pith. sign in

arxiv: 2606.24696 · v1 · pith:NSVKUHFRnew · submitted 2026-06-23 · ⚛️ physics.flu-dyn · cs.LG

A Physics-Informed Fourier-Wavelet Transformer for Multiscale Computational Fluid Dynamics Surrogate Modeling

Somyajit Chakraborty , Ming Pan , Xizhong Chen This is my paper

Pith reviewed 2026-06-25 22:45 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LG
keywords physics-informed surrogate modelingFourier-wavelet transformercomputational fluid dynamicsmultiscale flow reconstructioncylinder wakefluid-structure interactionself-supervised pretrainingPDE residual diagnostics
0
0 comments X

The pith

A Fourier-wavelet transformer that biases attention using PDE residuals reconstructs multiscale fluid velocity fields more accurately than prior surrogate models on standard benchmarks.

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

The paper develops a transformer that encodes flow fields with combined Fourier and wavelet bases and steers self-attention according to how much local regions violate the governing equations. It adds two self-supervised pretraining objectives that mask parts of the field and enforce consistency with the physics equations. On cylinder-wake and fluid-structure interaction cases the resulting model records the lowest normalized errors among tested approaches and recovers fine wake structures more reliably. A reader would care because such surrogates could shorten expensive CFD runs while preserving detail in wakes and near-body regions without manual retuning for each new geometry.

Core claim

The authors claim that hybrid Fourier-wavelet spectral encoding together with physics-biased self-attention derived from PDE residual diagnostics, plus Masked Physics Prediction and Equation Consistency Prediction pretraining, produces next-step velocity-field reconstructions that outperform spectral, transformer, operator-learning, and physics-informed neural-network baselines on two real benchmarks while recovering localized multiscale structures including near-body, wake-core, and far-wake features.

What carries the argument

Physics-biased self-attention derived from partial differential equation residual diagnostics, which re-weights attention to emphasize locations where the predicted field deviates from the flow equations.

If this is right

  • Component-wise comparisons show improved capture of near-body, wake-core, and far-wake localized structures.
  • The model achieves the lowest all-channel normalized mean-squared error on both cylinder-wake and fluid-structure interaction benchmarks under a shared evaluation protocol.
  • All-channel Pearson correlation reaches 0.97019 on the cylinder-wake case while maintaining a practical accuracy-cost balance.
  • The approach generalizes across the two tested real-world flow regimes without case-by-case hyperparameter changes.

Where Pith is reading between the lines

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

  • If the residual diagnostic generalizes, the same bias mechanism could be transferred to surrogate modeling of other PDE systems such as heat or mass transport.
  • The two pretraining tasks may reduce data requirements when the model is applied to flows outside the original training distribution.
  • Scale-separated error diagnostics could be used to design hybrid architectures that allocate different resolution levels to different physical regions.

Load-bearing premise

The physics-biased attention from PDE residuals improves recovery of localized structures without introducing new biases or requiring benchmark-specific tuning.

What would settle it

Apply the trained model to a third independent flow configuration and measure whether it still records the lowest all-channel normalized mean-squared error and visibly better scale-separated wake recovery than the strongest baseline.

Figures

Figures reproduced from arXiv: 2606.24696 by Ming Pan, Somyajit Chakraborty, Xizhong Chen.

Figure 1
Figure 1. Figure 1: PIBERT architecture and training objectives. (a) Raw CFD fields and PDE parameters defined on [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FSI-real validation convergence for the comparison runs. Panel (a) shows the logged fine-tuning [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Aggregate performance on the RealPDEBench cylinder-real benchmark for PIBERT, FNO2d, [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of ground truth, prediction, and signed error on a held-out RealPDEBench cylinder-real [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: PIBERT prediction against ground truth, with signed error, for [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: The same interpretation is also supported by the supplementary Cylinder-real [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Scale-separated and wake-line diagnostics for the RealPDEBench cylinder-real [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Aggregate performance on the RealPDEBench FSI-real benchmark for PIBERT, FNO2d, Deep [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of ground truth, prediction, and signed error on a held-out RealPDEBench FSI-real [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: PIBERT prediction against ground truth, with signed error, for [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scale-separated and wake-line diagnostics for the RealPDEBench FSI-real [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
read the original abstract

Physics-informed surrogate models can accelerate computational fluid dynamics simulations. However, many existing methods reproduce global flow patterns more reliably than localized multiscale structures. This study presents a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in real-world flow benchmarks. The proposed formulation combines hybrid Fourier-wavelet spectral encoding with physics-biased self-attention based on partial differential equation residual diagnostics. It also uses self-supervised pretraining through Masked Physics Prediction and Equation Consistency Prediction. The experiments are conducted on two real benchmark cases: cylinder-wake flow and fluid-structure interaction. All approaches are evaluated under a shared local protocol and compared with spectral, transformer-based, operator-learning, and physics-informed neural-network baselines. On the cylinder-wake benchmark, the proposed model achieves the best aggregate accuracy, with an all-channel normalized mean-squared error of 0.05875 and an all-channel Pearson correlation coefficient of 0.97019. On the fluid-structure-interaction benchmark, it gives the lowest all-channel normalized mean-squared error of $2.70 \times 10^{-4}$, compared with $4.02 \times 10^{-4}$ for the strongest baseline. Component-wise field comparisons and scale-separated diagnostics further show stronger recovery of localized wake structures, including near-body, wake-core, and far-wake features. The results demonstrate improved real-world flow reconstruction while maintaining a practical accuracy-cost tradeoff.

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 manuscript introduces a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in CFD surrogate modeling. It combines hybrid Fourier-wavelet spectral encoding, physics-biased self-attention derived from PDE residual diagnostics, and self-supervised pretraining (Masked Physics Prediction and Equation Consistency Prediction). On cylinder-wake and fluid-structure-interaction benchmarks, the model reports superior aggregate accuracy (all-channel NMSE 0.05875 / PCC 0.97019 on cylinder-wake; NMSE 2.70e-4 on FSI) versus spectral, transformer, operator-learning, and PINN baselines, with additional component-wise and scale-separated diagnostics.

Significance. If the performance gains are robustly attributable to the proposed components and the evaluation protocol is reproducible, the approach could advance multiscale flow surrogate modeling by improving recovery of localized wake structures while preserving practical accuracy-cost tradeoffs.

major comments (2)
  1. [Experiments / Results] The central attribution—that physics-biased self-attention improves localized multiscale recovery—lacks supporting evidence from controlled ablation; the manuscript reports only end-to-end benchmark comparisons without disabling the bias term while holding spectral encoding and pretraining objectives fixed.
  2. [Abstract / Experiments] Verification of the reported superiority is hindered by the absence of information on data splits, hyperparameter search procedures, statistical significance of metric differences, and whether baselines received equivalent tuning effort.
minor comments (1)
  1. [Abstract] The phrase 'shared local protocol' is used without definition or reference to supplementary material detailing the exact evaluation setup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Experiments / Results] The central attribution—that physics-biased self-attention improves localized multiscale recovery—lacks supporting evidence from controlled ablation; the manuscript reports only end-to-end benchmark comparisons without disabling the bias term while holding spectral encoding and pretraining objectives fixed.

    Authors: We agree that the current manuscript presents only end-to-end benchmark results and does not include controlled ablations that isolate the physics-biased self-attention by disabling the bias term while holding spectral encoding and pretraining fixed. To strengthen the attribution, we will add these ablation experiments to the revised manuscript, reporting the resulting changes in NMSE and scale-separated metrics on both benchmarks. revision: yes

  2. Referee: [Abstract / Experiments] Verification of the reported superiority is hindered by the absence of information on data splits, hyperparameter search procedures, statistical significance of metric differences, and whether baselines received equivalent tuning effort.

    Authors: We acknowledge that the manuscript lacks explicit details on these aspects. In the revision we will add a dedicated reproducibility section (or appendix) specifying the train/validation/test splits, the hyperparameter search procedure and ranges, statistical significance tests (e.g., paired t-tests or bootstrap confidence intervals) on the reported metric differences, and confirmation that all baselines were tuned under the same shared protocol with equivalent effort. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain; empirical results are independent of model inputs

full rationale

The manuscript presents an architecture combining hybrid Fourier-wavelet encoding, physics-biased self-attention from PDE residuals, and two self-supervised pretraining tasks, then reports end-to-end NMSE and PCC numbers on cylinder-wake and FSI benchmarks against external baselines. No equations, fitted parameters, or self-citations are shown that reduce the reported accuracy figures to quantities defined by the model itself. The performance claims rest on direct numerical comparisons under a shared protocol, satisfying the criterion of being self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; full text would be required to audit these.

pith-pipeline@v0.9.1-grok · 5791 in / 1058 out tokens · 32312 ms · 2026-06-25T22:45:13.173697+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

49 extracted references · 15 canonical work pages

  1. [1]

    Y. Liu, J. N. Kutz, S. L. Brunton, Hierarchical deep learning of multiscale differential equation time-steppers, Philosophical Transactions of the Royal Society A 380 (2229) (2022) 20210200

    2022

  2. [2]

    Y. Wang, N. Ye, Z. Li, Physics-informed surrogate for cardiovascular flow extrapolation through transductive learning, Engineering Applications of Artificial Intelligence 159 (2025) 111458.doi:10.1016/j.engappai.2025.111458

    work page doi:10.1016/j.engappai.2025.111458 2025

  3. [3]

    Q. Liu, W. Zhong, S. Koric, H. Meidani, Sequential neural operator transformer for high- fidelity surrogates of time-dependent non-linear partial differential equations, Engineering Applications of Artificial Intelligence 172 (2026) 114428.doi:10.1016/j.engappai. 2026.114428

    work page doi:10.1016/j.engappai 2026

  4. [4]

    Raissi, P

    M. Raissi, P. Perdikaris, N. Ahmadi, G. E. Karniadakis, Physics-informed neural networks and extensions, arXiv preprint arXiv:2408.16806 (2024)

    arXiv 2024

  5. [5]

    L. Yi, S. Yang, Y. Cui, Z. Lai, Transforming physics-informed machine learning to convex optimization, Engineering Applications of Artificial Intelligence 161 (2025) 112149. doi:https://doi.org/10.1016/j.engappai.2025.112149. URLhttps://www.sciencedirect.com/science/article/pii/S0952197625021578

    work page doi:10.1016/j.engappai.2025.112149 2025

  6. [6]

    Zhang, W

    W. Zhang, W. Suo, J. Song, W. Cao, Physics informed neural networks (pinns) as intelligent computing technique for solving partial differential equations: Limitation and future prospects, arXiv preprint arXiv:2411.18240 (2024)

    arXiv 2024

  7. [7]

    Rui, Z.-W

    E.-Z. Rui, Z.-W. Chen, Y.-Q. Ni, L. Yuan, G.-Z. Zeng, Reconstruction of 3d flow field around a building model in wind tunnel: a novel physics-informed neural network frame- work adopting dynamic prioritization self-adaptive loss balance strategy, Engineering Applications of Computational Fluid Mechanics 17 (1) (2023) 2238849

    2023

  8. [8]

    Penwarden, A

    M. Penwarden, A. D. Jagtap, S. Zhe, G. E. Karniadakis, R. M. Kirby, A unified scalable framework for causal sweeping strategies for physics-informed neural networks (pinns) and their temporal decompositions, Journal of Computational Physics 493 (2023) 112464

    2023

  9. [9]

    Abbasi, A

    J. Abbasi, A. D. Jagtap, B. Moseley, A. Hiorth, P. Østebø Andersen, Challenges and advancements in modeling shock fronts with physics-informed neural networks: A review and benchmarking study, Neurocomputing 657 (2025) 131440.doi:https: //doi.org/10.1016/j.neucom.2025.131440. URLhttps://www.sciencedirect.com/science/article/pii/S0925231225021125

    work page doi:10.1016/j.neucom.2025.131440 2025

  10. [10]

    Kovachki, Z

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, A. Anand- kumar, Neural operator: Learning maps between function spaces with applications to pdes, Journal of Machine Learning Research 24 (89) (2023) 1–97

    2023

  11. [11]

    Serrano, L

    L. Serrano, L. Le Boudec, A. Kassaï Koupaï, T. X. Wang, Y. Yin, J.-N. Vittaut, P. Gallinari, Operator learning with neural fields: Tackling pdes on general geometries, Advances in Neural Information Processing Systems 36 (2023) 70581–70611. 41

    2023

  12. [12]

    Raonic, R

    B. Raonic, R. Molinaro, T. De Ryck, T. Rohner, F. Bartolucci, R. Alaifari, S. Mishra, E. de Bézenac, Convolutional neural operators for robust and accurate learning of pdes, Advances in Neural Information Processing Systems 36 (2023) 77187–77200

    2023

  13. [13]

    G. Wen, Z. Li, K. Azizzadenesheli, A. Anandkumar, S. M. Benson, U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow, Advances in Water Resources 163 (2022) 104180

    2022

  14. [14]

    Sinha, B

    S. Sinha, B. Benton, P. Emami, On the effectiveness of neural operators at zero-shot weather downscaling, Environmental Data Science 4 (2025) e21

    2025

  15. [15]

    H. Wang, Y. Cao, Z. Huang, Y. Liu, P. Hu, X. Luo, Z. Song, W. Zhao, J. Liu, J. Sun, et al., Recent advances on machine learning for computational fluid dynamics: A survey, arXiv preprint arXiv:2408.12171 (2024)

    arXiv 2024

  16. [16]

    Q. Luo, W. Zeng, M. Chen, G. Peng, X. Yuan, Q. Yin, Self-attention and transformers: Driving the evolution of large language models, in: 2023 IEEE 6th International Confer- ence on Electronic Information and Communication Technology (ICEICT), IEEE, 2023, pp. 401–405

    2023

  17. [17]

    Z. Zhao, X. Ding, B. A. Prakash, Pinnsformer: A transformer-based framework for physics-informed neural networks, arXiv preprint arXiv:2307.11833 (2023)

    arXiv 2023

  18. [18]

    Lorsung, Z

    C. Lorsung, Z. Li, A. Barati Farimani, Physics informed token transformer for solving partial differential equations, Machine Learning: Science and Technology 5 (1) (2024) 015032.doi:10.1088/2632-2153/ad27e3. URLhttps://dx.doi.org/10.1088/2632-2153/ad27e3

    work page doi:10.1088/2632-2153/ad27e3 2024

  19. [19]

    P. Hu, H. Feng, H. Liu, T. Yan, W. Deng, T. Gao, R. Zheng, H. Zheng, C. Yu, C. Wang, K. Li, Z.-M. Ma, D. Zhou, X. Lu, D. Fan, T. Wu, RealPDEBench: A benchmark for complex physical systems with real-world data, in: The Fourteenth International Conference on Learning Representations, 2026. URLhttps://openreview.net/forum?id=y3oHMcoItR

    2026

  20. [20]

    Y. Luo, Y. Chen, Z. Zhang, Cfdbench: A large-scale benchmark for machine learning methods in fluid dynamics, arXiv preprint arXiv:2310.05963 (2023)

    arXiv 2023

  21. [21]

    G. Daly, J. Fieldsend, G. Hassall, G. Tabor, Data-driven plasma modelling: Fluorocarbon icp data set, dataset (2023).doi:10.5281/zenodo.7704879

    work page doi:10.5281/zenodo.7704879 2023

  22. [22]

    Janny, A

    S. Janny, A. Bénéteau, M. Nadri, J. Digne, N. Thome, C. Wolf, Eagle: Large-scale learning of turbulent fluid dynamics with mesh transformers, in: International Conference on Learning Representations, 2023, dataset and benchmark

    2023

  23. [23]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378 (2019) 686–707. 42

    2019

  24. [24]

    A. M. Roy, S. Guha, A data-driven physics-constrained deep learning computational framework for solving von mises plasticity, Engineering Applications of Artificial Intelli- gence 122 (2023) 106049.doi:10.1016/j.engappai.2023.106049

    work page doi:10.1016/j.engappai.2023.106049 2023

  25. [25]

    S. J. Anagnostopoulos, J. D. Toscano, N. Stergiopulos, G. E. Karniadakis, Residual-based attention in physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 421 (2024) 116805.doi:10.1016/j.cma.2024.116805

    work page doi:10.1016/j.cma.2024.116805 2024

  26. [26]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, Nature Machine Intelligence 3 (3) (2021) 218–229.doi:10.1038/s42256-021-00302-5. URLhttps://doi.org/10.1038/s42256-021-00302-5

    work page doi:10.1038/s42256-021-00302-5 2021

  27. [27]

    Ovadia, A

    O. Ovadia, A. Kahana, P. Stinis, E. Turkel, D. Givoli, G. E. Karniadakis, Vito: Vision transformer-operator, Computer Methods in Applied Mechanics and Engineering 428 (2024) 117109.doi:10.1016/j.cma.2024.117109

    work page doi:10.1016/j.cma.2024.117109 2024

  28. [28]

    Darwish, A

    W. Zhong, H. Meidani, Physics-informed geometry-aware neural operator, Computer Methods in Applied Mechanics and Engineering 434 (2025) 117540.doi:10.1016/j. cma.2024.117540

    work page doi:10.1016/j 2025

  29. [29]

    Z. Li, N. B. Kovachki, K. Azizzadenesheli, B. liu, K. Bhattacharya, A. Stuart, A. Anand- kumar, Fourier neural operator for parametric partial differential equations, in: Interna- tional Conference on Learning Representations, 2021. URLhttps://openreview.net/forum?id=c8P9NQVtmnO

    2021

  30. [30]

    G. Wen, Z. Li, K. Azizzadenesheli, A. Anandkumar, S. M. Benson, U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow, Advances in Water Resources 163 (2022) 104180. doi:https://doi.org/10.1016/j.advwatres. 2022.104180. URLhttps://www.sciencedirect.com/science/article/pii/S0309170822000562

    work page doi:10.1016/j.advwatres 2022

  31. [31]

    Q. Xu, N. Thuerey, Y. Shi, J. Bamber, C. Ouyang, X. X. Zhu, Physics-embedded fourier neural network for partial differential equations, arXiv preprint arXiv:2407.11158 (2024)

    arXiv 2024

  32. [32]

    Y. Li, L. Xu, S. Ying, Dwnn: Deep wavelet neural network for solving partial differential equations, Mathematics 10 (12) (2022).doi:10.3390/math10121976. URLhttps://www.mdpi.com/2227-7390/10/12/1976

    work page doi:10.3390/math10121976 2022

  33. [33]

    J. Su, J. Ma, S. Tong, E. Xu, M. Chen, Multiscale attention wavelet neural operator for capturing steep trajectories in biochemical systems, Proceedings of the AAAI Conference onArtificialIntelligence38(13)(2024)15100–15107. doi:10.1609/aaai.v38i13.29432. URLhttps://ojs.aaai.org/index.php/AAAI/article/view/29432

    work page doi:10.1609/aaai.v38i13.29432 2024

  34. [34]

    Wang, J.-S

    H. Wang, J.-S. Pan, H. Wu, F. Zhang, T. Wu, Fourierflow: Frequency-aware flow matching for generative turbulence modeling (2026). URLhttps://openreview.net/forum?id=a3sRspQ62b 43

    2026

  35. [35]

    P. Hu, R. Wang, X. Zheng, T. Zhang, H. Feng, R. Feng, L. Wei, Y. Wang, Z.-M. Ma, T. Wu, Wavelet diffusion neural operator, in: The Thirteenth International Conference on Learning Representations, 2025. URLhttps://openreview.net/forum?id=FQhDIGuaJ4

    2025

  36. [36]

    Hemmasian, A

    A. Hemmasian, A. Barati Farimani, Multi-scale time-stepping of partial differential equations with transformers, Computer Methods in Applied Mechanics and Engineering 426 (2024) 116983.doi:10.1016/j.cma.2024.116983

    work page doi:10.1016/j.cma.2024.116983 2024

  37. [37]

    T.-Y. Yang, J. Rosca, K. Narasimhan, P. J. Ramadge, Learning physics constrained dynamics using autoencoders, Advances in Neural Information Processing Systems 35 (2022) 17157–17172

    2022

  38. [38]

    Calvello, N

    E. Calvello, N. B. Kovachki, M. E. Levine, A. M. Stuart, Continuum attention for neural operators, Journal of Machine Learning Research 26 (300) (2025) 1–52

    2025

  39. [39]

    Cao, Choose a transformer: Fourier or galerkin, Advances in neural information processing systems 34 (2021) 24924–24940

    S. Cao, Choose a transformer: Fourier or galerkin, Advances in neural information processing systems 34 (2021) 24924–24940

    2021

  40. [40]

    H. Wu, H. Luo, H. Wang, J. Wang, M. Long, Transolver: A fast transformer solver for pdes on general geometries, in: International Conference on Machine Learning, PMLR, 2024, pp. 53681–53705

    2024

  41. [41]

    G. Berend, Masked latent semantic modeling: an efficient pre-training alternative to masked language modeling, in: Findings of the Association for Computational Linguistics: ACL 2023, 2023, pp. 13949–13962

    2023

  42. [42]

    Garnier, V

    P. Garnier, V. Lannelongue, J. Viquerat, E. Hachem, Meshmask: Physics-based simula- tions with masked graph neural networks, in: The Thirteenth International Conference on Learning Representations, 2025. URLhttps://openreview.net/forum?id=bFHR8hNk4I

    2025

  43. [43]

    Devlin, M.-W

    J. Devlin, M.-W. Chang, K. Lee, K. Toutanova, Bert: Pre-training of deep bidirectional transformers for language understanding, in: Proceedings of the 2019 conference of the North American chapter of the association for computational linguistics: human language technologies, volume 1 (long and short papers), 2019, pp. 4171–4186

    2019

  44. [44]

    M. V. Koroteev, Bert: a review of applications in natural language processing and understanding, arXiv preprint arXiv:2103.11943 (2021)

    arXiv 2021

  45. [45]

    S. Qin, F. Lyu, W. Peng, D. Geng, J. Wang, X. Tang, S. Leroyer, N. Gao, X. Liu, L. L. Wang, Toward a better understanding of fourier neural operators from a spectral perspective, arXiv preprint arXiv:2404.07200 (2024).arXiv:2404.07200

    arXiv 2024

  46. [46]

    Z.Hao, S.Liu, Y.Zhang, C.Ying, Y.Feng, H.Su, J.Zhu, Physics-informedmachinelearn- ing: A survey on problems, methods and applications, arXiv preprint arXiv:2211.08064 (2022). 44

    arXiv 2022

  47. [47]

    A. Zhou, A. B. Farimani, Masked autoencoders are PDE learners, Transactions on Machine Learning Research (2024). URLhttps://openreview.net/forum?id=rZNuiFwXVs

    2024

  48. [48]

    Taghizadeh, M

    M. Taghizadeh, M. A. Nabian, N. Alemazkoor, Multi-fidelity physics-informed generative adversarial network for solving partial differential equations, Journal of Computing and Information Science in Engineering 24 (11) (2024) 111003

    2024

  49. [49]

    H. Xue, A. Araujo, B. Hu, Y. Chen, Diffusion-based adversarial sample generation for improved stealthiness and controllability, Advances in Neural Information Processing Systems 36 (2023) 2894–2921. 45

    2023