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arxiv: 2606.27126 · v1 · pith:EX4452BEnew · submitted 2026-06-25 · 💻 cs.LG · physics.data-an· physics.flu-dyn

Kolmogorov Arnold networks (KAN) for aerodynamic prediction: a comparison with MLPs and GNNs

Miguel Jaraiz , Fermin Gutierrez , Pablo Yeste , Miguel S\'anchez-Dom\'inguez , Eusebio Valero , Gonzalo Rubio , Lucas Lacasa This is my paper

Pith reviewed 2026-06-26 05:10 UTC · model grok-4.3

classification 💻 cs.LG physics.data-anphysics.flu-dyn
keywords Kolmogorov-Arnold networksaerodynamic predictionpressure coefficientsairfoil surrogate modelingmachine learning comparisongraph neural networksmultilayer perceptrons
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The pith

Kolmogorov-Arnold networks predict airfoil pressure distributions with performance comparable to multilayer perceptrons but lower complexity.

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

The paper evaluates Kolmogorov-Arnold networks for predicting surface pressure on airfoils in subsonic and transonic flows. It finds that KANs perform well and can interpolate between different Mach numbers and angles of attack. However, their results are marginally inferior to those from multilayer perceptrons, while graph neural networks achieve the best accuracy at the cost of longer training times. KANs benefit from lower model complexity leading to faster training but are prone to instabilities that require careful hyperparameter tuning.

Core claim

KAN models show good performance in predicting the whole pressure coefficients and are able to interpolate across Mach numbers and angles of attack, however their performance is comparable --marginally inferior-- to a suitably trained MLP, where best performance is achieved by a GNN at the expense of requiring lengthier training. Optimal KAN models have typically much lower complexity than MLP and GNN hence resulting in faster training, but KANs suffer from training instabilities and their performance is highly dependent on proper hyperparameter optimisation.

What carries the argument

Kolmogorov-Arnold network, which uses trainable parameters to adapt activation functions rather than the coefficients of affine transformations.

If this is right

  • KANs can predict entire pressure coefficient distributions on airfoils.
  • KANs interpolate across varying Mach numbers and angles of attack.
  • Optimal KANs have lower complexity than MLPs and GNNs, enabling faster training.
  • GNNs provide the highest accuracy but require longer training times.
  • KAN performance depends heavily on hyperparameter optimization and can suffer instabilities.

Where Pith is reading between the lines

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

  • KANs could be advantageous in resource-constrained settings where model size matters more than peak accuracy.
  • Stabilizing training procedures might make KANs competitive in other physics simulation tasks.
  • Graph structures may better capture the spatial relationships in aerodynamic data than standard network architectures.
  • Future comparisons should ensure identical hyperparameter search efforts for fair assessment.

Load-bearing premise

Hyperparameter optimization was performed with equivalent rigor and computational budget for KAN, MLP, and GNN models.

What would settle it

Retraining all three model types with an identical extensive hyperparameter search procedure and comparing the resulting test errors on the same airfoil pressure dataset.

Figures

Figures reproduced from arXiv: 2606.27126 by Eusebio Valero, Fermin Gutierrez, Gonzalo Rubio, Lucas Lacasa, Miguel Jaraiz, Miguel S\'anchez-Dom\'inguez, Pablo Yeste.

Figure 1
Figure 1. Figure 1: Data split. The database available for this study was kindly shared by DLR [21], but we nonetheless provide details on how this was constructed for the sake of self-containedness. As the surface geometry, we use the supercritical NLR7301 airfoil introduced in Hines and Bekemeyer [21]. High-fidelity RANS-based CFD computations were performed with the DLR flow solver TAU [33], employing the Spalart-Allmaras … view at source ↗
Figure 2
Figure 2. Figure 2: Performance of trained KAN-based models (in terms of MSE loss in the validation set) for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spearman correlation analysis between all pairs of hyperparameters (and the validation loss) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Linear-log frequency histograms of MSE loss (in the validation set) for the KAN-based model [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Training and validation MSE loss curves as a function of the number of epochs for the MLP, [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: True versus predicted surface pressure coefficients for the three models. The color scale [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Surface pressure distribution at four different operating conditions of the test set. Each panel [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Correlation matrices for the test set: (top left) reference, (top right) MLP, (bottom left) KAN, [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the KAN model against a MLP of comparable complexity. (Left panel): [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

Kolmogorov Arnold networks (KAN) have recently been introduced as a (deep) neural network architecture whose trainable parameters adapt the activation functions, instead of the coefficients of the affine transformations at the core of traditional architectures such as deep multilayer perceptrons (MLPs). This architecture builds on the Kolmogorov-Arnold theorem, which endows it with universal approximation properties. While the advent of KANs has been received with excitement, there is a current debate about the possible KAN supremacy over deep multilayer perceptrons (MLPs) for classic fields such as symbolic regression, generic-purpose machine learning, natural language processing or computer vision. Here we assess the performance of KANs --and its nuanced comparison against MLPs and graph neural networks (GNNs)-- in the realm of fluid dynamics surrogate modelling. To that aim, we consider the task of predicting the surface pressure distribution over subsonic and transonic airfoils, a canonical task in aerodynamics. Our results show that KAN models show good performance in predicting the whole pressure coefficients and is able to interpolate across Mach numbers and angles of attack, however its performance is comparable --marginally inferior-- to a suitably trained MLP, where best performance is achieved by a GNN at the expense or requiring lengthier training. While the optimal KAN model have typically much lower complexity than MLP and GNN --hence resulting in faster training--, we find that KANs suffer from training instabilities, and their performance is highly dependent on a proper hyperparameter optimisation.

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

1 major / 1 minor

Summary. The paper evaluates Kolmogorov-Arnold Networks (KANs) as surrogate models for predicting surface pressure coefficient distributions on subsonic and transonic airfoils. It reports that KANs achieve good predictive performance and can interpolate across Mach numbers and angles of attack, but are marginally inferior to suitably trained MLPs; GNNs yield the best accuracy at the cost of longer training, while optimal KANs exhibit lower complexity (hence faster training) but suffer from training instabilities and strong dependence on hyperparameter choices.

Significance. If the comparative ordering is shown to be robust under equivalent hyperparameter optimization effort, the work supplies a useful empirical benchmark for architecture selection in aerodynamic surrogate modeling. The study is a pure empirical comparison with no derivations or self-referential claims, and its emphasis on training speed versus accuracy trade-offs is directly relevant to engineering applications where model complexity matters.

major comments (1)
  1. [Abstract] Abstract: the headline comparative claims (KAN marginally inferior to MLP; GNN best) rest on performance numbers obtained after tuning, yet the text explicitly states that KAN performance is highly dependent on proper hyperparameter optimisation and that KANs suffer from training instabilities. No information is supplied on search spaces, number of trials, grid sizes, early-stopping criteria, or wall-clock resources allocated to each architecture family, leaving the fairness of the comparison unverified and the reported ordering potentially sensitive to unequal tuning budgets.
minor comments (1)
  1. [Abstract] Abstract: minor grammatical issues ('or requiring lengthier training' should read 'of requiring'; 'optimal KAN model have' should read 'models have').

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback highlighting the need for greater transparency in our hyperparameter tuning procedures. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline comparative claims (KAN marginally inferior to MLP; GNN best) rest on performance numbers obtained after tuning, yet the text explicitly states that KAN performance is highly dependent on proper hyperparameter optimisation and that KANs suffer from training instabilities. No information is supplied on search spaces, number of trials, grid sizes, early-stopping criteria, or wall-clock resources allocated to each architecture family, leaving the fairness of the comparison unverified and the reported ordering potentially sensitive to unequal tuning budgets.

    Authors: We agree that the absence of explicit details on the hyperparameter optimization process limits the ability to verify the fairness of the reported performance ordering. The manuscript already notes the sensitivity of KANs, but does not provide the requested specifics. In the revised version we will add a new subsection (or appendix) that documents, for each architecture family: the hyperparameter search spaces explored, the number of trials performed, the search method employed, early-stopping criteria, and approximate wall-clock resources allocated to tuning. This addition will allow readers to assess whether the comparative results are robust under comparable tuning effort. We do not claim that the original tuning budgets were provably equal; the revision will make this limitation transparent. revision: yes

Circularity Check

0 steps flagged

Empirical benchmark with no derivation chain or self-referential steps

full rationale

The paper is a pure empirical comparison of KAN, MLP and GNN performance on airfoil pressure prediction. No equations, derivations, fitted parameters presented as predictions, or load-bearing self-citations appear in the abstract or described content. All claims rest on observed numerical results after training, with no reduction of outputs to inputs by construction. The Kolmogorov-Arnold theorem is cited externally as background, not as an internal self-definition. Hyperparameter sensitivity is noted as an experimental observation, not a circular premise.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an empirical architecture comparison study. No new mathematical axioms, free parameters, or invented physical entities are introduced; all claims rest on standard supervised learning assumptions and existing neural network theory.

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