Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

A Geometry-Aware Message Passing Neural Network for Modeling Aerodynamics over Airfoils

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Airfoil flow fields can be predicted by a neural network that encodes the airfoil's surface shape and propagates it to every mesh point, training on downsampled meshes without losing accuracy on full-resolution meshes.

desk verdict Competition-grade airfoil RANS surrogate; Surf2Vol message passing is a real contribution, and the untested full-mesh expressiveness question is a non-fatal gap. read the letter →

arxiv 2412.09399 v2 pith:KSBUPC7Y submitted 2024-12-12 cs.LG

Jacob Helwig , Xuan Zhang , Haiyang Yu , Shuiwang Ji This is my paper · ORCID
classification cs.LG
keywords geometry-awaremessagepassingairfoilaerodynamicsRANSsurrogatemodelsurface-to-volumemeshdownsamplinggeneralizationcoordinatesystemembeddingsinletvelocitycanonicalizationlog-pressuretransform
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a neural network can map an airfoil's shape and inflow conditions directly to the steady flow around it, replacing expensive CFD solves in design loops. Its model, GeoMPNN, first learns a latent graph representation of the airfoil surface, then propagates that representation to all volume points through directed surface-to-volume message passing. Because volume points never exchange messages with each other, the model can be trained on randomly downsampled meshes and evaluated on the full mesh without a distribution shift in neighborhood structure. The paper also shows that several physically motivated input treatments, including trailing-edge coordinates, hybrid polar-Cartesian features, sinusoidal and spherical-harmonic embeddings, rotation into an inlet-aligned frame, and a log transform of pressure, each improve accuracy. Validated on the AirfRANS benchmark, the method outperforms previous surrogates and produces accurate lift and drag estimates.

What carries the argument

The central mechanism is surface-to-volume message passing (Surf2Vol): a directed bipartite graph in which every volume point connects to its eight nearest neighbors on the airfoil surface. Surface points first exchange messages among themselves through four layers of a standard graph network to form a latent geometric representation; that representation is then injected into each directed edge and aggregated into the volume point's node embedding. This one mechanism carries the airfoil shape into the flow field, and by removing edges between volume points it makes the model resolution-independent during training. Supporting it are the hybrid coordinate systems (leading-edge and trailing-edge origins, polar angles computed against four reference axes), sinusoidal and spherical-harmonic basis embeddings of those coordinates, the inlet-velocity rotation, and the log-pressure target.

What would settle it

Run GeoMPNN and a variant that adds volume-volume message passing on the full-resolution test mesh, then compare errors in the wake region downstream of the trailing edge on high-angle-of-attack cases; if the variant is clearly more accurate there, the claim that surface-to-volume message passing alone suffices is wrong.

Watch

Extended reading notes

Core claim

GeoMPNN predicts the four steady-state RANS fields, namely two velocity components, reduced pressure, and turbulent viscosity, by treating the airfoil surface as the source of all geometric information. After four layers of message passing over surface points, each volume point aggregates features from its eight nearest surface neighbors, so the message entering a point encodes the shape of the airfoil region closest to it. The architecture deliberately omits volume-to-volume message passing; since each volume point's neighborhood consists only of surface points, the neighborhood is unchanged when the volume mesh is subsampled, which is why training on 32K sampled points transfers to full-resolution evaluation. The input representation gives the model explicit ways to separate freestream, airfoil, and downstream regions and to tell above from below, and rotating the coordinate system so the inlet velocity aligns with the x-axis makes the turbulent-viscosity field a much more concentrated target. Together these choices improve accuracy on the AirfRANS benchmark, with the largest gains in physics-compliance and out-of-distribution scores.

Load-bearing premise

The load-bearing premise is that volume-to-volume communication is unnecessary, meaning each mesh point's steady-state flow value is determined by the nearby airfoil surface and its own position, and the paper supports this with one ablation on a subsampled test set rather than a first-principles argument.

Editorial extensions

If this is right

  • Training can be done on roughly 32K randomly sampled mesh points per example per epoch, with full-mesh prediction accuracy matching subsampled accuracy, lowering memory and compute costs.
  • Models trained this way transfer to out-of-distribution airfoil geometries and flow conditions, with the largest relative gains in physics-compliance and OOD scores.
  • Accurate pressure prediction under the log transform yields better lift-coefficient estimates, which is what makes the surrogate usable for airfoil design optimization.
  • The small benefit of adding volume-volume message passing suggests that for steady-state RANS over airfoils, surface-surface and surface-volume interactions dominate.
  • The inlet-velocity canonicalization reduces the difficulty of modeling turbulent viscosity, implying a preprocessing step that other rotation-sensitive flow surrogates could adopt.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If surface-to-volume message passing is truly sufficient for this problem, the same recipe may apply to other steady-state PDEs whose inputs are mostly positional with global parameters, such as steady heat conduction or Darcy flow; this is a testable extension the paper does not run.
  • Because the architecture's neighborhoods are independent of volume-mesh density, it may naturally support adaptive mesh refinement or variable-resolution training without retraining, which the paper does not demonstrate.
  • The inlet-velocity rotation is a form of frame canonicalization; a natural follow-up is to test whether rotating other fields or their gradients into the same frame improves accuracy beyond turbulent viscosity and pressure.
  • The paper evaluates on the AirfRANS dataset only; a stronger test of the resolution-invariance claim would be to run the same architecture on a different mesh family or on three-dimensional geometries, where wake interactions are richer.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes GeoMPNN, a message passing neural network for surrogate modeling of steady-state incompressible RANS flows over airfoils. The architecture first constructs a latent graph on the airfoil surface via standard message passing, then propagates this representation to all volume mesh points through directed surface-to-volume edges, deliberately omitting volume-to-volume message passing. The model adds physically motivated input features: a trailing-edge coordinate system, polar and spherical-harmonic angle embeddings, sinusoidal positional embeddings, an inlet-velocity canonicalization of coordinates, and a log transform for the pressure target. The method is evaluated on the AirfRANS benchmark in the NeurIPS 2024 ML4CFD Competition, where it won the Best Student Submission award and placed fourth overall. The authors report progressive improvements from MLP and GNN baselines through the S2V, TRAIL, POLAR, SINE, SPH, INLET, and final GeoMPNN variants, with each experiment repeated 8 times, and they demonstrate that training on 32K subsampled mesh points does not degrade accuracy when evaluating on the full mesh.

Significance. If the claims hold, GeoMPNN is a practical and effective neural surrogate for airfoil aerodynamics, with a simple and computationally efficient mechanism for training on subsampled meshes without test-time distribution shift. The paper's strengths include controlled ablations that isolate each design choice, repeated runs on a public benchmark, a public code release in the AIRS library, and a strong competition result. The inlet-velocity canonicalization is a deterministic coordinate transformation rather than a fitted parameter, and no derivation is fit to the target, so circularity is not a concern. The main weaknesses are that the central architectural claim about the unimportance of volume-volume interactions is tested only at training-like resolution, not at the advertised full-mesh deployment regime, and that some close incremental comparisons lack statistical or effect-size support.

major comments (4)
  1. [Section 4.2.1, Figure 6] The conclusion that "the most important interactions in this problem are surface-surface and surface-volume" is based on comparing S2V with S2V+GNN on the test set randomly subsampled to 32K points, matching the training resolution. This does not test the deployment regime advertised in the abstract and Section 3.1, namely evaluation on the full mesh. At full resolution the volume graph is denser, and because the pressure field in incompressible steady RANS is determined by a globally coupled constraint, volume-volume message passing could in principle have a larger benefit at full resolution than at 32K. Please report the same ablation at full test resolution, or at a sweep of resolutions, or explicitly restrict the architectural claim to the subsampled setting. This is load-bearing because the omission of volume-volume message passing is a central design decision.
  2. [Sections 4.4 and 4.5, Figures 15 and 17] Several incremental claims are supported only by point estimates or box plots with substantial overlap across the 8 runs. For example, in Figure 15 the statement that "the SINE model achieves a better ML Score" and "the SPH model achieves a new best error on the pressure field" would be more convincing with significance tests or confidence intervals, especially because the differences appear comparable to run-to-run variability. The same applies to the comparisons in Figure 17. Please add statistical summaries or explicitly characterize the magnitude and uncertainty of the reported improvements.
  3. [Figures 3 and 5, Section 4.1.2 and 4.2.1] The turbulent viscosity field is omitted from both resolution-shift figures, with an explanation given only for Figure 3 (overfitting). Section 4.2.1 states that "the SURF2VOL model error does not change with increased resolution," but this is not empirically verified for the turbulent viscosity field, which is one of the four predicted quantities and is specifically targeted by the inlet-velocity canonicalization in Section 4.5.1. Either report the νt result at subsampled and full resolution or explicitly scope the resolution-invariance claim to the three shown fields.
  4. [Equations (9) and (10)] The stated formula for the m=0 spherical harmonics, Y_l^0(θ) = sqrt((2l+1)!/4π) P_l(cos θ), is not the standard normalization; the conventional expression is sqrt((2l+1)/(4π)) P_l(cos θ). The same issue appears in the odd-harmonic definition in Equation (10). Please correct the formula and confirm that the code and the paper use the same normalization, as this affects reproducibility of the SPH and downstream models.
minor comments (5)
  1. [Section 4.2.1] The hyperparameters of the added volume message passing layers in the S2V+GNN ablation (radius, number of layers, hidden dimensions, neighbor sampling threshold) are not specified, which makes the ablation difficult to reproduce exactly.
  2. [Figure 3 caption] The caption explains that turbulent viscosity is omitted due to overfitting, but the corresponding text in Section 4.1.2 says the error increase is observed "on each field" without this caveat; please align the text with the figure.
  3. [Section 4.4.1] The notation PE(x) is used both for scalars and, via Equation (8), for 2D vectors. This is understandable but slightly confusing; a distinct symbol for the vectorized version would improve clarity.
  4. [Section 4.5.1] The matrix Rv is described as belonging to O(2), but the displayed form is a proper rotation with determinant +1, i.e., an element of SO(2); this is a minor mathematical wording issue.
  5. [General] The manuscript uses "ML Score", "PHYSICS SCORE", "OOD Score", and "GLOBAL SCORE" with inconsistent capitalization; please standardize for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: GeoMPNN's design choices are validated on held-out AirfRANS benchmark data, with deterministic input reparameterizations and external competition results rather than fitted predictions.

full rationale

The paper's central claim is that the GeoMPNN architecture improves aerodynamic surrogate modeling on AirfRANS. Each proposed component—Surf2Vol message passing, trailing-edge coordinates, hybrid polar-Cartesian features, sinusoidal and spherical-harmonic embeddings, inlet-velocity canonicalization, and log-pressure prediction—is either a fixed, closed-form transformation of the input features or is assessed by controlled comparisons on held-out in-distribution and OOD test sets. No parameter is fitted to the target fields and then renamed as a prediction. The inlet-velocity rotation Rv is a deterministic function of the input vector v∞, and the log-pressure transform is invertible, so neither encodes the answer by construction. The volume-volume ablation (Figure 6) is performed on a subsampled test set, which is a limitation in the strength of an empirical conclusion, not a definitional circularity: the claim that volume-volume interactions are unnecessary is supported by experiment rather than being equivalent to the model's inputs. Self-citations (Helwig et al. 2023; Zhang et al. 2024; Lin et al. 2024; Liu et al. 2022) appear only as background references for FNOs, SinENet, frame averaging, and spherical message passing; none is load-bearing, and no uniqueness theorem is imported from the authors' prior work. The NeurIPS 2024 ML4CFD Competition result is an external benchmark, providing independent validation. No circular step can be quoted or exhibited.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper contributes an empirical architecture; no new physical entities are introduced. Its central claim rests on several hand-chosen hyperparameters (k, r, L, M, nbasis), a hand-chosen log-pressure transform, and domain assumptions about the AirfRANS dataset and the sufficiency of surface-to-volume information flow. These are plausible but not independently grounded.

free parameters (6)
  • k (Surf2Vol nearest neighbors) = 8
    Number of surface points aggregated into each volume node message; chosen by hand, central to Surf2Vol message passing (Section 4.2).
  • Surface radius graph radius r = 0.05
    Radius for building the latent surface graph and GNN baseline neighbor graph (Sections 3.1 and 4.2).
  • Surface message passing layers L = 4
    Depth of the geometric encoding GNN over the airfoil surface (Section 4.2).
  • Neighbor sampling threshold M = 8 (surface graph) / 4 (baseline GNN)
    Max neighbors sampled in radius graphs (Sections 3.1 and 4.2).
  • Basis dimension nbasis = 8
    Dimension of sinusoidal and spherical harmonic embedding bases (Section 4.4).
  • Pressure log-transform = q = sign(p) * log(|p|+1)
    Hand-chosen nonlinear reparameterization of the pressure target to reduce variance (Section 4.5.2).
assumptions (5)
  • domain assumption Steady-state incompressible RANS equations govern the data
    The AirfRANS dataset is generated by solving these equations; the surrogate inherits any inaccuracies in the solver (Sections 1-2).
  • domain assumption AirfRANS training set (103 solutions) is representative
    The model is trained only on these solutions; the OOD test set may include geometries and conditions outside this range (Section 2).
  • ad hoc to paper Volume-volume interactions can be neglected
    The SURF2VOL model omits message passing between volume nodes; this is justified empirically in Section 4.2.1 but is a modeling choice that could limit accuracy if false.
  • ad hoc to paper Inlet-velocity canonicalization is a valid symmetry
    Rotating inputs by R_v is assumed to preserve all information needed to predict scalar fields and to simplify the learning task (Section 4.5.1).
  • standard math Standard ML components work as expected
    Message passing, sinusoidal embeddings, and spherical harmonics are taken as given (Sections 4.2-4.4).

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Geometry-Aware Message Passing Neural Network for Modeling Aerodynamics over Airfoils." pith.science (2026). https://pith.science/paper/KSBUPC7Y

@misc{pith2026241209399,
  author       = {Pith},
  title        = {Pith review of: A Geometry-Aware Message Passing Neural Network for Modeling Aerodynamics over Airfoils},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSBUPC7Y}},
  note         = {Machine review of arXiv:2412.09399}
}
read the original abstract

Computational modeling of aerodynamics is a key problem in aerospace engineering, often involving flows interacting with solid objects such as airfoils. Deep surrogate models have emerged as purely data-driven approaches that learn direct mappings from simulation conditions to solutions based on either simulation or experimental data. Here, we consider modeling of incompressible flows over solid objects, wherein geometric structures are a key factor in determining aerodynamics. To effectively incorporate geometries, we propose a message passing scheme that efficiently and expressively integrates the airfoil shape with the mesh representation. Under this framework, we first obtain a representation of the geometry in the form of a latent graph on the airfoil surface. We subsequently propagate this representation to all collocation points through message passing on a directed, bipartite graph. We demonstrate that this framework supports efficient training by downsampling the solution mesh while avoiding distribution shifts at test time when evaluated on the full mesh. To enable our model to be able to distinguish between distinct spatial regimes of dynamics relative to the airfoil, we represent mesh points in both a leading edge and trailing edge coordinate system. We further enhance the expressiveness of our coordinate system representations by embedding our hybrid Polar-Cartesian coordinates using sinusoidal and spherical harmonics bases. We additionally find that a change of basis to canonicalize input representations with respect to inlet velocity substantially improves generalization. Altogether, these design choices lead to a purely data-driven machine learning framework known as GeoMPNN, which won the Best Student Submission award at the NeurIPS 2024 ML4CFD Competition, placing 4th overall. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Figures

Figures reproduced from arXiv: 2412.09399 by the authors.

Figure 1
Figure 1. Full mesh (left) and subsampled mesh (right). To reduce training costs, we randomly sample 32K mesh points for each training example at each epoch. Evaluation is conducted at the full resolution. The NeurIPS 2024 ML4CFD Competition follows the LIPS framework [Leyli Abadi et al., 2022] to evaluate model performance. Models are assessed along three dimensions, resulting in an overall GLOBAL SCORE out of 100. On the in… view at source ↗
Figure 2
Figure 2. Scores for the MLP and GNN baselines across 8 runs. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Difference in errors between subsampled and full resolution for MLP and GNN. We analyze [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Comparison of the SURF2VOL model to the GNN. Top: MSE errors on each field, where lower is better, and the targets are normalized as discussed in Section 2.2. Bottom: Scores in each category, where higher is better. the resolution, we compare the error on each field at…
Figure 5
Figure 5. Figure 5: Difference in errors between subsampled and full resolution for [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Results for the SURF2VOL model compared to the SURF2VOL+GNN on the subsampled test set. Although the SURF2VOL+GNN adds additional operations in the form of message passing between volume nodes following each SURF2VOL MESSAGE PASSING layer, it does not offer a substanti…
Figure 7
Figure 7. Figure 7: Spatial regimes of dynamics. Interactions between the airfoil and flow are unique in [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Leading and trailing edge coordinate systems. The leading and trailing edge coordinate [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Regions by angle in the leading edge coordinate system. Polar angles can be used to derive [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Angles in the leading edge coordinate system with respect to four different reference [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Comparison of the T [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: Sinusoidal embeddings of varying frequency for distance in the leading edge coordinate [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: Spherical harmonics embeddings of angles in the leading edge coordinate system with [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Odd harmonics embeddings of angles in the leading edge coordinate system with [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 15
Figure 15. Figure 15: Comparison of the SINE and SPH models to the POLAR model. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]
Figure 16
Figure 16. Figure 16: Turbulent viscosity and inlet velocity in the leading edge coordinate system ( [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]
Figure 17
Figure 17. Figure 17: Comparison of the INLET model to the SINE and SPH models, which previously had the best error on turbulent viscosity and the best GLOBAL SCORE, respectively [PITH_FULL_IMAGE:figures/full_fig_p014_17.png]
Figure 18
Figure 18. Figure 18: Pressure field with the least range (left) and with the greatest range (right). the x-axis. This effectively canonicalizes the coordinate system with respect to the inlet velocity, a technique which has been applied in machine learning to improve generalization [Puny …
Figure 19
Figure 19. Figure 19: Distribution of normalized pressures [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]
Figure 20
Figure 20. Figure 20: Log-transformed pressure field with the least range ( [PITH_FULL_IMAGE:figures/full_fig_p015_20.png]
Figure 21
Figure 21. Figure 21: Comparison of the GEOMPNN model to the SPH and INLET models, which previously had the best error on pressure and the best GLOBAL SCORE, respectively. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_21.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. NeurIPS 2024 ML4CFD Competition: Results and Retrospective Analysis

    cs.LG 2025-06 conditional novelty 4.0 of 10

    A retrospective of the ML4CFD competition shows a Gaussian-process-based entry outranking deep learning models and the OpenFOAM solver on a tailored multi-criteria score.

Reference graph

Works this paper leans on

28 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [1]

    P. W. Battaglia, J. B. Hamrick, V. Bapst, A. Sanchez-Gonzalez, V. Zambaldi, M. Malinowski, A. Tacchetti, D. Raposo, A. Santoro, R. Faulkner, et al. Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261, 2018

    arXiv 2018

  2. [2]

    Bonnet, J

    F. Bonnet, J. Mazari, P. Cinnella, and P. Gallinari. Airfrans: High fidelity computational fluid dynamics dataset for approximating reynolds-averaged navier--stokes solutions. Advances in Neural Information Processing Systems, 35: 0 23463--23478, 2022

    work page 2022

  3. [3]

    S. Cao. Choose a transformer: Fourier or galerkin. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors, Advances in Neural Information Processing Systems, 2021. URL https://openreview.net/forum?id=ssohLcmn4-r

    work page 2021

  4. [4]

    Casenave, B

    F. Casenave, B. Staber, and X. Roynard. Mmgp: a mesh morphing gaussian process-based machine learning method for regression of physical problems under nonparametrized geometrical variability. Advances in Neural Information Processing Systems, 36, 2024

    work page 2024

  5. [5]

    Fey and J

    M. Fey and J. E. Lenssen. Fast graph representation learning with PyTorch Geometric . In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019

    2019

  6. [6]

    Gasteiger, J

    J. Gasteiger, J. Groß, and S. Günnemann. Directional message passing for molecular graphs. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=B1eWbxStPH

    work page 2020

  7. [7]

    Gilmer, S

    J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl. Neural message passing for quantum chemistry. In International conference on machine learning, pages 1263--1272. PMLR, 2017

    work page 2017

  8. [8]

    J. K. Gupta and J. Brandstetter. Towards multi-spatiotemporal-scale generalized PDE modeling. Transactions on Machine Learning Research, 2023. ISSN 2835-8856. URL https://openreview.net/forum?id=dPSTDbGtBY

    work page 2023

Show all 28 references
  1. [9]

    Z. Hao, Z. Wang, H. Su, C. Ying, Y. Dong, S. Liu, Z. Cheng, J. Song, and J. Zhu. Gnot: A general neural operator transformer for operator learning. In International Conference on Machine Learning, pages 12556--12569. PMLR, 2023

    2023

  2. [10]

    Helwig, X

    J. Helwig, X. Zhang, C. Fu, J. Kurtin, S. Wojtowytsch, and S. Ji. Group equivariant fourier neural operators for partial differential equations. In International Conference on Machine Learning, pages 12907--12930. PMLR, 2023

    2023

  3. [11]

    D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In ICLR (Poster), 2015. URL http://arxiv.org/abs/1412.6980

    2015 arXiv

  4. [12]

    Leyli Abadi, A

    M. Leyli Abadi, A. Marot, J. Picault, D. Danan, M. Yagoubi, B. Donnot, S. Attoui, P. Dimitrov, A. Farjallah, and C. Etienam. Lips-learning industrial physical simulation benchmark suite. Advances in Neural Information Processing Systems, 35: 0 28095--28109, 2022

    2022

  5. [13]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar. Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485, 2020

    2003 arXiv

  6. [14]

    Z. Li, N. B. Kovachki, K. Azizzadenesheli, B. liu, K. Bhattacharya, A. Stuart, and A. Anandkumar. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=c8P9NQVtmnO

    2021

  7. [15]

    Z. Li, K. Meidani, and A. B. Farimani. Transformer for partial differential equations operator learning. Transactions on Machine Learning Research, 2023. ISSN 2835-8856. URL https://openreview.net/forum?id=EPPqt3uERT

    2023

  8. [16]

    Y. Lin, J. Helwig, S. Gui, and S. Ji. Equivariance via minimal frame averaging for more symmetries and efficiency. In Forty-first International Conference on Machine Learning, 2024. URL https://openreview.net/forum?id=guFsTBXsov

    2024

  9. [17]

    T. Liu. Evolutionary understanding of airfoil lift. Advances in Aerodynamics, 3 0 (1): 0 37, 2021

    2021

  10. [18]

    Y. Liu, L. Wang, M. Liu, Y. Lin, X. Zhang, B. Oztekin, and S. Ji. Spherical message passing for 3d molecular graphs. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=givsRXsOt9r

    2022

  11. [19]

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3 0 (3): 0 218--229, 2021

    2021

  12. [20]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019

    2019

  13. [21]

    O. Puny, M. Atzmon, E. J. Smith, I. Misra, A. Grover, H. Ben-Hamu, and Y. Lipman. Frame averaging for invariant and equivariant network design. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=zIUyj55nXR

    2022

  14. [22]

    L. N. Smith and N. Topin. Super-convergence: Very fast training of neural networks using large learning rates. In Artificial intelligence and machine learning for multi-domain operations applications, volume 11006, pages 369--386. SPIE, 2019

    2019

  15. [23]

    A. Tran, A. Mathews, L. Xie, and C. S. Ong. Factorized fourier neural operators. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=tmIiMPl4IPa

    2023

  16. [24]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems, 2017

    2017

  17. [25]

    E. W. Weisstein. Spherical harmonic. https://mathworld. wolfram. com/, 2004

    2004

  18. [26]

    Yagoubi, D

    M. Yagoubi, D. Danan, M. Leyli-Abadi, J.-P. Brunet, J. A. Mazari, F. Bonnet, A. Farjallah, P. Cinnella, P. Gallinari, M. Schoenauer, et al. Neurips 2024 ml4cfd competition: Harnessing machine learning for computational fluid dynamics in airfoil design. arXiv preprint arXiv:240...

    2024 arXiv

  19. [27]

    Zhang, L

    X. Zhang, L. Wang, J. Helwig, Y. Luo, C. Fu, Y. Xie, M. Liu, Y. Lin, Z. Xu, K. Yan, et al. Artificial intelligence for science in quantum, atomistic, and continuum systems. arXiv preprint arXiv:2307.08423, 2023

    2023 arXiv

  20. [28]

    Zhang, J

    X. Zhang, J. Helwig, Y. Lin, Y. Xie, C. Fu, S. Wojtowytsch, and S. Ji. Sinenet: Learning temporal dynamics in time-dependent partial differential equations. In The Twelfth International Conference on Learning Representations, 2024. URL https://openreview.net/forum?id=LSYhE2hLWG

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.