). Every fine-tuned model (the standard fine-tuning control, the final topology-inspired objective, and the other loss configurations of the loss study) shares the same pretrained CNN initialization, 2688-sample fine-tuning set, optimizer, and schedule, and differs only in its auxiliary loss terms. All models, including the pretrained CNN and GAN, are evaluated on the same 168-field benchmark: fidelity and domain measures on the full fields, and persistence-diagram and merge-tree measures on a fixed 160×160160\times 160 crop.
Figure 3: Fixed-pair supervision used in the final objective. For each ground-truth training field, TTK computes the persistence diagram once, offline, and stores the two endpoint vertices of each supervised persistence pair together with the ground-truth birth value bb , death value dd , and persistence pp . During training, the predicted speed s^\hat{s} is read at those fixed vertices; LCVL_{\mathrm{CV}} penalizes endpoint-value errors and LpersL_{\mathrm{pers}} penalizes the error in the predicted persistence p^=∣d^−b^∣\hat{p}=|\hat{d}-\hat{b}| . Gradients flow only through ordinary scalar-field values. Because no diagram of the prediction is computed or matched, this is a differentiable proxy for persistence agreement, not an optimization of a persistence-diagram distance.
FamilyMetricPreferCNN winsGAN wins
DirectPSNR- uvuv↑\uparrow1680
DirectSSIM-speed↑\uparrow1653
Wind powerWPD bias ∣⋅∣|\cdot|↓\downarrow11553
Wind powerWPD MAE↓\downarrow1680
Wind powerWPD RMSE↓\downarrow1671
Wind powerWPD W1↓\downarrow11949
Table 2: Per-field win counts for the pretrained CNN and GAN on the 168-field benchmark. Higher is better for PSNR and SSIM; lower is better for all listed errors and distances. No ties occurred in these comparisons.
 Mean distanceReduction in mean distanceFinal lower than
DistanceCNNControlFinalControl vs. CNNFinal vs. CNNFinal vs. controlCNNControl
Bottleneck dBd_{B}3.1243.2882.150−- 5.2%++ 31.2%++ 34.6%151/168156/168
W2,∞W_{2,\infty}19.1521.0614.33−- 10.0%++ 25.2%++ 32.0%167/168168/168
W2,2W_{2,2}24.5527.5117.83−- 12.0%++ 27.4%++ 35.2%166/168168/168
Merge tree5.8686.0125.657−- 2.5%++ 3.6%++ 5.9%98/168124/168
All three PD lower      150/168156/168
Table 3: Final objective (G+FP) versus the pretrained CNN and the standard fine-tuning control on the 168-field benchmark. Means are shown; percentages are relative reductions in mean distance (positive: lower, i.e., better). “Control vs. CNN” shows the change from fine-tuning with the reconstruction loss alone; “Final vs. control” the change from adding the auxiliary losses under otherwise identical fine-tuning. The last two columns count the fields on which the final model has the lower distance.
Figure 4: Loss study. Each cell gives the percent reduction in a mean distance relative to the standard fine-tuning control (positive and blue: lower distance; red: higher). Configurations are named by the losses added to LuvL_{uv} : speed (S), gradient (G), soft level-set (L), local maximum (M), and fixed-pair supervision (FP). Full values are in Appendix A .
Figure 5: Mean PD distance versus mean merge-tree distance for all topology-bearing methods (lower-left is better), for each of the three PD distances. Outlined markers lie on the Pareto front of that pair of distances (dashed). Labels mark the pretrained CNN, the standard fine-tuning control, and the configurations with gradient (G), local-maximum (M), fixed-pair (FP), and gradient plus fixed-pair (G+FP, final) supervision. The pretrained GAN lies outside the plotted range; its coordinates are given in each panel.
FamilyMetricPreferCNNControlFinalControl vs. CNNFinal vs. CNNFinal vs. control
DirectPSNR- uvuv↑\uparrow31.192533.789232.4949+2.60+2.60 dB+1.30+1.30 dB−1.29-1.29 dB
DirectSSIM-speed↑\uparrow0.74120.81340.7932+9.8%+9.8\%+7.0%+7.0\%−2.5%-2.5\%
DirectSpeed MAE↓\downarrow0.69410.49580.5899+28.6%+28.6\%+15.0%+15.0\%−19.0%-19.0\%
DirectSpeed RMSE↓\downarrow1.10780.85550.9891+22.8%+22.8\%+10.7%+10.7\%−15.6%-15.6\%
Wind powerWPD bias ∣⋅∣|\cdot|↓\downarrow35.343931.303322.6377+11.4%+11.4\%+36.0%+36.0\%+27.7%+27.7\%
Wind powerWPD MAE↓\downarrow231.6709158.9769190.6897+31.4%+31.4\%+17.7%+17.7\%−19.9%-19.9\%
Table 4: Non-topology metrics on the 168-field benchmark. Higher is better for PSNR and SSIM; lower is better otherwise. The last three columns give relative changes (positive: better), with PSNR differences in dB: the standard fine-tuning control relative to the CNN, the final model relative to the CNN, and the final model relative to the control.
Figure 6: Qualitative comparison for Sample 78, a field on which the final model and the pretrained CNN are nearly tied in conventional error. (a–d) Wind speed on the 160×160160\times 160 topology crop for the ground truth, pretrained CNN, standard fine-tuning control, and topology-inspired final model (G+FP), on a shared color scale; RMSE is computed on the crop. (e–h) Persistence diagrams (finite D0D_{0} and D1D_{1} pairs combined): the ground-truth pairs (open circles) alone and overlaid with each model’s pairs (filled). Only pairs with persistence of at least 3 m/s are drawn, but the distances below each panel ( W2,2W_{2,2} , W2,∞W_{2,\infty} , dBd_{B} ) use the complete finite diagrams; percentages are changes relative to the pretrained CNN. (i) Number of pairs whose persistence is at least tt .
Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

 Mean distance% reduction vs. control 
MethoddBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MTdBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MTRanks
Pretrained CNN3.12419.1524.555.868++ 5.0++ 9.1++ 10.7++ 2.413/13/13/7
Pretrained GAN5.05914.6817.468.348−- 53.9++ 30.3++ 36.5−- 38.918/4/1/18
Standard fine-tuning control3.28821.0627.516.012––––17/18/18/12
+ Speed3.26221.0127.436.000++ 0.8++ 0.2++ 0.3++ 0.215/16/16/10
+ Level-set3.27021.0327.466.008++ 0.5++ 0.2++ 0.2++ 0.116/17/17/11
Appendix
Table 5: All topology-bearing methods on the 168-field benchmark (means; lower is better). Configurations are named by the losses added to the reconstruction loss LuvL_{uv} ; loss weights are fixed across configurations (Sec. 3.4 ). The middle columns give the percent reduction in mean distance relative to the standard fine-tuning control (blue: lower distance; red: higher). Best value per column in bold, second best underlined. Ranks are over all 18 methods for dBd_{B} / W2,∞W_{2,\infty} / W2,2W_{2,2} /MT.
LossUnweighted meanCoefficientWeightedRelative to LuvL_{uv}
LuvL_{uv}0.03175910.031759100%
LspeedL_{\mathrm{speed}}1.2705210.010.01270540%
LgradL_{\mathrm{grad}}0.5752920.050.02876591%
LlevelsetL_{\mathrm{levelset}}0.0343490.250.00858727%
LmaxL_{\mathrm{max}}6.0646710.0010.00606519%
Appendix
Table 6: Loss-scale calibration at the pretrained CNN checkpoint before fine-tuning. Means are computed over all 168 fields of the fixed benchmark in 42 batches of four, evaluating the losses without updating model weights. Shown are the mean unweighted loss, its coefficient, and the weighted contribution relative to LuvL_{uv} . Coefficients compensate for the very different numerical scales of the unweighted losses.
EffectdBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MT
(a) Speed ×\times gradient ×\times level-set: all eight combinations
Speed−- 0.024−- 0.212−- 0.283−- 0.030
Gradient++ 0.613++ 5.322++ 7.993−- 0.186
Level-set−- 0.004−- 0.098−- 0.119++ 0.011
Speed ×\times gradient−- 0.078−- 0.343−- 0.480−- 0.068
Speed ×\times level-set++ 0.032++ 0.121++ 0.170++ 0.081
Appendix
Table 7: Factorial effects on the four topology distances (positive: lowers the distance on average). (a) All eight combinations of the speed, gradient, and level-set losses. (b) The local-maximum loss crossed with fixed-pair supervision, both added to speed + gradient + level-set. (c) The level-set loss crossed with fixed-pair supervision, both added to gradient supervision.
MethoddBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MT
Pretrained CNN2.91318.81624.0575.712
Pretrained GAN4.19613.51815.8357.587
Standard fine-tuning control3.18821.29027.8415.972
+ Speed3.18921.25927.8665.991
+ Level-set3.19221.28027.8725.970
+ Speed + Level-set3.14121.10127.5825.913
Appendix
Table 8: Median distances across the 168 fields (lower is better). The main conclusions agree with the means in Table 5 , although some close rankings change (for example, G+FP has a lower median W2,∞W_{2,\infty} than G+L+FP): G+FP has the lowest median dBd_{B} , FP alone the lowest median MT, and every fixed-pair configuration improves on the control and the CNN in all four medians.
 vs. pretrained CNNvs. control
MethoddBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MT3PD+MTdBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MT3PD+MT
Pretrained CNN––––––1021461511049367
Pretrained GAN3915515820391643167168214316
Standard fine-tuning control62221764117––––––
+ Speed65201761116102129133958049
+ Level-set62221864115103120127896738
Appendix
Table 9: Number of the 168 fields on which each method has a lower distance than the pretrained CNN (left) and than the standard fine-tuning control (right). “3PD”: all three PD distances lower; “+MT”: all three PD distances and MT lower.
ChangedBd_{B}W2,∞W_{2,\infty}W2,2W_{2,2}MT
Speed, added to control++ 0.80%++ 0.25%++ 0.29%++ 0.20%
Level-set, added to control++ 0.54%++ 0.16%++ 0.19%++ 0.07%
Gradient, added to control++ 22.63%++ 28.10%++ 32.05%−- 0.73%
Local-max, added to control++ 14.31%++ 4.25%++ 4.59%++ 5.36%
Fixed-pair, added to control++ 31.97%++ 27.99%++ 30.57%++ 6.95%
Local-max, added to S+G+L++ 7.55%++ 0.88%++ 0.80%++ 1.31%
Appendix
Table 10: Percent reduction in mean distance from adding or removing losses (positive: the change lowers the distance). S: speed, G: gradient, L: level-set, M: local maximum, FP: fixed-pair supervision. These are descriptive contrasts between trained models, not causal effect estimates.
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