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arxiv: 2505.12167 · v2 · submitted 2025-05-17 · 💻 cs.LG · cs.CR

FABLE: A Localized, Targeted Adversarial Attack on Weather Forecasting Models

Pith reviewed 2026-05-22 14:01 UTC · model grok-4.3

classification 💻 cs.LG cs.CR
keywords adversarial attackweather forecastingdeep learningwavelet decompositiontargeted attackmodel robustnessspatio-temporal data
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The pith

FABLE uses wavelet decomposition to create localized adversarial inputs that steer deep learning weather forecasts to targeted outcomes.

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

The paper sets out to demonstrate that deep learning weather forecasting models can be manipulated toward specific forecast results through carefully controlled adversarial changes to the input data. It introduces FABLE, which applies 3D discrete wavelet decomposition to break apart spatial and temporal elements and then limits perturbation sizes differently in each part. A reader would care because these models now outperform traditional physics simulations for weather prediction, so hidden ways to shift their outputs could affect trust in forecasts used for planning and safety. The method focuses on keeping the altered inputs visually and physically close to real data while achieving the desired model behavior.

Core claim

FABLE performs a 3D discrete wavelet decomposition to disentangle the spatial and temporal components of the data. By regulating the magnitude of adversarial perturbations across different components, FABLE produces adversarial inputs that remain closely aligned with the original inputs while steering the DLWF models toward generating the targeted forecast outcomes. Experimental results on real-world weather datasets demonstrate the effectiveness of FABLE over baseline methods across various metrics.

What carries the argument

3D discrete wavelet decomposition that separates spatial and temporal components, combined with separate control over perturbation magnitudes in each component to enable targeted steering of model outputs.

If this is right

  • Adversarial inputs can achieve specific forecast targets while staying closely aligned with original weather data.
  • The approach works without access to the internal details of the DLWF model being attacked.
  • Localized control via wavelets improves attack success compared with standard perturbation methods on these models.
  • DLWF models may need new robustness techniques focused on multi-scale spatial-temporal features.

Where Pith is reading between the lines

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

  • Similar wavelet-based localization could be tested on other spatio-temporal forecasting tasks such as traffic or energy prediction.
  • Operational weather systems might incorporate wavelet-domain checks to detect subtle input manipulations.
  • The separation of components suggests a path to more interpretable robustness testing in environmental deep learning models.

Load-bearing premise

Regulating perturbation magnitudes separately across wavelet components will reliably produce targeted forecast changes without requiring model-specific knowledge or breaking the physical plausibility of the input data.

What would settle it

Running FABLE on a DLWF model with a chosen target forecast and finding that the output forecast does not shift to the target or that the perturbed input shows clear physical inconsistencies with real weather patterns.

Figures

Figures reproduced from arXiv: 2505.12167 by Asadullah Hill Galib, Jack Gunn, Lifeng Luo, Pang-Ning Tan, Xin Lan, Yue Deng.

Figure 1
Figure 1. Figure 1: The top two panels show the original input data and [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance comparison of existing adversarial attack [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of adversarial predictors X′ (rows 2-5, column 2) generated by different attack methods when applied to the original predictor X (row 1, column 2) of the NLDAS precipitation dataset [7]. The first column shows the magnitude of perturbations (bounded by 2.5) introduced by different attack methods. The third column shows the original forecast Yˆ (row 1) and adversarial forecasts g(X′ ) (rows 2-5) pr… view at source ↗
Figure 4
Figure 4. Figure 4: Impact of existing adversarial attack methods on the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of varying penalty weights for perturbations [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

Deep learning-based weather forecasting (DLWF) models have recently demonstrated significant performance gains over gold-standard physics-based simulation tools. However, these models are potentially vulnerable to adversarial attacks, which raises concerns about their trustworthiness. In this paper, we investigate the feasibility and challenges of applying existing adversarial attack methods to DLWF models and propose a novel framework called FABLE (Forecast Alteration By Localized targeted advErsarial attack) to address them. FABLE performs a 3D discrete wavelet decomposition to disentangle the spatial and temporal components of the data. By regulating the magnitude of adversarial perturbations across different components, FABLE produces adversarial inputs that remain closely aligned with the original inputs while steering the DLWF models toward generating the targeted forecast outcomes. Experimental results on real-world weather datasets demonstrate the effectiveness of FABLE over baseline methods across various metrics.

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 / 2 minor

Summary. The manuscript introduces FABLE, a framework for localized targeted adversarial attacks on deep learning-based weather forecasting (DLWF) models. It performs 3D discrete wavelet decomposition to disentangle spatial and temporal components of input data, then regulates the magnitude of perturbations separately across these components to generate adversarial examples that remain close to the originals while steering the DLWF model outputs toward specified target forecasts. The authors report that experiments on real-world weather datasets demonstrate FABLE's effectiveness relative to baseline attack methods across multiple metrics.

Significance. If the empirical claims hold after addressing the points below, the work is significant because DLWF models are increasingly competitive with physics-based simulators yet their robustness has received limited attention. The wavelet-based localization offers a structured way to control perturbations that may better respect the multi-scale nature of meteorological fields than generic L_p attacks. The manuscript earns credit for conducting the evaluation on real-world datasets rather than synthetic data alone.

major comments (2)
  1. [§3.2] §3.2 (FABLE framework): the central mechanism of independently bounding perturbation magnitudes per wavelet component is presented as sufficient to produce targeted forecast changes while preserving physical plausibility, yet the text provides no description of post-inverse-transform verification for meteorological constraints (e.g., non-negative precipitation, approximate mass conservation). This assumption is load-bearing for the claim that the generated inputs remain physically realistic.
  2. [§4] §4 (Experimental setup) and Table 1: the superiority over baselines is asserted across metrics, but the section does not report error bars, the number of random seeds, or the precise implementation details (e.g., whether baselines were re-implemented with the same wavelet preprocessing or used their original code). Without these, the quantitative advantage cannot be assessed reliably.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by including at least one concrete quantitative result (e.g., a success rate or perturbation norm) rather than the qualitative statement of effectiveness.
  2. [§3.1] Notation for the 3D discrete wavelet coefficients and the per-component magnitude bounds should be introduced with an explicit equation or table to avoid ambiguity when reading the perturbation regulation step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. We address each major comment point by point below, indicating the revisions we plan to incorporate to strengthen the work.

read point-by-point responses
  1. Referee: [§3.2] §3.2 (FABLE framework): the central mechanism of independently bounding perturbation magnitudes per wavelet component is presented as sufficient to produce targeted forecast changes while preserving physical plausibility, yet the text provides no description of post-inverse-transform verification for meteorological constraints (e.g., non-negative precipitation, approximate mass conservation). This assumption is load-bearing for the claim that the generated inputs remain physically realistic.

    Authors: We agree that the manuscript does not include an explicit description of post-inverse-transform verification for meteorological constraints. While the 3D wavelet decomposition and per-component magnitude bounding are intended to produce localized, small perturbations that remain close to the original inputs (thereby implicitly respecting physical properties such as non-negativity in precipitation fields), we acknowledge that this is not sufficient without verification. In the revised manuscript we will add a dedicated paragraph in §3.2 that describes the post-processing checks performed after the inverse transform, including clipping of negative precipitation values to zero and a simple mass-conservation diagnostic on integrated fields, along with any observed impact on attack success rates. revision: yes

  2. Referee: [§4] §4 (Experimental setup) and Table 1: the superiority over baselines is asserted across metrics, but the section does not report error bars, the number of random seeds, or the precise implementation details (e.g., whether baselines were re-implemented with the same wavelet preprocessing or used their original code). Without these, the quantitative advantage cannot be assessed reliably.

    Authors: We concur that the absence of error bars, seed counts, and baseline implementation details limits the interpretability of Table 1. The experiments were run with multiple random seeds and the baselines were re-implemented from their original public code but evaluated on the identical real-world datasets and preprocessing pipeline (without forcing wavelet decomposition onto methods that do not use it). In the revision we will (i) add error bars (standard deviation across seeds) to all metrics in Table 1, (ii) state that results are averaged over five independent random seeds, and (iii) expand the implementation subsection to clarify exactly how each baseline was obtained and executed. revision: yes

Circularity Check

0 steps flagged

No circularity: FABLE is an empirical method proposal validated by experiments

full rationale

The paper proposes FABLE, a framework that applies 3D discrete wavelet decomposition to weather data and regulates perturbation magnitudes across components to create targeted adversarial inputs for DLWF models. All central claims rest on experimental results comparing FABLE to baselines across metrics on real-world datasets. No equations, derivations, or first-principles results are presented that reduce by construction to fitted parameters, self-defined quantities, or self-citation chains. The method is self-contained as a practical attack design whose success is measured externally rather than tautologically.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method relies on the standard properties of discrete wavelet transforms and the existence of differentiable DLWF models; no new physical constants or invented entities are introduced. A small number of attack hyperparameters (wavelet decomposition levels, per-component magnitude bounds) are chosen by the authors.

free parameters (1)
  • per-component perturbation magnitude bounds
    Chosen to keep adversarial examples close to originals while achieving targeted forecast changes.
axioms (1)
  • domain assumption 3D discrete wavelet decomposition cleanly separates spatial and temporal frequency components in weather data without significant information loss for the forecasting task.
    Invoked to justify regulating perturbations separately across components.

pith-pipeline@v0.9.0 · 5684 in / 1308 out tokens · 27709 ms · 2026-05-22T14:01:30.620131+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Guided Diffusion Sampling for Precipitation Forecast Interventions

    cs.LG 2026-05 unverdicted novelty 7.0

    Gradient-guided diffusion sampling reduces extreme precipitation forecasts in data-driven weather models while producing more physically plausible changes than adversarial perturbations.

  2. Adversarial Attacks on Downstream Weather Forecasting Models: Application to Tropical Cyclone Trajectory Prediction

    cs.LG 2025-10 unverdicted novelty 6.0

    Cyc-Attack uses a differentiable surrogate for TC detection, a skewness-aware loss, and gradient weighting to perturb DLWF inputs and steer downstream TC trajectory predictions toward specified targets with higher suc...

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 2 Pith papers · 6 internal anchors

  1. [1]

    Earthformer: exploring space-time transformers for earth system forecasting,

    Z. Gao, X. Shi, H. Wang, Y . Zhu, Y . B. Wang, M. Li, and D.-Y . Yeung, “Earthformer: exploring space-time transformers for earth system forecasting,”Advances in Neural Information Processing Systems, vol. 35, pp. 25 390–25 403, 2022

  2. [2]

    Conditional local convolution for spatio-temporal meteorological forecasting,

    H. Lin, Z. Gao, Y . Xu, L. Wu, L. Li, and S. Z. Li, “Conditional local convolution for spatio-temporal meteorological forecasting,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, no. 7, 2022, pp. 7470–7478

  3. [3]

    Learning skillful medium-range global weather forecasting,

    R. Lam, A. Sanchez-Gonzalez, M. Willson, P. Wirnsberger, M. Fortunato, F. Alet, S. Ravuri, T. Ewalds, Z. Eaton-Rosen, W. Huet al., “Learning skillful medium-range global weather forecasting,”Science, vol. 382, no. 6677, pp. 1416–1421, 2023

  4. [4]

    Accurate medium- range global weather forecasting with 3d neural networks,

    K. Bi, L. Xie, H. Zhang, X. Chen, X. Gu, and Q. Tian, “Accurate medium- range global weather forecasting with 3d neural networks,”Nature, vol. 619, no. 7970, pp. 533–538, 2023

  5. [5]

    Adversarial Attacks and Defences: A Survey

    A. Chakraborty, M. Alam, V . Dey, A. Chattopadhyay, and D. Mukhopad- hyay, “Adversarial attacks and defences: a survey,”ArXiv Preprint ArXiv:1810.00069, 2018

  6. [6]

    Targeted adversarial attacks on wind power forecasts,

    R. Heinrich, C. Scholz, S. V ogt, and M. Lehna, “Targeted adversarial attacks on wind power forecasts,”Machine Learning, vol. 113, no. 2, pp. 863–889, 2024

  7. [7]

    The multi-institution north american land data assimilation system (nldas): Utilizing multiple gcip products and partners in a continental distributed hydrological modeling system,

    K. E. Mitchell, D. Lohmann, P. R. Houser, E. F. Wood, J. C. Schaake, A. Robock, B. A. Cosgrove, J. Sheffield, Q. Duan, L. Luoet al., “The multi-institution north american land data assimilation system (nldas): Utilizing multiple gcip products and partners in a continental distributed hydrological modeling system,”Journal of Geophysical Research: Atmospher...

  8. [8]

    Explaining and Harnessing Adversarial Examples

    I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,”ArXiv Preprint ArXiv:1412.6572, 2014

  9. [9]

    Towards Deep Learning Models Resistant to Adversarial Attacks

    A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning models resistant to adversarial attacks,”ArXiv Preprint ArXiv:1706.06083, 2017

  10. [10]

    The operational cmc–mrb global environmental multiscale (gem) model. part i: Design considerations and formulation,

    J. C ˆot´e, S. Gravel, A. M ´ethot, A. Patoine, M. Roch, and A. Staniforth, “The operational cmc–mrb global environmental multiscale (gem) model. part i: Design considerations and formulation,”Monthly Weather Review, vol. 126, no. 6, pp. 1373–1395, 1998

  11. [11]

    A description of the advanced research wrf version 3,

    W. C. Skamarock, J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, M. G. Duda, X.-Y . Huang, W. Wang, J. G. Powerset al., “A description of the advanced research wrf version 3,”NCAR Technical Note, vol. 475, no. 125, pp. 10–5065, 2008

  12. [12]

    The era5 global reanalysis,

    H. Hersbach, B. Bell, P. Berrisford, S. Hirahara, A. Hor ´anyi, J. Mu ˜noz- Sabater, J. Nicolas, C. Peubey, R. Radu, D. Scheperset al., “The era5 global reanalysis,”Quarterly Journal of the Royal Meteorological Society, vol. 146, no. 730, pp. 1999–2049, 2020

  13. [13]

    Probabilistic weather forecasting with machine learning,

    I. Price, A. Sanchez-Gonzalez, F. Alet, T. R. Andersson, A. El-Kadi, D. Masters, T. Ewalds, J. Stott, S. Mohamed, P. Battagliaet al., “Probabilistic weather forecasting with machine learning,”Nature, vol. 637, no. 8044, pp. 84–90, 2025

  14. [14]

    Spatiotemporal vision transformer for short time weather forecasting,

    A. Bojesomo, H. Al-Marzouqi, and P. Liatsis, “Spatiotemporal vision transformer for short time weather forecasting,” in2021 IEEE Interna- tional Conference on Big Data (Big Data). IEEE, 2021, pp. 5741–5746

  15. [15]

    Spatiotemporal swin-transformer network for short time weather fore- casting

    A. Bojesomo, H. Al-Marzouqi, P. Liatsis, G. Cong, and M. Ramanath, “Spatiotemporal swin-transformer network for short time weather fore- casting.” inCIKM Workshops, 2021

  16. [16]

    A novel transformer network with shifted window cross-attention for spatiotemporal weather forecasting,

    A. Bojesomo, H. AlMarzouqi, and P. Liatsis, “A novel transformer network with shifted window cross-attention for spatiotemporal weather forecasting,”IEEE Journal of Selected Topics in Applied Earth Observa- tions and Remote Sensing, 2023

  17. [17]

    Deep- extrema: a deep learning approach for forecasting block maxima in time series data,

    A. H. Galib, A. McDonald, T. Wilson, L. Luo, and P.-N. Tan, “Deep- extrema: a deep learning approach for forecasting block maxima in time series data,”In Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI, 2022

  18. [18]

    Boosting adversarial attacks with momentum,

    Y . Dong, F. Liao, T. Pang, H. Su, J. Zhu, X. Hu, and J. Li, “Boosting adversarial attacks with momentum,” inProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 9185–9193

  19. [19]

    Practical adversarial attacks on spa- tiotemporal traffic forecasting models,

    F. Liu, H. Liu, and W. Jiang, “Practical adversarial attacks on spa- tiotemporal traffic forecasting models,”Advances in Neural Information Processing Systems, vol. 35, pp. 19 035–19 047, 2022

  20. [20]

    Imperceptible adversarial attack via invertible neural networks,

    Z. Chen, Z. Wang, J.-J. Huang, W. Zhao, X. Liu, and D. Guan, “Imperceptible adversarial attack via invertible neural networks,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, no. 1, 2023, pp. 414–424

  21. [21]

    Efficient robustness assessment via adversarial spatial-temporal focus on videos,

    X. Wei, S. Wang, and H. Yan, “Efficient robustness assessment via adversarial spatial-temporal focus on videos,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 45, no. 9, pp. 10 898– 10 912, 2023

  22. [22]

    Imperceptible adversarial attack with multi-granular spatio-temporal attention for video action recognition,

    G. Wu, Y . Xu, J. Li, Z. Shi, and X. Liu, “Imperceptible adversarial attack with multi-granular spatio-temporal attention for video action recognition,”IEEE Internet of Things Journal, 2023

  23. [23]

    A gradient-based wind power forecasting attack method considering point and direction selection,

    R. Jiao, Z. Han, X. Liu, C. Zhou, and M. Du, “A gradient-based wind power forecasting attack method considering point and direction selection,” IEEE Transactions on Smart Grid, 2023

  24. [24]

    On vulnerability of renewable energy forecasting: adversarial learning attacks,

    J. Ruan, Q. Wang, S. Chen, H. Lyu, G. Liang, J. Zhao, and Z. Y . Dong, “On vulnerability of renewable energy forecasting: adversarial learning attacks,”IEEE Transactions on Industrial Informatics, 2023

  25. [25]

    Adversarial diffusion attacks on graph-based traffic prediction models,

    L. Zhu, K. Feng, Z. Pu, and W. Ma, “Adversarial diffusion attacks on graph-based traffic prediction models,”IEEE Internet of Things Journal, 2023

  26. [26]

    Evaluating adversarial attacks on federated learning for temperature forecasting,

    K. Chichifoi, F. Merizzi, and M. Colajanni, “Evaluating adversarial attacks on federated learning for temperature forecasting,”arXiv preprint arXiv:2512.13207, 2025

  27. [27]

    Adversarial observations in weather forecasting,

    E. Imgrund, T. Eisenhofer, and K. Rieck, “Adversarial observations in weather forecasting,” inProceedings of the 2025 ACM SIGSAC Conference on Computer and Communications Security, 2025, pp. 3579– 3590

  28. [28]

    Forecasting fails: Unveiling evasion attacks in weather prediction models,

    H. Arif, P.-Y . Chen, A. Gittens, J. Diffenderfer, and B. Kailkhura, “Forecasting fails: Unveiling evasion attacks in weather prediction models,” arXiv preprint arXiv:2512.08832, 2025

  29. [29]

    Notes on continuous stochastic phenomena,

    P. A. Moran, “Notes on continuous stochastic phenomena,”Biometrika, vol. 37, no. 1/2, pp. 17–23, 1950

  30. [30]

    P. J. Brockwell and R. A. Davis,Introduction to time series and forecasting. Springer, 2002

  31. [31]

    FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

    J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheliet al., “Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators,”arXiv preprint arXiv:2202.11214, 2022

  32. [32]

    A foundation model for the earth system,

    C. Bodnar, W. P. Bruinsma, A. Lucic, M. Stanley, A. Allen, J. Brandstetter, P. Garvan, M. Riechert, J. A. Weyn, H. Donget al., “A foundation model for the earth system,”Nature, pp. 1–8, 2025

  33. [33]

    Prithvi wxc: Foundation model for weather and climate.arXiv preprint arXiv:2409.13598, 2024

    J. Schmude, S. Roy, W. Trojak, J. Jakubik, D. S. Civitarese, S. Singh, J. Kuehnert, K. Ankur, A. Gupta, C. E. Phillipset al., “Prithvi wxc: Foun- dation model for weather and climate,”arXiv preprint arXiv:2409.13598, 2024

  34. [34]

    Lower bounds on adversarial robustness from optimal transport,

    A. N. Bhagoji, D. Cullina, and P. Mittal, “Lower bounds on adversarial robustness from optimal transport,”Advances in Neural Information Processing Systems, vol. 32, 2019

  35. [35]

    On Evaluating Adversarial Robustness

    N. Carlini, A. Athalye, N. Papernot, W. Brendel, J. Rauber, D. Tsipras, I. Goodfellow, A. Madry, and A. Kurakin, “On evaluating adversarial robustness,”arXiv preprint arXiv:1902.06705, 2019

  36. [36]

    Towards the worst-case robustness of large language models,

    H. Chen, Y . Dong, Z. Wei, H. Su, and J. Zhu, “Towards the worst-case robustness of large language models,”arXiv preprint arXiv:2501.19040, 2025

  37. [37]

    Debnath and F

    L. Debnath and F. A. Shah,Wavelet transforms and their applications. Springer, 2015, vol. 434

  38. [38]

    Adam: A Method for Stochastic Optimization

    D. P. Kingma, “Adam: a method for stochastic optimization,”ArXiv Preprint ArXiv:1412.6980, 2014

  39. [39]

    Principal component analysis,

    H. Abdi and L. J. Williams, “Principal component analysis,”Wiley interdisciplinary reviews: computational statistics, vol. 2, no. 4, pp. 433–459, 2010

  40. [40]

    Isolation forest,

    F. T. Liu, K. M. Ting, and Z.-H. Zhou, “Isolation forest,” in2008 eighth ieee international conference on data mining. IEEE, 2008, pp. 413–422

  41. [41]

    Lof: identifying density-based local outliers,

    M. M. Breunig, H.-P. Kriegel, R. T. Ng, and J. Sander, “Lof: identifying density-based local outliers,” inProceedings of the 2000 ACM SIGMOD international conference on Management of data, 2000, pp. 93–104

  42. [42]

    Regularizing autoencoders with wavelet transform for sequence anomaly detection,

    Y . Yao, J. Ma, and Y . Ye, “Regularizing autoencoders with wavelet transform for sequence anomaly detection,”Pattern Recognition, vol. 134, p. 109084, 2023. 13 APPENDIXA HAARWAVELETTRANSFORM Section III-B presents the decomposition of a one- dimensional signal using the discrete wavelet transform with the Haar wavelet basis. In this section, we extend th...