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REVIEW 3 major objections 5 minor 23 references

Four deep-learning operator models are benchmarked for predicting surface pressure in aerodynamics, and the ranking points to Transolver and BSMS-GNN as the most promising surrogates.

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2026-08-02 03:27 UTC pith:XG5O6PWY

load-bearing objection The 2D airfoil benchmark is careful and useful, but the 3D ranking reversal is a single-run artifact with unequal training budgets — don't take BSMS-GNN's lead over Transolver++ seriously yet. the 3 major comments →

arxiv 2607.13866 v1 pith:XG5O6PWY submitted 2026-07-15 physics.flu-dyn

Evaluation of State-of-the-Art Deep Learning Architectures for Aerodynamical Predictions

Jan Scherz , Derrick Hines , Philipp Bekemeyer This is my paper
classification physics.flu-dyn
keywords neural operatorsoperator learningsurrogate modelsaerodynamicssurface pressure predictiongraph neural networkstransformersshock prediction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper benchmarks four state-of-the-art operator-learning architectures—BSMS-GNN, Transolver(++), UPT, and GAOT—on aerodynamic surface pressure prediction, using a 2D airfoil dataset and an industrial-scale 3D NASA CRM aircraft dataset. On the 2D dataset, Transolver achieves the highest accuracy, closely followed by BSMS-GNN, then GAOT, with UPT clearly last. On the 3D dataset, the order reverses at the top: BSMS-GNN scores lowest errors, followed by Transolver++, then GAOT and UPT, with much larger gaps between models. The paper argues that BSMS-GNN and Transolver(++) are the most promising surrogate models for aerospace applications, while also noting that the best-performing models tend to be the most computationally lightweight.

Core claim

On the paper's own terms, the central discovery is a performance ranking: on the 2D airfoil dataset, the highest accuracy is achieved by Transolver, followed tightly by BSMS-GNN, then GAOT, and somewhat behind, UPT; on the NASA CRM dataset, BSMS-GNN achieves the lowest errors across all metrics, followed by Transolver++, then GAOT, and UPT performs markedly worse. The authors further find that the best results in both settings are produced by the most lightweight models in terms of parameters and training time, and they highlight BSMS-GNN and Transolver(++) as promising surrogate models for aerodynamical applications.

What carries the argument

The comparison is carried by the architectural mechanisms underlying each model: BSMS-GNN's bi-stride multi-scale graph construction with a U-Net structure permits efficient long-range information exchange on user-provided graphs; Transolver's Physics attention groups mesh points into a small number of learned slices, making self-attention linear in the number of points; UPT uses a GNO encoder to pool input points into latent tokens processed by a transformer, with predictions via cross-attention; GAOT uses a multiscale attentional GNO (MAGNO) encoder to map input clouds to a regular latent grid. The benchmark protocol itself—hyperparameter optimization and ten repeated instantiations on 2D,

Load-bearing premise

The 3D ranking assumes that a single training run per model, with hyperparameters chosen by transferring results from 2D or scaling them ad hoc, is enough to fairly establish the NASA CRM hierarchy, so an unlucky or favorable configuration for any architecture could change the reported order.

What would settle it

Retrain each model on the NASA CRM dataset with multiple seeds and a per-architecture hyperparameter optimization (e.g., systematically varying learning rate, batch size, and pooling radii). If, across repetitions, Transolver++ or another model achieves lower MAE and RMSE than BSMS-GNN with non-overlapping confidence intervals, the paper's 3D ranking is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Practitioners building surrogate models for similar aerospace pressure-prediction tasks can prioritize BSMS-GNN and Transolver(++) over UPT and GAOT, based on the reported error metrics.
  • The reversal of the top-two ranking between the 2D and 3D datasets indicates that architecture choice should depend on geometry complexity and dataset size, not just on general-purpose claims.
  • Since the most accurate models were also the most computationally efficient, large parameter counts are not necessary for good performance on these aerodynamic datasets.
  • All models degrade in the presence of strong shocks, especially near design-space boundaries, so shock prediction remains a shared weakness that future work would need to address.
  • The NASA CRM dataset's data-scarcity setting suggests that BSMS-GNN and Transolver++ can generalize from roughly 100 training samples to industrial-scale meshes, which is relevant to real engineering workflows.

Where Pith is reading between the lines

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

  • The 3D ranking rests on a single training run per model with hyperparameters largely transferred from the 2D study or scaled manually; a fairer multi-seed comparison with per-model tuning on the CRM dataset could plausibly change the BSMS-GNN/Transolver++ gap.
  • The fact that the simpler, physics-attention-based Transolver matches or beats heavier hybrid architectures on 2D suggests that explicit multiscale graph construction is not always necessary for learning surface pressure on modest geometries.
  • The consistent misplacement of shocks across models points to a concrete research direction: incorporating shock-capturing or discontinuity-aware loss terms into operator learning could improve aerospace surrogate accuracy more than further architecture search.
  • If the reported 3D ordering holds under repeated runs, it implies that for high-resolution surface meshes with scarce training data, a deliberately coarsened graph with multi-scale message passing may generalize better than transformer-heavy latent-token designs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper benchmarks four operator-learning architectures (BSMS-GNN, Transolver/Transolver++, UPT, GAOT) for surface-pressure prediction on two aerodynamic datasets: a 2D airfoil dataset (598 CST-perturbed RAE2822 geometries) and the industrial-scale 3D NASA CRM dataset (149 samples, 454k surface points each). The 2D experiment performs per-model hyperparameter optimization on a validation split, trains ten instantiations per model, and reports 95% confidence intervals; the ranking is Transolver > BSMS-GNN > GAOT > UPT. The 3D experiment trains a single instantiation per model with 2D-derived or ad hoc hyperparameters and reports BSMS-GNN > Transolver++ > GAOT > UPT, with the two best models separated by 32–34% in several error metrics. The paper concludes that BSMS-GNN and Transolver(++) are the most promising surrogate models.

Significance. If the results hold, the paper provides a valuable practitioner-oriented comparison for a class of problems (aerodynamic surface pressure prediction) where public benchmarks are scarce. The 2D experiment is a model of good empirical practice: hyperparameter search on a held-out validation split, ten retrained instantiations, 95% confidence intervals, a local error analysis as a function of flow parameters, and explicit discussion of failure cases, including shock-capture behavior. The derivation of supernode pooling radii in Appendix A.3 is a useful methodological detail. However, the 3D experiment, which drives the reversal of the ranking and the final recommendation, is based on one run per model without a statistically valid comparison. The paper itself disclaims any statistical statement for the 3D results (Section 5.1). The load-bearing 3D claim therefore needs either additional repetitions or a substantially softened interpretation.

major comments (3)
  1. [Section 5.1, Table 9] The central claim that BSMS-GNN surpasses Transolver++ on the NASA CRM dataset rests on a single training run per model. Section 5 states that the authors "abstained from doing repetition runs" and Section 5.1 states that the results "do not allow us to make any statistical statement." The paper attempts to substitute the lack of confidence intervals by appealing to gaps "multiple times larger" than the 2D confidence intervals, but this does not control for run-to-run variance in the 3D setting, where training is more unstable and hyperparameters were not optimized. The ranking reversal and the conclusion "BSMS-GNN and Transolver(++) as promising surrogate models" (Section 7) depend on this single-run comparison. At minimum, the 3D ranking should be reported as preliminary, and the conclusion should be correspondingly qualified.
  2. [Section 5.1, Table 10] The comparison between BSMS-GNN and Transolver++ is confounded by unequal numbers of gradient updates. BSMS-GNN, UPT, and GAOT were trained with batch size 1, while Transolver++ used batch size 4 to "keep its training time feasible." Over 10,000 epochs with 105 training samples, this yields roughly 1.05M parameter updates for BSMS-GNN versus roughly 262.5k for Transolver++ — a factor of 4. The reported 32–34% error gap (Section 5.1) may therefore reflect training effort rather than architectural superiority. A fair comparison should equalize the number of gradient updates (or use repetitions over several update budgets) and should report the update count for each model.
  3. [Section A.1 and Section 6] The 3D setup choices are not demonstrably fair to all architectures. Section 6 notes that prior experience with BSMS-GNN "exceeds the experience present with all other three architectures," and Appendix A.1 shows that BSMS-GNN uses the dataset-specific adjacency matrix, while the other models' 3D hyperparameters (e.g., UPT's 20,480 supernodes, GAOT's 131,072 supernodes, Transolver++'s four layers and hidden size 256) are chosen by "previous experiences" or "to the best of our knowledge." Because the paper recommends BSMS-GNN, the absence of a documented process for equal tuning effort (e.g., a small validation-based search or ablations for all models) is a load-bearing concern. The 3D ranking should be labeled as conditional on the selected hyperparameters, or the authors should provide evidence that the ranking is robust to reasonable alternative settings.
minor comments (5)
  1. [Appendix A.2] The RMSE and Rel. L2 metrics are defined as the mean over samples of the per-sample values. This is an unconventional choice; readers may expect a pooled RMSE over all points. Please state explicitly that these are per-sample averages and justify why this is preferable for this benchmark.
  2. [Section 4.1] UPT received a second round of hyperparameter optimization and a custom exponential-decay scheduler, while the other models did not receive comparable additional tuning. This is a reasonable response to the initially poor performance, but the asymmetry should be discussed as a possible source of bias, and the final ranking should be interpreted with this in mind.
  3. [Section 5] The caveat that the 3D experiment has no hyperparameter optimization and no repetition runs appears only in Section 5. Given that the abstract and conclusion emphasize the 3D results, this limitation should be stated in the abstract and revisited in the conclusion so that readers are not misled.
  4. [Throughout] Several typos and minor language issues: "quadradic" (Section 2), "archtitecture" (Section 1), "tupel" (Section 2), "noticable" (Section 5.1), "opimization" (Section 4.1). A careful proofread is needed.
  5. [Tables 11–18] The appendix tables are detailed and helpful. However, the connection between the entries and the corresponding paragraphs in Section A.1 could be made clearer by referring to table numbers in the text (e.g., "see Table 11 for BSMS-GNN").

Circularity Check

0 steps flagged

No circularity: an empirical benchmark with fresh held-out evaluations; self-citations are to the models/datasets under test, not load-bearing support.

full rationale

This paper is an empirical benchmark, not a derivation, so the equation-level circularity patterns do not apply. The central ranking claims are produced by training the compared architectures on the stated datasets and evaluating on held-out test samples (2D: ten instantiations with confidence intervals; 3D: single runs explicitly disclosed as statistically non-decisive). No fitted parameter is renamed as a prediction, and no result is defined in terms of itself. The self-citations (Hines and Bekemeyer 2026 for BSMS-GNN; Bekemeyer et al. 2025 for the datasets) reference the architectures and data objects under evaluation rather than supplying evidence for the ranking; the comparisons are run fresh against public baselines. The paper even flags its own limitations (“we abstained from doing repetition runs”, “the results presented in Table 9 do not allow us to make any statistical statement”, and the greater prior experience with BSMS-GNN). These are legitimate robustness/fairness concerns, not circularity. The 3D single-run and unequal-batch-size issues are threats to the strength of the ranking, but they do not make the benchmark’s output equivalent to its inputs by construction. Accordingly, no circular step is present and the score is 0.

Axiom & Free-Parameter Ledger

6 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities. Its conclusions rest on many hand-picked experimental settings: unequal training budgets, transferred hyperparameters, pooling-radius choices, and single-run 3D evaluations. The counts above show that the central benchmark claim is sensitive to configuration choices as much as to architecture quality.

free parameters (6)
  • 2D training epochs per model = Transolver 2,500; BSMS-GNN 10,000; UPT 15,000; GAOT 5,000
    Chosen from initial runs or HPO to reach a 'fully trained state'; unequal training budgets affect the fair comparison of final MAE (Table 7, Appendix A.1).
  • 3D batch size = BSMS-GNN/UPT/GAOT = 1; Transolver++ = 4
    Batch sizes were chosen for memory/time feasibility, not matched across models; this affects gradient noise and convergence and can change the ranking (Section 5, Table 10).
  • 3D UPT supernodes/latent tokens = 20,480 supernodes, 5,120 latent tokens
    Scaled by factor 10 from Alkin et al.'s settings based on mesh-point count; not optimized for the NASA CRM task and may disadvantage UPT (Appendix A.1, Table 15).
  • 3D GAOT supernodes grid and pooling radius = Grid [64,64,32], radius 0.066
    Taken from author suggestions and 2D experience; authors note attempts to increase radius/supernodes had the opposite effect, but no 3D HPO was run (Table 17, Section A.1).
  • 2D UPT/GAOT pooling radii = UPT 0.062; GAOT 0.066/0.072
    Derived to provide ~24 neighbors per supernode under an arc-length approximation and uniform-node assumption; the performance comparison depends on this design choice (Appendix A.3).
  • BSMS-GNN 3D graph construction = Dataset adjacency (~4 neighbors), 11 scales, min-distance-to-center seeding
    Chosen from prior in-house experience with the model; the authors have more tuning experience with BSMS-GNN than with the other architectures (Tables 11-12; Section 6).
axioms (4)
  • domain assumption The DLR TAU RANS + Spalart-Allmaras CFD solutions in the two datasets are treated as ground truth for pressure and lift.
    No independent verification or uncertainty quantification of the CFD target data is provided; all accuracy claims are relative to these simulations (Sections 4 and 5, Tables 1 and 8).
  • ad hoc to paper The ~24-neighbors-per-supernode guidance from Alkin et al. (2024) is a valid design choice for UPT and GAOT.
    Appendix A.3 uses this guidance to set pooling radii; if the guidance is not appropriate for these geometries, the models' performance is not representative.
  • ad hoc to paper 2D hyperparameter findings transfer to the 3D NASA CRM dataset, with ad hoc scaling where needed.
    Section 5 explicitly skips 3D hyperparameter optimization and transfers/adjusts settings; this is an untested assumption that directly affects the 3D ranking.
  • ad hoc to paper A single training run per model on NASA CRM, with gaps much larger than the 2D confidence intervals, is sufficient to infer the 3D ranking.
    Section 5.1 states no statistical statement is possible but uses gap magnitudes as evidence; if run-to-run variance is large, the ordering could flip.

pith-pipeline@v1.3.0-alltime-deepseek · 36045 in / 13491 out tokens · 127789 ms · 2026-08-02T03:27:48.642742+00:00 · methodology

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read the original abstract

Surrogate models are used to substitute classical numerical solvers in engineering applications where the computational cost of the latter becomes infeasible. For instance, in aerodynamics such models offer cost-effective alternatives to computational fluid dynamics in problems such as shape optimization and load analysis, which oftentimes require high-fidelity simulations for a multitude of different parameter combinations. A specific class of deep learning-based surrogate models termed operator learning models directly approximates the solution operators to the partial differential equations underlying the physical phenomenon, thereby learning to replicate solutions to entire families of problems. However, while nowadays numerous architectures of this type get published, corresponding benchmark studies remain scarce. In this article, we advance the study of AI-based surrogate methods by thoroughly benchmarking four state-of-the-art operator learning models on their aptitude for applications in aerospace engineering. In two experiments, we assess the models' capabilities of predicting the surface pressure distribution on two-dimensional airfoil shapes of varying complexity and on an industrial-scale three-dimensional aircraft configuration. Thereby, we evaluate the models' abilities to fulfill frequent requirements in aerodynamics such as capturing discontinuities (shocks) in the solutions, scalability towards excessive amounts of mesh points and handling of data scarcity. Accompanied by a careful analysis, our findings drive forward the field of AI-based surrogate modeling by providing detailed insights into the strengths and weaknesses of the individual architectures, thus allowing to identify priorities for future developments. In particular the Bi-Stride Multi-Scale Graph Neural Network and Transolver(++) are highlighted as promising surrogate models for aerodynamical applications.

Figures

Figures reproduced from arXiv: 2607.13866 by Derrick Hines, Jan Scherz, Philipp Bekemeyer.

Figure 1
Figure 1. Figure 1: Parallel coordinates plots visualizing the hyperparameter optimizations for the individual models. Each colored line corresponds to a specific combination of hyperparameters and the resulting MAE on the validation set. : Preprint submitted to Elsevier Page 25 of 45 [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Predictions of surface pressure distributions on three chosen test samples of the 2D airfoil dataset by UPT, GAOT, Transolver and BSMS-GNN. Each plot shows the ground truth (solid line) and the mean of the predictions (dashed lines) of ten fully trained instantiations of each model with optimal hyperparameters in accordance with Section 4.1. The test samples correspond to the flow parameters 𝑀 ∶ 0.490, 𝐴𝑜𝐴… view at source ↗
Figure 3
Figure 3. Figure 3: Predictions and variance of surface pressure distributions by UPT, GAOT, Transolver and BSMS-GNN on the test sample corresponding to the flow parameters 𝑀 ∶ 0.734, 𝐴𝑜𝐴 ∶ 3.758, 𝑅𝑒 ∶ 4.701 ∗ 106 . Each plot shows the ground truth (solid line) and the predictions (dashed lines) of ten fully trained instantiations of the respective model. : Preprint submitted to Elsevier Page 27 of 45 [PITH_FULL_IMAGE:figure… view at source ↗
Figure 4
Figure 4. Figure 4: Average of the MAEs attained by the ten fully trained instantiations of each model on all 100 test samples of the 2D airfoil dataset (log plot). The MAEs are displayed in dependence of the angle of attack 𝐴𝑜𝐴 and the Mach number 𝑀 of the samples, neglecting the influence of the airfoil geometry and the Reynolds number. : Preprint submitted to Elsevier Page 28 of 45 [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Predictions of the lift coefficient 𝐶𝑙 of BSMS-GNN, Transolver, UPT and GAOT and difference Δ between the predicted values and the ground truth on 17 additional datasamples corresponding to the original RAE2822 airfoil geometry at 𝑀 = 0.2, 𝑅𝑒 = 106 and 𝐴𝑜𝐴 varying between −3.0 and 5.0. The plotted values correspond to the mean of the 𝐶𝑙 predictions of the ten fully trained instantiations of each architectu… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the training loss (log plot) of the individual models on the full 2D airfoil training set. Each curve corresponds to the loss of the respective model’s instantiation achieving the one below median MAE in the ten repetition runs evaluated in Section 4.2. BSMS-GNN, UPT and GAOT were trained based on mean squared loss; for Transolver relative 𝐿2 -loss was used. : Preprint submitted to Elsevier Pa… view at source ↗
Figure 7
Figure 7. Figure 7: Predictions and errors of all models on a test sample corresponding to a Mach number of 0.547, an angle of attack of 1.265, an outboard aileron angle of 14.240, an inboard aileron angle of 11.603, an elevator angle of 9.835 and an HTP angle of −0.864. The upper left plot shows the ground truth pressure distribution on the right wing of the aircraft configuration, the two plots to its right display the mode… view at source ↗
Figure 8
Figure 8. Figure 8: Predictions and errors of all models on a test sample corresponding to a Mach number of 0.722, an angle of attack of 3.457, an outboard aileron angle of 1.360, an inboard aileron angle of −12.420, an elevator angle of 2.893 and an HTP angle of 0.959. : Preprint submitted to Elsevier Page 32 of 45 [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Predictions and errors of all models on a test sample corresponding to a Mach number of 0.877, an angle of attack of 4.383, an outboard aileron angle of 1.648, an inboard aileron angle of 2.274, an elevator angle of −9.654 and an HTP angle of −1.740. : Preprint submitted to Elsevier Page 33 of 45 [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mean absolute errors of the fully trained instantiations of BSMS-GNN, Transolver++, UPT and GAOT on all 44 test samples of the NASA CRM dataset (log plot); plotted in dependence of the angle of attack 𝐴𝑜𝐴 and the Mach number 𝑀. : Preprint submitted to Elsevier Page 34 of 45 [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the training loss (log plot) of the individual models on the NASA CRM dataset. For all models mean squared loss was used as the loss function [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The arc 𝐴 of the surface curve 𝛾 lying within the pooling radius 𝑟 around the supernode ̃𝑥. : Preprint submitted to Elsevier Page 35 of 45 [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: a) Surface point 𝑥 with furthest possible distance 𝑑̂ = √1 2 𝑑 to a supernode ̃𝑥. b) The arc 𝐴 of the surface curve 𝛾 lying within the pooling radius 𝑟 around the supernode ̃𝑥0 . : Preprint submitted to Elsevier Page 36 of 45 [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗

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Reference graph

Works this paper leans on

23 extracted references · 2 canonical work pages

  1. [3]

    Ab-upt: Scaling neural cfd surrogates for high-fidelity automotive aerodynamics simulations via anchored-branched universal physics transformers.https://doi.org/10.48550/arXiv.2502. 09692. Bonnet, F., Mazari, J., Cinnella, P., Gallinari, P.,

    doi:10.48550/arxiv.2502

  2. [5]

    URL:https://mlanthology.org/iclr/2022/brandstetter2022iclr-message/

    Message passing neural pde solvers, in: International Conference on Learning Representations. URL:https://mlanthology.org/iclr/2022/brandstetter2022iclr-message/. :Preprint submitted to Elsevier Page 22 of 45 Evaluation of SOTA Deep Learning Architectures for Aerodynamical Predictions Cao, S.,

    2022

  3. [8]

    72525–72624

    Poseidon: Efficient foundation models for pdes, in: Advances in Neural Information Processing Systems, pp. 72525–72624. doi:10.52202/079017-2311. Hines, D., Bekemeyer, P.,

    doi:10.52202/079017-2311

  4. [9]

    Liu, S., Yu, Y., Zhang, T., Liu, H., Liu, X., Meng, D.,

    doi:10.1145/3648506. Liu, S., Yu, Y., Zhang, T., Liu, H., Liu, X., Meng, D.,

    doi:10.1145/3648506

  5. [11]

    JournalofMachineLearningResearch24,1–97

    Neural operator: Learning maps between functionspaceswithapplicationstopdes. JournalofMachineLearningResearch24,1–97. URL:http://jmlr.org/papers/v24/21-1524. html. Li,Z.,Huang,D.Z.,Liu,B.,Anandkumar,A.,2023a. Fourierneuraloperatorwithlearneddeformationsforpdesongeneralgeometries. Journalof Machine Learning Research 24, 1–26. URL:http://jmlr.org/papers/v24...

    Pith/arXiv arXiv doi:10.48550/arxiv.2010.08895 2010

  6. [12]

    2205.13671

    Transformer for partial differential equations’ operator learning.https://doi.org/10.48550/arXiv. 2205.13671. Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, B., Azizzadenesheli, K., Anandkumar, A.,

    Pith/arXiv arXiv doi:10.48550/arxiv.2205.13671

  7. [14]

    Neurocomputing 648, 130518

    Architectures, variants, and performance of neural operators: A comparative review. Neurocomputing 648, 130518. doi:https://doi.org/10.1016/j.neucom.2025.130518. Lu, L., Jin, P., Karniadakis, G.,

    arXiv 2025

  8. [16]

    150039–150101

    Rigno: A graph-based framework for robust and accurate operator learning for pdes on arbitrary domains, in: Advances in Neural Information Processing Systems, pp. 150039–150101. URL:https:// proceedings.neurips.cc/paper_files/paper/2025/file/dcb91f43033bb1d367d1848806dee98d-Paper-Conference.pdf. Pfaff, T., Fortunato, M., Sanchez-Gonzalez, A., Battaglia, P.,

    2025

  9. [18]

    JournalofComputationalPhysics378,686–707

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problemsinvolvingnonlinearpartialdifferentialequations. JournalofComputationalPhysics378,686–707. doi:https://doi.org/10.1016/ j.jcp.2018.10.045. Ranade,R.,Nabian,M.,Tangsali,K.,Kamenev,A.,Hennigh,O.,R.,C.,Choudhry,S.,2025. Domino:Adecomposablemulti-scaleiterativen...

    Pith/arXiv arXiv doi:10.48550/arxiv.2501.13350 2018

  10. [19]

    doi:10.52202/075280-3376

    Convolutional neural operators for robustandaccuratelearningofpdes,in:AdvancesinNeuralInformationProcessingSystems,pp.77187–77200. doi:10.52202/075280-3376. Roubíček, T.,

    work page doi:10.52202/075280-3376

  11. [20]

    V03BT03A049

    Blendednet: A blended wing body aircraft dataset and surrogate model for aerodynamic predictions, in: ASME 2025 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, p. V03BT03A049. doi:10.1115/DETC2025-168977. Vaswani,A.,Shazeer,N.,Parmar,N.,Uszkoreit,J.,Jones,L.,Gomez,A.N.,Kaiser,L.,Polosukhin,I....

    doi:10.48550/arxiv.2505.18781 2025

  12. [21]

    Zhou, H., Wu, H., Shangguan, H., Ma, Y., Weng, H., Wang, J., Long, M.,

    Transolver: A fast transformer solver for pdes on general geometries.https: //doi.org/10.48550/arXiv.2402.02366. Zhou, H., Wu, H., Shangguan, H., Ma, Y., Weng, H., Wang, J., Long, M.,

    Pith/arXiv arXiv doi:10.48550/arxiv.2402.02366

  13. [22]

    Transolver-3: Scaling up transformer solvers to industrial-scale geometries.https://doi.org/10.48550/arXiv.2602.04940. :Preprint submitted to Elsevier Page 24 of 45 Evaluation of SOTA Deep Learning Architectures for Aerodynamical Predictions Figure 1:Parallel coordinates plots visualizing the hyperparameter optimizations for the individual models. Each co...

    Pith/arXiv arXiv doi:10.48550/arxiv.2602.04940

  14. [72]

    Hyperparameters (2nd round) Values Max. learning rate 0.0001,0.0002, 0.001 (Epochs, Decay rate) (5000, 0.00080), (5000, 0.00120), (10000, 0.00040), (10000, 0.00060), (15000, 0.00026),(15000, 0.00040) Table 5 Evaluated hyperparameter combinations for GAOT. The choices leading to the best result marked in bold. Hyperparameters Values Max. learning rate 0.00...

    2025

  15. [1998]

    American Mathematical Society, Providence, Rhode Island

    Partial Differential Equations. American Mathematical Society, Providence, Rhode Island. Hao,Z.,Wang,Z.,Su,H.,Ying,C.,Dong,Y.,Liu,S.,Cheng,Z.,Song,J.,Zhu,J.,2023. GNOT:Ageneralneuraloperatortransformerforoperator learning, in: Proceedings of the 40th International Conference on Machine Learning, pp. 12556–12569. URL:https://proceedings.mlr. press/v202/hao...

    2023

  16. [2019]

    Luo,H.,Wu,H.,Zhou,H.,Xing,L.,Di,Y.,Wang,J.,Long,M.,2025

    Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators.https://doi.org/10.1038/s42256-021-00302-5. Luo,H.,Wu,H.,Zhou,H.,Xing,L.,Di,Y.,Wang,J.,Long,M.,2025. Transolver++:Anaccurateneuralsolverforpdesonmillion-scalegeometries. https://doi.org/10.48550/arXiv.2502.02414. Mousavi,...

    Pith/arXiv arXiv doi:10.48550/arxiv.2502.02414 2025

  17. [2020]

    Quarteroni, A., Valli, A.,

    Learning mesh-based simulation with graph networks.https://doi.org/ 10.48550/arXiv.2010.03409. Quarteroni, A., Valli, A.,

    Pith/arXiv arXiv doi:10.48550/arxiv.2010.03409 2010

  18. [2021]

    24924–24940

    Choose a transformer: Fourier or galerkin, in: Advances in Neural Information Processing Systems, pp. 24924–24940. URL: https://proceedings.neurips.cc/paper_files/paper/2021/file/d0921d442ee91b896ad95059d13df618-Paper.pdf. Cao,Y.,Chai,M.,Li,M.,Jiang,C.,2023. Efficientlearningofmesh-basedphysicalsimulationwithbi-stridemulti-scalegraphneuralnetwork,in: Proc...

    doi:10.48550/arxiv.2512.07847 2021

  19. [2022]

    23463–23478

    Airfrans: High fidelity computational fluid dynamics dataset for approximating reynolds- averaged navier–stokes solutions, in: Advances in Neural Information Processing Systems, pp. 23463–23478. doi:10.52202/068431-1705. Brandstetter, J., Worrall, D., Welling, M.,

    work page doi:10.52202/068431-1705

  20. [2023]

    Aerospace Science and Technology 137, 108268

    Graph neural networks for the prediction of aircraft surface pressure distributions. Aerospace Science and Technology 137, 108268. doi:https://doi.org/10.1016/j.ast.2023.108268. Hines, D., Bekemeyer, P.,

    arXiv 2023

  21. [2024]

    25152–25194

    Universal physics transformers: A framework for efficiently scaling neural operators, in: Advances in Neural Information Processing Systems, pp. 25152–25194. doi:10.52202/079017-0793. Ashton, N., Mockett, C., Fuchs, M., Fliessbach, L., Hetmann, H., Knacke, T., Schonwald, N., Skaperdas, V., Fotiadis, G., Walle, A., Hupertz, B., Maddix,D.,2024. Drivaerml:Hi...

    Pith/arXiv arXiv doi:10.48550/arxiv.2408.11969 2024

  22. [2025]

    doi:10.2514/6.2025-0036

    Introduction of applied aerodynamics surrogate modeling benchmark cases, in: AIAA SCITECH 2025 Forum. doi:10.2514/6.2025-0036. Bleeker, M., Dorfer, M., Kronlachner, T., Sonnleitner, R., Alkin, B., Brandstetter, J.,

    doi:10.2514/6.2025-0036 2025

  23. [2026]

    Computers & Fluids 308, 106979

    Prediction of surface pressure distributions of non-parametric airfoils using geometric deep learning methods. Computers & Fluids 308, 106979. doi:https://doi.org/10.1016/j.compfluid.2026.106979. Kovachki, N., Li, Z., Liu, B., Azizzadenesheli, K., Bhattacharya, K., Stuart, A., Anandkumar, A.,

    arXiv 2026