pith. sign in

arxiv: 2604.10328 · v1 · submitted 2026-04-11 · 💻 cs.LG · cs.AI

A Diffusion-Contrastive Graph Neural Network with Virtual Nodes for Wind Nowcasting in Unobserved Regions

Pith reviewed 2026-05-10 15:43 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords graph neural networkswind nowcastingvirtual nodesdiffusion contrastive learningunobserved regionsself-supervised learningweather predictionspatial interpolation
0
0 comments X

The pith

Virtual nodes in a diffusion-contrastive graph neural network infer wind conditions in unobserved regions from nearby stations alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a self-supervised graph neural network that inserts virtual nodes into unobserved locations to predict wind speed, gusts, and direction without any direct measurements there. The model combines diffusion to spread information across the station graph with contrastive learning to refine representations, training solely on data from existing weather stations. On high-temporal-resolution data from the Netherlands, this yields 30 to 46 percent lower mean absolute error than interpolation or regression baselines for short-term forecasts in the gaps. The result matters because weather stations cannot be placed everywhere, leaving many areas without reliable nowcasts that affect energy planning, farming, and hazard alerts.

Core claim

By adding virtual nodes at unobserved sites inside a graph neural network that performs diffusion and contrastive learning, the model learns to generate accurate wind nowcasts at those sites using only signals from observed stations, delivering more than 30 percent error reduction over conventional methods.

What carries the argument

Virtual nodes embedded in a diffusion-contrastive graph neural network, which propagate wind information from observed stations to fill spatial gaps via graph structure and self-supervised signals.

If this is right

  • Nowcasts become feasible in any region that has at least some nearby stations, without new sensor installations.
  • Error reductions above 30 percent directly improve short-term wind forecasts used for renewable energy output planning.
  • The self-supervised training removes the need for ground-truth labels at the virtual node locations.
  • The same graph structure supports joint prediction of multiple wind variables (speed, gusts, direction) in one pass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The virtual-node approach could be tested on other sparse spatial prediction tasks such as air-quality mapping or flood forecasting.
  • Adding even light physics-based constraints to the loss might further stabilize predictions in complex terrain, though the paper does not explore this.
  • Performance on datasets from different climates or station densities would show whether the 30-46 percent gain generalizes beyond the Netherlands.

Load-bearing premise

Virtual nodes placed in unobserved locations can accurately infer wind conditions solely through graph diffusion and contrastive signals from observed stations, without direct measurements or explicit atmospheric physics constraints.

What would settle it

Applying the trained model to a fresh collection of stations treated as fully unobserved and finding that the mean absolute error improvement over baselines falls below 20 percent would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2604.10328 by Jie Shi, Siamak Mehrkanoon.

Figure 1
Figure 1. Figure 1: Overview of meteorological station locations and the spatiotemporal coverage of training and testing datasets [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the model architecture and input graph generation process. Step A: Graph construction with real [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graph diffusion and virtual node dependence. (a) Diffusion process transforming the spatial graph structure and variable distribution across the Netherlands. (b) Influence distribution showing how virtual nodes aggregate information from multiple nearby real stations. 4.4. Evaluation To rigorously assess the performance of our proposed model, we conducted an evaluation based on predictions at eight specifi… view at source ↗
Figure 4
Figure 4. Figure 4: Model performance across metrics, lead times, and test stations. (a) MAE/ [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Seasonal MAE results. (a) Lead–time MAE for wind direction, wind speed, and wind gust. (b) Station-wise [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Case studies of short-horizon wind forecasts. (a) Wind-direction roses at horizon [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Embedding distance dynamics and projections. (a,b) Positive and negative pair-distance evolution for aug [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

Accurate weather nowcasting remains one of the central challenges in atmospheric science, with critical implications for climate resilience, energy security, and disaster preparedness. Since it is not feasible to deploy observation stations everywhere, some regions lack dense observational networks, resulting in unreliable short-term wind predictions across those unobserved areas. Here we present a deep graph self-supervised framework that extends nowcasting capability into such unobserved regions without requiring new sensors. Our approach introduces "virtual nodes" into a diffusion and contrastive-based graph neural network, enabling the model to learn wind condition (i.e., speed, direction and gusts) in places with no direct measurements. Using high-temporal resolution weather station data across the Netherlands, we demonstrate that this approach reduces nowcast mean absolute error (MAE) of wind speed, gusts, and direction in unobserved regions by more than 30% - 46% compared with interpolation and regression methods. By enabling localized nowcasts where no measurements exist, this method opens new pathways for renewable energy integration, agricultural planning, and early-warning systems in data-sparse regions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents a self-supervised diffusion-contrastive graph neural network that incorporates virtual nodes to enable wind nowcasting (speed, gusts, direction) in unobserved regions without new sensors. Using high-temporal-resolution weather station data from the Netherlands, it claims MAE reductions of 30-46% relative to standard interpolation and regression baselines.

Significance. If the quantitative gains hold under stronger controls, the virtual-node approach could meaningfully extend nowcasting into data-sparse areas, supporting applications in renewable energy and disaster preparedness. The self-supervised framing is a reasonable direction for graph-based spatio-temporal tasks.

major comments (2)
  1. [Abstract] Abstract: the headline result (30-46% MAE reduction in unobserved regions) is reported exclusively against interpolation and regression. The same held-out-station protocol must be run against competitive spatio-temporal baselines (e.g., other message-passing GNNs or temporal graph networks) to isolate the contribution of virtual nodes plus contrastive diffusion; without this, the central claim that the proposed architecture is the key advance remains under-supported.
  2. [Abstract] Abstract: no details are supplied on experimental setup, baseline implementations, statistical significance testing, spatial cross-validation strategy, or controls for temporal/spatial autocorrelation. These omissions make the reported error reductions difficult to interpret or reproduce.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'high-temporal resolution' is used without specifying sampling interval, number of stations, or exact time span covered by the Netherlands dataset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We agree that strengthening the experimental comparisons and providing clearer details on the setup will improve the manuscript. Below we respond point by point and commit to the indicated revisions.

read point-by-point responses
  1. Referee: [Abstract] Abstract: the headline result (30-46% MAE reduction in unobserved regions) is reported exclusively against interpolation and regression. The same held-out-station protocol must be run against competitive spatio-temporal baselines (e.g., other message-passing GNNs or temporal graph networks) to isolate the contribution of virtual nodes plus contrastive diffusion; without this, the central claim that the proposed architecture is the key advance remains under-supported.

    Authors: We agree that comparisons limited to interpolation and regression do not fully isolate the specific contributions of the virtual-node mechanism and the diffusion-contrastive objective. To address this, we will add experiments using additional competitive spatio-temporal baselines, including standard message-passing GNNs (such as GAT and GraphSAGE) and temporal graph networks, all evaluated under the identical held-out-station protocol on the Netherlands dataset. These results will be included in the revised manuscript to better support the architectural claims. revision: yes

  2. Referee: [Abstract] Abstract: no details are supplied on experimental setup, baseline implementations, statistical significance testing, spatial cross-validation strategy, or controls for temporal/spatial autocorrelation. These omissions make the reported error reductions difficult to interpret or reproduce.

    Authors: We acknowledge that the abstract itself contains no such details, which limits immediate interpretability. The full manuscript provides the experimental protocol, dataset description, and baseline implementations in the Experiments section, along with the held-out station evaluation. To improve clarity and reproducibility, we will expand the abstract with a concise summary of the setup, cross-validation approach, and significance testing. We will also add explicit text on autocorrelation controls (e.g., temporal blocking and spatial partitioning) in the revised Experiments section. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical model evaluation on held-out stations

full rationale

The paper proposes a GNN architecture with virtual nodes, diffusion, and contrastive learning for wind nowcasting, then reports MAE reductions on real weather-station data from the Netherlands. The central result is an empirical comparison against interpolation and regression baselines on unobserved locations; no derivation chain, uniqueness theorem, or fitted parameter is presented as a first-principles prediction that reduces to its own inputs by construction. The evaluation protocol (train on observed stations, test on held-out stations) is externally falsifiable and does not rely on self-referential definitions or self-citation load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim depends on the unstated assumption that graph connectivity and self-supervised signals suffice to represent unobserved wind fields accurately; no free parameters, axioms, or additional invented entities beyond virtual nodes are specified in the abstract.

invented entities (1)
  • virtual nodes no independent evidence
    purpose: Represent locations without sensors to enable learning of wind conditions via graph propagation
    Introduced to extend the model to unobserved regions; no independent evidence provided.

pith-pipeline@v0.9.0 · 5489 in / 1156 out tokens · 49791 ms · 2026-05-10T15:43:20.242907+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    Sweeney, R

    C. Sweeney, R. J. Bessa, J. Browell, P. Pinson, The future of forecasting for renewable energy, Wiley Interdisciplinary Reviews: Energy and Environment 9 (2019). URLhttps://api.semanticscholar.org/CorpusID:203136752

  2. [2]

    Meenal, D

    R. Meenal, D. Binu, K. C. Ramya, P. A. Michael, K. V . Kumar, E. Rajasekaran, B. Sangeetha, Weather forecasting for renewable energy system: A review, Archives of Computational Methods in Engineering 29 (2022) 2875 – 2891. URLhttps://api.semanticscholar.org/CorpusID:246340491

  3. [3]

    K. E. Ukhurebor, C. O. Adetunji, O. T. Olugbemi, W. Nwankwo, A. S. Olayinka, C. Umezu- ruike, D. I. Hefft, Precision agriculture: Weather forecasting for future farming, AI, Edge and IoT-based Smart Agriculture (2022). URLhttps://api.semanticscholar.org/CorpusID:244438612

  4. [4]

    G. Hu, S. L. Dance, A. Fowler, D. Simonin, J. Waller, T. Auligne, S. Healy, D. Hotta, U. Löhnert, T. Miyoshi, et al., On methods for assessment of the value of observations in convection-permitting data assimilation and numerical weather forecasting, Quarterly Journal of the Royal Meteorological Society 151 (768) (2025) e4933

  5. [5]

    G. F. L. R. Bernardes, R. Ishibashi, A. Ivo, V . Rosset, B. Y . L. Kimura, Prototyping low- cost automatic weather stations for natural disaster monitoring, Digit. Commun. Networks 9 (2021) 941–956. URLhttps://api.semanticscholar.org/CorpusID:231855238

  6. [6]

    Price, A

    I. Price, A. Sanchez-Gonzalez, F. Alet, T. R. Andersson, A. El-Kadi, D. Masters, T. Ewalds, J. Stott, S. Mohamed, P. Battaglia, et al., Probabilistic weather forecasting with machine learning, Nature 637 (8044) (2025) 84–90

  7. [7]

    Ham, J.-H

    Y .-G. Ham, J.-H. Kim, S.-K. Min, D. Kim, T. Li, A. Timmermann, M. F. Stuecker, Anthro- pogenic fingerprints in daily precipitation revealed by deep learning, Nature 622 (7982) (2023) 301–307

  8. [8]

    Trebing, T

    K. Trebing, T. Sta ´nczyk, S. Mehrkanoon, SmaAt-UNet: Precipitation nowcasting using a small attention-unet architecture, Pattern Recognition Letters 145 (2021) 178–186

  9. [9]

    Reulen, J

    E. Reulen, J. Shi, S. Mehrkanoon, GA-SmaAt-GNet: Generative adversarial small attention GNet for extreme precipitation nowcasting, Knowledge-Based Systems 305 (2024) 112612

  10. [10]

    Vatamány, S

    L. Vatamány, S. Mehrkanoon, Graph dual-stream convolutional attention fusion for precip- itation nowcasting, Engineering Applications of Artificial Intelligence 141 (2025) 109788. 21

  11. [11]

    I. A. Abdellaoui, S. Mehrkanoon, Symbolic regression for scientific discovery: an applica- tion to wind speed forecasting, in: 2021 IEEE symposium series on computational intelli- gence (SSCI), IEEE, 2021, pp. 01–08

  12. [12]

    Aykas, S

    D. Aykas, S. Mehrkanoon, Multistream graph attention networks for wind speed forecast- ing, in: 2021 IEEE Symposium Series on Computational Intelligence (SSCI), IEEE, 2021, pp. 1–8

  13. [13]

    Allen, S

    A. Allen, S. Markou, W. Tebbutt, J. Requeima, W. P. Bruinsma, T. R. Andersson, M. Her- zog, N. D. Lane, M. Chantry, J. S. Hosking, et al., End-to-end data-driven weather predic- tion, Nature 641 (8065) (2025) 1172–1179

  14. [14]

    Alzubaidi, J

    L. Alzubaidi, J. Bai, A. Al-Sabaawi, J. Santamaría, A. S. Albahri, B. S. N. Al-Dabbagh, M. A. Fadhel, M. Manoufali, J. Zhang, A. H. Al-Timemy, et al., A survey on deep learning tools dealing with data scarcity: definitions, challenges, solutions, tips, and applications, Journal of Big Data 10 (1) (2023) 46

  15. [15]

    Gasteiger, S

    J. Gasteiger, S. Weißenberger, S. Günnemann, Diffusion improves graph learning, Ad- vances in neural information processing systems 32 (2019)

  16. [16]

    T. Chen, S. Kornblith, M. Norouzi, G. Hinton, A simple framework for contrastive learning of visual representations, in: International conference on machine learning, PmLR, 2020, pp. 1597–1607

  17. [17]

    H. J. Miller, Tobler’s first law and spatial analysis, Annals of the association of American geographers 94 (2) (2004) 284–289

  18. [18]

    la Gasse, N

    L. la Gasse, N. Meinen, I. Wijnant, C. de Valk, H. van den Brink, A meteorological ap- proach for the derivation of design gust wind speeds for engineering structures in the nether- lands (2025)

  19. [19]

    Mehrkanoon, Deep shared representation learning for weather elements forecasting, Knowledge-Based Systems 179 (2019) 120–128

    S. Mehrkanoon, Deep shared representation learning for weather elements forecasting, Knowledge-Based Systems 179 (2019) 120–128

  20. [20]

    L. Chen, Y . Cao, L. Ma, J. Zhang, A deep learning-based methodology for precipitation nowcasting with radar, Earth and Space Science 7 (2) (2020) e2019EA000812

  21. [21]

    Ravuri, K

    S. Ravuri, K. Lenc, M. Willson, D. Kangin, R. Lam, P. Mirowski, M. Fitzsimons, M. Athanassiadou, S. Kashem, S. Madge, et al., Skilful precipitation nowcasting using deep generative models of radar, Nature 597 (7878) (2021) 672–677

  22. [22]

    J. G. Fernández, S. Mehrkanoon, Broad-unet: Multi-scale feature learning for nowcasting tasks, Neural Networks 144 (2021) 419–427

  23. [23]

    Z. Gao, X. Shi, H. Wang, D.-Y . Yeung, W.-c. Woo, W.-K. Wong, Deep learning and the weather forecasting problem: Precipitation nowcasting, Deep Learning for the Earth Sci- ences: a comprehensive approach to remote sensing, climate science, and geosciences (2021) 218–239

  24. [24]

    Y . Yang, S. Mehrkanoon, AA-Transunet: Attention augmented transunet for nowcasting tasks, in: 2022 international joint conference on neural networks (IJCNN), IEEE, 2022, pp. 01–08. 22

  25. [25]

    J. G. Fernández, I. A. Abdellaoui, S. Mehrkanoon, Deep coastal sea elements forecasting using unet-based models, Knowledge-Based Systems 252 (2022) 109445

  26. [26]

    A. B. Upadhyay, S. R. Shah, R. A. Thakkar, Theoretical assessment for weather nowcasting using deep learning methods, Archives of Computational Methods in Engineering 31 (7) (2024) 3891–3900

  27. [27]

    Trebing, S

    K. Trebing, S. Mehrkanoon, Wind speed prediction using multidimensional convolutional neural networks, in: 2020 IEEE symposium series on computational intelligence (SSCI), IEEE, 2020, pp. 713–720

  28. [28]

    Sta ´nczyk, S

    T. Sta ´nczyk, S. Mehrkanoon, Deep graph convolutional networks for wind speed prediction, in: ESANN 2021 Proceedings-29th European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning, i6doc. com publication, 2021, pp. 147– 152

  29. [29]

    Cressie, The origins of kriging, Mathematical Geology 22 (3) (1990) 239–252

    N. Cressie, The origins of kriging, Mathematical Geology 22 (3) (1990) 239–252

  30. [30]

    Shepard, A two-dimensional interpolation function for irregularly spaced data, Proceed- ings of the 1968 23rd ACM National Conference (1968) 517–524

    D. Shepard, A two-dimensional interpolation function for irregularly spaced data, Proceed- ings of the 1968 23rd ACM National Conference (1968) 517–524

  31. [31]

    D. S. Wilks, Statistical Methods in the Atmospheric Sciences, Academic Press, 2011

  32. [32]

    Q. Wu, H. Zheng, X. Guo, G. Liu, Promoting wind energy for sustainable development by precise wind speed prediction based on graph neural networks, Renewable Energy 199 (2022) 977–992

  33. [33]

    X. Yang, F. Zhang, P. Sun, X. Li, Z. Du, R. Liu, A spatio-temporal graph-guided convo- lutional lstm for tropical cyclones precipitation nowcasting, Applied Soft Computing 124 (2022) 109003

  34. [34]

    arXiv preprint arXiv:2410.05431 , year=

    M. Andrae, T. Landelius, J. Oskarsson, F. Lindsten, Continuous ensemble weather fore- casting with diffusion models, arXiv preprint arXiv:2410.05431 (2024)

  35. [35]

    H. Dong, A. Prabowo, H. Xue, F. D. Salim, Double-diffusion: Diffusion conditioned diffu- sion probabilistic model for air quality prediction, arXiv preprint arXiv:2506.23053 (2025)

  36. [36]

    J. Han, Q. Wang, L. Zhao, Multi-scale adaptive diffusion graph neural network for spa- tiotemporal forecasting, Information Fusion 104 (2024) 102057

  37. [37]

    Kumar, P

    P. Kumar, P. Rawat, S. Chauhan, Contrastive self-supervised learning: review, progress, challenges and future research directions, International Journal of Multimedia Information Retrieval 11 (4) (2022) 461–488

  38. [38]

    Huang, Y

    J. Huang, Y . Sun, X. Dong, H. Zhou, C. Xie, C. Zhu, Z. Ma, Cross-modal contrastive learning for net load prediction based on public weather forecasts, IEEE Transactions on Smart Grid (2025)

  39. [39]

    Y . Gong, T. He, M. Chen, B. Wang, L. Nie, Y . Yin, Spatio-temporal enhanced contrastive and contextual learning for weather forecasting, IEEE Transactions on Knowledge and Data Engineering 36 (8) (2024) 4260–4274. 23

  40. [40]

    L. Bai, L. Yao, C. Li, X. Wang, C. Wang, Adaptive graph convolutional recurrent net- work for traffic forecasting, Advances in neural information processing systems 33 (2020) 17804–17815

  41. [41]

    S. Luan, C. Hua, M. Xu, Q. Lu, J. Zhu, X.-W. Chang, J. Fu, J. Leskovec, D. Precup, When do graph neural networks help with node classification? investigating the homophily principle on node distinguishability, Advances in Neural Information Processing Systems 36 (2023) 28748–28760

  42. [42]

    L. Page, S. Brin, R. Motwani, T. Winograd, The pagerank citation ranking: Bringing order to the web., Tech. rep., Stanford infolab (1999)

  43. [43]

    Zhang, L

    Z. Zhang, L. Lin, S. Gao, J. Wang, H. Zhao, H. Yu, A machine learning model for hub- height short-term wind speed prediction, Nature Communications 16 (1) (2025) 3195

  44. [44]

    Z. Chen, J. Zhang, S. Zhou, Z. Zhao, Y . Liu, Ultra-short-term prediction of spatio-temporal wind speed based on a hybrid deep learning model, Frontiers in Earth Science 13 (2025) 1580945

  45. [45]

    K. He, H. Fan, Y . Wu, S. Xie, R. Girshick, Momentum contrast for unsupervised visual representation learning, in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp. 9729–9738

  46. [46]

    Chung, The heat kernel as the pagerank of a graph, Proceedings of the National Academy of Sciences 104 (50) (2007) 19735–19740

    F. Chung, The heat kernel as the pagerank of a graph, Proceedings of the National Academy of Sciences 104 (50) (2007) 19735–19740

  47. [47]

    Zhang, H

    S. Zhang, H. Tong, J. Xu, R. Maciejewski, Graph convolutional networks: a comprehensive review, Computational Social Networks 6 (1) (2019) 1–23

  48. [48]

    D. S. K. Karunasingha, Root mean square error or mean absolute error? use their ratio as well, Information Sciences 585 (2022) 609–629

  49. [49]

    E. J. Hannan, L. Kavalieris, Regression, autoregression models, Journal of Time Series Analysis 7 (1) (1986) 27–49

  50. [50]

    James, D

    G. James, D. Witten, T. Hastie, R. Tibshirani, J. Taylor, Linear regression, in: An Intro- duction to Statistical Learning: With Applications in Python, Springer, Cham, 2023, pp. 69–134

  51. [51]

    Zhang, Introduction to machine learning: k-nearest neighbors, Annals of translational medicine 4 (11) (2016) 218

    Z. Zhang, Introduction to machine learning: k-nearest neighbors, Annals of translational medicine 4 (11) (2016) 218

  52. [52]

    G. Y . Lu, D. W. Wong, An adaptive inverse-distance weighting spatial interpolation tech- nique, Computers & geosciences 34 (9) (2008) 1044–1055

  53. [53]

    Gryning, R

    S.-E. Gryning, R. Floors, Investigating predictability of offshore winds using a mesoscale model driven by forecast and reanalysis data, Meteorologische Zeitschrift 29 (2) (2020) 117–130

  54. [54]

    Baatsen, R

    M. Baatsen, R. J. Haarsma, A. J. Van Delden, H. De Vries, Severe autumn storms in future western europe with a warmer atlantic ocean, Climate Dynamics 45 (3) (2015) 949–964. 24

  55. [55]

    Wever, G

    N. Wever, G. Groen, Improving potential wind for extreme wind statistics, Tech. Rep. WR 2009-02, Royal Netherlands Meteorological Institute (KNMI), De Bilt, The Netherlands (Mar. 2009)

  56. [56]

    B. R. Cheneka, S. J. Watson, S. Basu, Quantifying the impacts of synoptic weather patterns on north sea wind power production and ramp events under a changing climate, Energy and Climate Change 4 (2023) 100113

  57. [57]

    T. T. Cai, R. Ma, Theoretical foundations of t-sne for visualizing high-dimensional clus- tered data, Journal of Machine Learning Research 23 (301) (2022) 1–54

  58. [58]

    Frontiers in Artificial Intelligence, 5:843038

    B. Ghojogh, A. Ghodsi, F. Karray, M. Crowley, Uniform manifold approximation and projection (umap) and its variants: tutorial and survey, arXiv preprint arXiv:2109.02508 (2021). 25