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REVIEW 4 major objections 6 minor 34 references

EFKAN: A KAN-Integrated Neural Operator For Efficient Magnetotelluric Forward Modeling

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read EFKAN predicts magnetotelluric apparent resistivity and phase more accurately than the MLP-based EFNO, while running about 1000 times faster than finite-difference forward modeling.

desk verdict A KAN-for-MLP swap in EFNO for 2D MT forward modeling; the speedup is real, but the accuracy claim over EFNO fails on the paper's own generalization tests. read the letter →

arxiv 2502.02195 v2 pith:FAKI3CNJ submitted 2025-02-04 physics.geo-ph cs.LG

Feng Wang , Hong Qiu , Yingying Huang , Xiaozhe Gu , Renfang Wang , Bo Yang This is my paper · ORCID
classification physics.geo-phcs.LG
keywords MagnetotelluricforwardmodelingKolmogorov-ArnoldnetworkNeuraloperatorFourierApparentresistivityPhaseFinitedifferencemethodGaussianrandomfield
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

Forward modeling is the computational bottleneck of magnetotelluric (MT) inversion, which converts surface electromagnetic measurements into subsurface resistivity images. This paper tries to establish that a neural operator called EFKAN can serve as a practical replacement for the finite-difference solver by pairing a Fourier neural operator (FNO) branch with a Kolmogorov-Arnold network (KAN) trunk. On synthetic test data, EFKAN is reported to reach lower relative $\ell^1$ errors in apparent resistivity and phase than the EFNO baseline (FNO with an MLP trunk), and to run roughly 1000 times faster than FDM. If true, the result matters because a fast, accurate forward model would allow MT inversion to explore far more conductivity models in the same wall-clock time, improving both resolution and uncertainty assessment.

What carries the argument

The load-bearing object is the EFKAN architecture itself: the DeepONet decomposition $G(\sigma)(y,z,f) \approx \sum_k b_k(\sigma)\, t_k(y,z,f)$, with $b_k$ computed by an FNO branch (six Fourier layers, $k_{\max}=18$, GELU activations) and $t_k$ by a KAN trunk (three layers, widths 2–256–4096, grid size 5, spline order 3). KAN replaces each linear weight by a learnable univariate function, here a weighted sum of a basis function and a B-spline; the paper credits this locality and adjustability for the improved accuracy and training stability. The synthetic training distribution is produced with the spectral method using power spectrum $P(k)\propto |k|^{-\beta/2}$ for $\beta\in\{3,\ldots,7\}$, aimed at mimicking smooth geological conductivity variations.

What would settle it

Average the relative $\ell^1$ errors over all 100 samples of testing dataset B: Table 4 already shows EFKAN at 0.0884 versus EFNO at 0.0819, so if that reversal is representative, the claim that EFKAN is more accurate than EFNO fails for the unseen-anomaly case.

Watch

Extended reading notes

Core claim

EFKAN is a DeepONet-style operator in which a six-layer Fourier neural operator serves as the branch, encoding the 2-D resistivity model $\rho(y,z)$, and a three-layer Kolmogorov-Arnold network serves as the trunk, encoding the queried surface coordinates and frequencies $(y,z,f)$; the apparent resistivities $\rho_{xy},\rho_{yx}$ and phases $\phi_{xy},\phi_{yx}$ are recovered as the branch–trunk inner product. The paper's central claim is that this KAN-for-MLP swap improves accuracy: EFKAN reports lower mean relative $\ell^1$ error than EFNO on in-distribution test set A and on entirely unseen frequencies (test set C), and better fits to the finite-difference ground truth on rectangular-anomaly models, while taking about 0.01 seconds per model versus about 10 seconds for FDM. The paper also reports smoother convergence and retention of accuracy when training is downsampled to 22 frequencies/coordinates or downscaled to 5,000 samples.

Load-bearing premise

All accuracy claims rest on the assumption that the synthetic training distribution—smooth Gaussian random fields with spectral exponents $\beta=3$–7, plus rectangular anomalies in testing—is representative enough of real subsurface conductivity structures for the reported gains to carry over to field data.

Editorial extensions

If this is right

  • An MT inversion could afford orders of magnitude more forward solves per iteration, making global or stochastic searches feasible at a fraction of the current cost.
  • Because EFKAN outputs responses at arbitrary query points and frequencies, inversions can be evaluated exactly at the field stations and periods of an observed dataset without retraining the surrogate.
  • The reported accuracy from training on 22 frequencies/coordinates or on 5,000 samples suggests the surrogate can be deployed in data-scarce geoscience settings with modest synthetic training sets.
  • The smoother loss curves and lower final loss than EFNO indicate that KAN trunks are a viable drop-in replacement for MLP trunks in operator learning for PDEs.

Reading between the lines

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

  • Because B-spline KANs are not GPU-parallelized, EFKAN is slower per query than EFNO (roughly 1.1 s vs 0.2 s per 100 models in the paper's tables); testing the paper's suggested alternatives—radial basis functions, wavelets, or Jacobi polynomials—would show whether the accuracy gain can be combined with EFNO-level speed.
  • The accuracy gain may come from the trunk's learnable univariate functions rather than from KAN as such; an ablation comparing EFKAN with another learnable-activation trunk at matched parameter count and training budget would isolate the mechanism.
  • The training distribution is smooth Gaussian random fields, so the decisive out-of-distribution test is real-world structures with sharp faults, salt flanks, or thin layers; evaluating on such models or on field MT data would show whether the reported accuracy survives where it matters.
  • The same FNO-branch/KAN-trunk split should transfer to 3-D MT forward modeling, but the paper demonstrates only 2-D; a 3-D extension is the natural next step to test before use in 3-D inversion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes EFKAN, a neural operator for 2D magnetotelluric forward modeling that combines a Fourier neural operator (FNO) as the branch network with a Kolmogorov-Arnold network (KAN) as the trunk network. The trunk maps the branch output to apparent resistivity and phase at arbitrary receiver locations and frequencies. The method is trained on synthetic resistivity models generated with the spectral method (Gaussian random fields, beta 3-7) plus rectangular anomaly test cases, and is compared against EFNO, an FNO-with-MLP-trunk baseline, with finite-difference method (FDM) solutions as ground truth. The paper claims higher accuracy than EFNO and roughly 1000x speedup over FDM, and reports experiments on the original training dataset, a downsampled frequency/coordinate dataset, and a reduced-sample dataset.

Significance. If the accuracy claims held across all test distributions, EFKAN would be a useful surrogate for rapid 2D MT forward modeling, with the practical advantage of evaluating responses at arbitrary locations and frequencies. The paper has clear strengths: predictions are benchmarked against FDM solutions (an external reference, so the evaluation is not circular), the source code is publicly available, and the experimental design includes several meaningful generalization tests (unseen rectangular anomalies, unseen frequencies, downsampled training data, and reduced training data). The central weakness is that the unqualified accuracy advantage over EFNO is not supported by the paper's own aggregate and per-component results on the out-of-distribution tests, and the baseline comparison is not controlled for model capacity or optimization variance.

major comments (4)
  1. [Table 4 and Discussion] The Discussion states without qualification that EFKAN 'can achieve higher accuracy in apparent resistivity and phase measurements compared to the EFNO,' but Table 4 contradicts this on the out-of-distribution tests: on testing dataset B, EFKAN has larger average l1 error than EFNO (0.0884 vs 0.0819), and on dataset C the two are essentially tied (0.0776 vs 0.0785). The per-component errors reported in Section 3.2 also favor EFNO for several quantities on datasets B and C, for example rho_xy 0.0908 vs 0.0826 and rho_yx 0.0969 vs 0.0867 on B, and rho_xy 0.0451 vs 0.0391 and phi_yx 0.1485 vs 0.1349 on C. The accuracy claim needs to be re-scoped to the same-distribution setting, and the paper should discuss why the in-domain gain does not transfer to the generalization tests that are most relevant for field deployment.
  2. [Section 2.2.3 and Tables 1-2] The EFNO baseline is uncontrolled: the MLP trunk architecture, layer widths, and parameter count of EFNO are never reported, while the KAN trunk used in EFKAN has 4096 hidden neurons in its final layer (Table 2). The same-distribution accuracy gain on testing dataset A could therefore reflect larger model capacity or different optimization behavior rather than a KAN-specific advantage. A fair comparison should match the trunk capacity or explicitly report parameter counts for both models and demonstrate that the gain survives similar-capacity settings.
  3. [Section 3.2 and Tables 4-6] All accuracy comparisons are based on single runs: no repeated seeds, no standard deviations, and no statistical significance testing. Given that the aggregate differences on dataset C are tiny (0.0776 vs 0.0785), and that the per-component errors on datasets B and C frequently favor EFNO, the claim of higher accuracy requires at least a few random restarts and reporting of the resulting error distributions before it can be considered established.
  4. [Section 4] The Discussion claims that KAN endows EFKAN with 'a certain degree of interpretability' and that KAN reduces overfitting, but neither is demonstrated in the experiments. No learned spline functions are analyzed, and the generalization results on dataset B (where EFKAN is worse than EFNO) do not support an overfitting-reduction claim. These interpretive claims should either be supported with specific evidence or removed.
minor comments (6)
  1. [Section 3] The note that color bars differ between the EFNO and EFKAN error plots makes visual comparisons unreliable; the paper should use a common color scale or state explicitly that visual differences are not directly comparable.
  2. [Table 1 and Section 3.2] There are typos in the manuscript: 'bacth size' in Table 1, 'the the original training dataset dataset' in Section 3.2 and in figure captions, and 'ENKAN' in the Discussion.
  3. [Figure 15 caption] The caption reads 'The 1-D profiles in from Fig. 14' and should be corrected.
  4. [Section 4] The explanation for testing loss being lower than training loss is not convincing: Eq. (17) defines an average per-sample loss, so the larger number of training samples does not explain a higher average training loss. The authors should compare loss on a matched-size random subset of the training set or revise the explanation.
  5. [Data availability] The data availability statement says 'No data was used for the research described in the article,' but the study generates and uses synthetic datasets; this should be clarified to avoid confusion about whether the generated data are shared.
  6. [Abstract and Conclusion] The abstract and conclusion should carry the same qualification as the revised Discussion, namely that the accuracy advantage over EFNO is demonstrated on same-distribution data but not consistently on unseen rectangular anomalies and unseen frequencies.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: EFKAN's predictions are trained and scored against independent FDM solutions; the disputed accuracy claims stem from uncontrolled baseline comparisons and conflicting Table 4 numbers, not from a self-referential derivation.

full rationale

The paper's core pipeline is a supervised surrogate-model evaluation: synthetic resistivity models are generated by a spectral/GRF method, labeled electromagnetic responses are produced by an external finite-difference solver, and both EFNO and EFKAN are trained to minimize a relative L1 loss against those FDM labels. The reported test errors on datasets A, B, and C are measured against the same external FDM ground truth, so the target quantities are not constructed from the model outputs or from fitted constants that are later renamed as predictions. The Kolmogorov-Arnold representation theorem and the KAN architecture are adopted from independent literature (Liu et al., 2024), and no uniqueness or forcing theorem from the present authors is invoked to justify the architecture. The only self-citation that appears in a load-bearing-looking sentence is the remark that test loss below training loss 'has also occurred in our previous work (Peng et al., 2022)', but that is anecdotal context and not an argument that establishes the paper's central accuracy claim. The legitimate concerns raised by the manuscript's own numbers — EFKAN is worse than EFNO on testing dataset B (0.0884 vs 0.0819) and essentially tied on dataset C (0.0776 vs 0.0785) in Table 4, while the MLP trunk of the EFNO baseline is never specified — are threats to the validity of the comparative accuracy claim, but they are experimental-design and internal-consistency issues, not circular derivations. No equation in the paper reduces to its inputs, no fitted parameter is relabeled as a prediction, and no self-citation chain carries the logical weight of the conclusion. Therefore, the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard operator-learning assumptions, on several domain assumptions about the MT problem, and on hand-chosen data generation and architecture parameters. No new physical entities are introduced. The learned network weights themselves are fitted to synthetic FDM labels, which is the normal empirical basis for this kind of surrogate model.

free parameters (4)
  • Fourier mode truncation kmax = 18
    Truncation of high Fourier modes in the FNO branch; chosen by hand and stated in Section 2.2.3.
  • KAN architecture hyperparameters = 3 layers, widths 2/256/4096, grid 5, spline order 3
    Trunk network configuration from Table 2; chosen by hand and not justified by an ablation study.
  • Spectral method beta values = 3, 4, 5, 6, 7
    Scale exponents for synthetic conductivity generation (Eq. 15); chosen by hand and they define the training distribution.
  • Conductivity range and boundary values = 1e-4 to 1 S/m; air 1e-9; lower boundary 1e-2
    Domain limits for the synthetic models in Section 3.1; they affect the input distribution the network must learn.
assumptions (5)
  • standard math Kolmogorov-Arnold representation theorem holds and is a useful basis for the trunk network.
    Invoked in Section 2.2.2 to justify KAN; the theorem is standard but the practical benefit is empirical.
  • domain assumption Quasi-stationary field assumption and neglect of displacement current are valid for the MT frequency range.
    Section 2.1, Eq. 2; this reduces Maxwell's equations to the Helmholtz equation used by the surrogate.
  • domain assumption The 2D MT problem decouples into xy and yx modes with Dirichlet boundary conditions.
    Section 2.1, Eqs. 3-4; the paper trains and tests only on 2D synthetic sections.
  • domain assumption FDM solutions at the chosen 64x64 grid are accurate enough to serve as ground truth.
    Section 3.1; all training labels and test references come from the FDM solver, so EFKAN error is measured against FDM, not against exact solutions.
  • domain assumption GRF-generated synthetic resistivity models are representative of real MT targets.
    Section 3.1 and Discussion; the authors acknowledge this may not fully capture sharp discontinuities and fault-like structures.

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Cite this review

Pith. "Pith review of EFKAN: A KAN-Integrated Neural Operator For Efficient Magnetotelluric Forward Modeling." pith.science (2026). https://pith.science/paper/FAKI3CNJ

@misc{pith2026250202195,
  author       = {Pith},
  title        = {Pith review of: EFKAN: A KAN-Integrated Neural Operator For Efficient Magnetotelluric Forward Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FAKI3CNJ}},
  note         = {Machine review of arXiv:2502.02195}
}
read the original abstract

Magnetotelluric (MT) forward modeling is fundamental for improving the accuracy and efficiency of MT inversion. Neural operators (NOs) have been effectively used for rapid MT forward modeling, demonstrating their promising performance in solving the MT forward modeling-related partial differential equations (PDEs). Particularly, they can obtain the electromagnetic field at arbitrary locations and frequencies. In these NOs, the projection layers have been dominated by multi-layer perceptrons (MLPs), which may potentially reduce the accuracy of solution due to they usually suffer from the disadvantages of MLPs, such as lack of interpretability, overfitting, and so on. Therefore, to improve the accuracy of MT forward modeling with NOs and explore the potential alternatives to MLPs, we propose a novel neural operator by extending the Fourier neural operator (FNO) with Kolmogorov-Arnold network (EFKAN). Within the EFKAN framework, the FNO serves as the branch network to calculate the apparent resistivity and phase from the resistivity model in the frequency domain. Meanwhile, the KAN acts as the trunk network to project the resistivity and phase, determined by the FNO, to the desired locations and frequencies. Experimental results demonstrate that the proposed method not only achieves higher accuracy in obtaining apparent resistivity and phase compared to the NO equipped with MLPs at the desired frequencies and locations but also outperforms traditional numerical methods in terms of computational speed.

Figures

Figures reproduced from arXiv: 2502.02195 by the authors.

Figure 1
Figure 1. (a) The full architecture of FNO (Li et al., 2021): 𝑃 is the lifting layer, and 𝑄 represents the projecting layer. (b) Fourier layers:  and  −1 are the Fourier and inverse Fourier transform, respectively, and 𝑊 denotes linear transform. F. Wang, H. Qiu, Y. Huang, X. Gu, R. Wang, and B. Yang: Preprint submitted to Elsevier Page 23 of 20 [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. The diagram of EFKAN: (a) The branch network consist of lifting layer 𝑃 , Fourier layer, and projecting layer 𝑄. The input of the branch network is the resistivity model; (b) the input of the trunk network is the frequencies and coordinates. The output of EFKAN contains of apparent resistivity {𝜌𝑥𝑦, 𝜌𝑦𝑥} and phase {𝜙𝑥𝑦, 𝜙𝑦𝑥}. F. Wang, H. Qiu, Y. Huang, X. Gu, R. Wang, and B. Yang: Preprint submitted to Elsevier Page… view at source ↗
Figure 3
Figure 3. An example of the resistivity model generated through the GRF method. F. Wang, H. Qiu, Y. Huang, X. Gu, R. Wang, and B. Yang: Preprint submitted to Elsevier Page 25 of 20 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: (a) The training loss of EFNO and EFKAN on the the original training dataset dataset; (b) The testing loss of EFNO and EFKAN on testing dataset A. F. Wang, H. Qiu, Y. Huang, X. Gu, R. Wang, and B. Yang: Preprint submitted to Elsevier Page 26 of 20 [PITH_FULL_IMAGE:fig…
Figure 5
Figure 5. Figure 5: Examples for demonstrating the effectiveness of EFKAN: (a) The smooth resistivity model randomly sampled from testing dataset A; (b) The smooth resistivity model with rectangular anomalies sampled from testing dataset B; (c) The smooth resistivity model with rectangula…
Figure 6
Figure 6. Figure 6: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model ( [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model with rectangular anomalies ( [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]
Figure 9
Figure 9. Figure 9: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]
Figure 10
Figure 10. Figure 10: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model with rectangular anomalies ( [PITH_FULL_IMAGE:figures/full_fig_p032_10.png]
Figure 11
Figure 11. Figure 11: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]
Figure 12
Figure 12. Figure 12: (a) The training loss of EFNO and EFKAN on the downsampling original training dataset dataset; (b) The testing loss of EFNO and EFKAN on testing dataset A. F. Wang, H. Qiu, Y. Huang, X. Gu, R. Wang, and B. Yang: Preprint submitted to Elsevier Page 34 of 20 [PITH_FULL…
Figure 13
Figure 13. Figure 13: Examples for demonstrating the effectiveness of EFKAN trained by the downsampling original training dataset: (a) The smooth resistivity model randomly sampled from testing dataset A; (b) The smooth resistivity model with rectangular anomalies sampled from testing data…
Figure 14
Figure 14. Figure 14: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model ( [PITH_FULL_IMAGE:figures/full_fig_p036_14.png]
Figure 15
Figure 15. Figure 15: The 1-D profiles in from [PITH_FULL_IMAGE:figures/full_fig_p037_15.png]
Figure 16
Figure 16. Figure 16: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model with rectangular anomalies ( [PITH_FULL_IMAGE:figures/full_fig_p038_16.png]
Figure 17
Figure 17. Figure 17: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p039_17.png]
Figure 18
Figure 18. Figure 18: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model with rectangular anomalies ( [PITH_FULL_IMAGE:figures/full_fig_p040_18.png]
Figure 19
Figure 19. Figure 19: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p041_19.png]
Figure 20
Figure 20. Figure 20: (a) The training loss of EFNO and EFKAN on the downscaling original training dataset dataset; (b) The testing loss of EFNO and EFKAN on testing dataset A. F. Wang, H. Qiu, Y. Huang, X. Gu, R. Wang, and B. Yang: Preprint submitted to Elsevier Page 42 of 20 [PITH_FULL_…
Figure 21
Figure 21. Figure 21: Examples for demonstrating the effectiveness of EFKAN trained on the downscaling original training dataset: (a) The smooth resistivity model randomly sampled from testing dataset A; (b) The smooth resistivity model with rectangular anomalies sampled from testing datas…
Figure 22
Figure 22. Figure 22: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model ( [PITH_FULL_IMAGE:figures/full_fig_p044_22.png]
Figure 23
Figure 23. Figure 23: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p045_23.png]
Figure 24
Figure 24. Figure 24: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model with rectangular anomalies ( [PITH_FULL_IMAGE:figures/full_fig_p046_24.png]
Figure 25
Figure 25. Figure 25: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p047_25.png]
Figure 26
Figure 26. Figure 26: Comparison of the ground truth obtained by FDM and the predicted apparent resistivity and phase by EFNO and EFKAN for the smooth resistivity model with rectangular anomalies ( [PITH_FULL_IMAGE:figures/full_fig_p048_26.png]
Figure 27
Figure 27. Figure 27: The 1-D profiles in [PITH_FULL_IMAGE:figures/full_fig_p049_27.png]

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Reviewed August 9, 2026 · model on record in the stance chip above.