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REVIEW 4 major objections 7 minor 1 cited by

Scaled-cPIKANs: Domain Scaling in Chebyshev-based Physics-informed Kolmogorov-Arnold Networks

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read By rescaling PDE spatial variables to $[-1,1]^d$ before training, Scaled-cPIKAN solves highly oscillatory PDEs over extended domains with substantially higher accuracy and faster convergence than unscaled Chebyshev PIKANs, scaled…

desk verdict A simple, plausible domain-scaling idea for Chebyshev PIKANs, with promising experiments but missing formulas and an overclaimed abstract. read the letter →

arxiv 2501.02762 v1 pith:S2ANE6OU submitted 2025-01-06 math.NA cs.NA

Farinaz Mostajeran , Salah A Faroughi This is my paper · ORCID
classification math.NAcs.NA MSC 65M7068T07
keywords Physics-informedneuralnetworksKolmogorov-ArnoldChebyshevpolynomialsDomainscalingVariableOscillatoryPDEsInverseproblemsScientificmachinelearning
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 introduces Scaled-cPIKAN, a physics-informed Kolmogorov-Arnold network whose Chebyshev basis functions are paired with an explicit rescaling of the PDE's spatial variables onto the $[-1,1]^d$ interval that Chebyshev polynomials naturally live on. The authors argue that this domain scaling, applied before training, stabilizes the loss and lets the network resolve oscillatory solutions on large domains, where unscaled Chebyshev PIKANs diverge and MLP-PINNs lose accuracy. Across the diffusion, Helmholtz, Allen-Cahn, and reaction-diffusion benchmarks, the method reports relative errors often several orders of magnitude lower than the baselines, with faster loss convergence in forward, inverse, and noisy-data settings.

What carries the argument

The load-bearing construction is the scaled PDE: the spatial coordinates are transferred to $\tilde{\Omega}=[-1,1]^d$, the differential operator becomes $N^s$, and the loss terms are evaluated on the scaled collocation data, so the network trains on the Chebyshev-native domain instead of the original extended domain. This is combined with the cKAN architecture, in which each univariate activation is a truncated Chebyshev expansion $\sum_i c_i T_i(\xi)$ with $\tanh$ used to keep layer inputs in $[-1,1]$. The domain scaling is the mechanism claimed to remove the loss divergence observed for high-frequency Helmholtz problems and to give the reported accuracy gap.

What would settle it

An independent rerun of the four benchmarks with matched parameter counts and loss weights, in which the affine map and the chain-rule form of $N^s$ are written out explicitly, would settle the claim; if Scaled-cPIKAN's advantage shrinks or disappears under that check, the reported gains are not attributable to domain scaling.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the spatial domain should be treated as part of the PDE formulation when using Chebyshev-based KANs: mapping the spatial variables to $[-1,1]^d$ and solving the correspondingly scaled differential equation yields a physics-informed network that stays accurate and converges quickly for oscillatory problems over domains such as $[-6,6]$, while the unscaled cPIKAN and both PINN variants degrade sharply as the domain grows. The scaling is presented not as an extra architecture but as a reformulation of the PDE in Eq. (12), with the scaled operator $N^s$ and scaled data sets, so the improved behavior is attributed to the domain transformation rather than to added network capacity.

Load-bearing premise

The claim stands or falls on whether the improvement is really caused by the spatial rescaling rather than by favorable hyperparameter choices; the paper defines the scaled PDE in Eq. (12) but does not write out the explicit coordinate map and the chain-rule form of $N^s$, so that step cannot be verified from the text alone.

Editorial extensions

If this is right

  • If the scaling claim holds, a one-line variable transformation is enough to stabilize Chebyshev PIKANs on extended domains, with no increase in network size.
  • Unscaled cPIKAN and PINN errors grow as the domain widens from $M=2$ to $M=6$, while Scaled-cPIKAN errors stay roughly constant, implying the method's accuracy is largely domain-independent.
  • Scaled-cPIKAN estimates the unknown reaction coefficient in the inverse reaction-diffusion problem with about 99% lower relative error than the baselines even under noisy data, suggesting the scaling also improves parameter identification.
  • Because Scaled-PINN outperforms PINN in several tests, the scaling is useful beyond KANs, although the Chebyshev version gains the most.

Reading between the lines

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

  • An explicit affine map would let future work test whether the scaling's benefit is purely a conditioning effect: oscillatory solutions such as $\sin(a\pi x)$ on $[-M,M]$ become $\sin(a\pi M s)$ on $s\in[-1,1]$, so the method is effectively asking Chebyshev polynomials to resolve a higher-frequency function on their natural spectral grid.
  • The same scaling idea presumably applies to other orthogonal-polynomial KAN bases that also have a native interval, and to temporal variables, which the paper keeps fixed at $[0,1]$; testing those extensions is a natural next step.
  • The paper notes that very large domains beyond its tested range would need decomposition; combining per-subdomain affine maps with domain decomposition is an evident continuation that the current experiments do not yet cover.
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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 / 7 minor

Summary. The paper proposes Scaled-cPIKAN, a Chebyshev-based physics-informed Kolmogorov-Arnold network that first rescales the spatial coordinates of a PDE to [-1,1]^d before training. The method is compared against cPIKAN, PINN, and a domain-scaled PINN on the diffusion, Helmholtz, Allen-Cahn, and forward/inverse reaction-diffusion equations, with relative L2 errors and training-loss histories reported for domain half-widths M = 2, 4, and 6. The authors conclude that Scaled-cPIKAN consistently outperforms the baselines, particularly on larger oscillatory domains and in noisy inverse problems.

Significance. If the reported improvements are reproducible, the contribution is a simple and practically relevant preprocessing step: an affine rescaling of the PDE before training a Chebyshev PIKAN. The paper deserves credit for matching parameter counts across methods in most comparisons and for evaluating accuracy against external analytic solutions, so the empirical comparisons are not circular. The breadth of evidence across four benchmark families is a strength. The main weaknesses are that the core coordinate transformation is never written down, all experiments are single runs without error bars, and one table contradicts the abstract's 'all test cases' claim; these issues currently prevent the reader from verifying the central mechanism.

major comments (4)
  1. [Section 3, Eqs. (12)-(18)] The central mechanism, the affine scaling of spatial variables, is never explicitly defined. The text introduces the scaled PDE in Eq. (12) with x̃ ∈ [-1,1]^d, but it does not give the map x → x̃, the chain-rule form of the scaled operators N^s and B^s, or the inverse map used to evaluate the network output on the original domain. Because the cKAN forward pass in Eq. (10) already applies tanh to its inputs, the only substantive difference between cPIKAN and Scaled-cPIKAN is precisely this coordinate transformation; if it is applied inconsistently to the residual, boundary, initial, or measurement terms, the comparisons in Tables 1-6 do not establish the claimed mechanism. This is especially acute in the inverse reaction-diffusion example of Section 4.4, where the scaling of the noisy data and source term is not specified and the reported κ errors differ from the baselines by three orders of magnitude; the reader cannot verify that this gap is caused by the scaling rather than by a different implicit treatment of the scaled data. Please write out the transformation and the scaled operator for at least one worked example, and specify how the network prediction is mapped back to the original coordinates before computing the error in Eq. (21).
  2. [Abstract; Table 5] The abstract claims that Scaled-cPIKAN 'significantly outperforms existing methods in all test cases,' but Table 5 (reaction-diffusion, forward, M = 2) reports a relative L2 error of 2.5 × 10^-2 for cPIKAN versus 2.7 × 10^-2 for Scaled-cPIKAN, i.e., the unscaled method is slightly better in that configuration. This contradiction should be resolved by either tempering the abstract and conclusion claims or explaining why this single M = 2 case is an exception. As written, the 'all test cases' statement is factually incorrect.
  3. [Tables 1-6] All tables report a single training run per configuration, with no standard deviations, seeds, or repeated trials. Given that the comparisons involve Adam optimization, randomly sampled collocation points, and loss landscapes that are sensitive to initialization, the reported differences—especially the three-order-of-magnitude gap in the inverse κ estimates of Table 6—cannot be distinguished from run-to-run variance. Please report statistics over multiple independent runs (at least 3-5 seeds) or clearly state that the tables are single-run exploratory results and adjust the strength of the conclusions accordingly.
  4. [Section 4.4, Eqs. (27)-(29)] In the inverse reaction-diffusion problem, Table 6 reports relative L2 errors for the source term f, but the text does not state how the predicted f is obtained from the network (e.g., by differentiating the approximate solution and evaluating N^s[ũ] = f̃) nor how the noisy scaled source term f^δ enters the loss. Please specify these details, because the scaling of f is part of the transformation that the paper claims is responsible for the improvement.
minor comments (7)
  1. [Section 4, Eq. (21)] The error metric is computed on the test set T* in the original domain; please state explicitly that the Scaled-cPIKAN output is first mapped back to the original coordinates before computing uP, and give the inverse mapping formula.
  2. [Table 2] The second block of Table 2 changes the header notation from (N_l, N_n, k) to (L, N, D); please unify the notation.
  3. [Table 4] PINN and Scaled-PINN report exactly the same relative L2 error (5.1 × 10^-1) for M = 2, 4, and 6; this suggests the models collapse to a trivial prediction, and the manuscript should comment on this behavior or verify the entries.
  4. [Section 4.4] The inverse problem uses only N_data = 10 measurement points; please specify how these points are selected and whether this is sufficient for the parameter estimation claims.
  5. [Figures 1-9 captions] The captions state that 'the first plot in the top row shows the ground truth solution,' which conflicts with the preceding description of the left column as the loss function; please correct the captions or the figure layout.
  6. [Section 4.1] The claim of 'several orders of magnitude higher accuracy' is stronger than the data for M = 2 (e.g., Table 1: Scaled-cPIKAN 4.3 × 10^-3 vs. cPIKAN 1.3 × 10^-2); please qualify the claim to be consistent with the reported orders of magnitude.
  7. [Section 3, Eq. (12)] The notation N^s and B^s is defined only verbally; at minimum, provide the definitions for one scalar example (e.g., the diffusion operator) to illustrate the chain rule.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity found: the claimed accuracy gains are measured against independent analytic solutions, and self-citations are not load-bearing.

full rationale

The paper's central claim is empirical: Scaled-cPIKAN is compared against analytic or manufactured ground-truth solutions for the diffusion, Helmholtz, Allen-Cahn, and reaction-diffusion problems. The relative L2 error in Eq. (21) is evaluated on a test set T* distinct from training points, against uE supplied by closed-form solutions such as u(x,t)=sin(pi x)exp(-pi^2 D t) and u(x,y)=sin(a1 pi x) sin(a2 pi y). No learned parameter is fit to these benchmark errors, and no reported 'prediction' is statistically forced by the training loss. The domain-scaling idea is described as a transformation in Eq. (12), but the paper never states the coordinate map or the chain-rule form of N^s; this omission makes the mechanism hard to reproduce, but it is not circular because the scaled operator is not defined in terms of the target solution or the reported error. Self-citations to the authors' earlier work (e.g., [33] in the survey of KAN implementations) are contextual and are not used to justify the scaling result. The abstract's 'all test cases' claim is internally weakened by Table 5, where cPIKAN reports 2.5e-2 and Scaled-cPIKAN reports 2.7e-2 at M=2, but this is an accuracy-claim inconsistency, not a derivation that consumes its own conclusion. Accordingly, no circular step can be exhibited with a specific reduction, and the appropriate finding is no significant circularity.

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

The central claim rests on standard approximation-theoretic assumptions and on hand-chosen experimental hyperparameters. No new physical entities are introduced; the scaled PDE is an affine reformulation of the original equation.

free parameters (5)
  • Residual loss weight lambda_res = 0.01, 0.5, or 1.0 per example
    Hand-selected in each experiment (Secs. 4.1-4.4); no sensitivity study, so gains may depend on these choices.
  • Data, initial, and boundary loss weights = 1.0 or 0.0 depending on problem
    Set per problem in Sec. 4; default values are not justified by any analysis.
  • Collocation and data counts = e.g., 2000/800, 4000/4000, 50000/400/200, 1200/10
    Chosen per benchmark and not swept; the number of training points directly affects the reported errors.
  • Network size (Nl, Nn, k) = e.g., (2,8,5), (4,8,3), (4,20,5)
    Chosen so all methods have similar parameter counts, but architecture choices can bias the comparison.
  • Learning rate and training epochs = 1e-3 or 1e-4; 10,000 to 150,000 epochs
    Vary by experiment; no tuning analysis or learning-rate sensitivity is reported.
assumptions (4)
  • standard math Kolmogorov-Arnold representation theorem: any continuous multivariate function can be written as sums and compositions of univariate functions.
    Invoked in Sec. 2.2, Eq. (4), as the basis of KAN architectures.
  • domain assumption Chebyshev polynomials are defined on [-1,1]; inputs must be normalized to that interval for stable training.
    Motivates the whole scaling method; cited from [29,32,13] in Secs. 1 and 2.2.
  • domain assumption Manufactured benchmark solutions and their derived source terms are representative of real PDE challenges.
    All four test problems use hand-made exact solutions; no real-world or independent solver validation is included.
  • domain assumption Adam optimization with fixed learning rate is sufficient to compare methods fairly.
    Assumed throughout Sec. 4 without per-method tuning, convergence criteria, or repeated seeds.

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Cite this review

Pith. "Pith review of Scaled-cPIKANs: Domain Scaling in Chebyshev-based Physics-informed Kolmogorov-Arnold Networks." pith.science (2026). https://pith.science/paper/S2ANE6OU

@misc{pith2026250102762,
  author       = {Pith},
  title        = {Pith review of: Scaled-cPIKANs: Domain Scaling in Chebyshev-based Physics-informed Kolmogorov-Arnold Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2ANE6OU}},
  note         = {Machine review of arXiv:2501.02762}
}
abstract

Partial Differential Equations (PDEs) are integral to modeling many scientific and engineering problems. Physics-informed Neural Networks (PINNs) have emerged as promising tools for solving PDEs by embedding governing equations into the neural network loss function. However, when dealing with PDEs characterized by strong oscillatory dynamics over large computational domains, PINNs based on Multilayer Perceptrons (MLPs) often exhibit poor convergence and reduced accuracy. To address these challenges, this paper introduces Scaled-cPIKAN, a physics-informed architecture rooted in Kolmogorov-Arnold Networks (KANs). Scaled-cPIKAN integrates Chebyshev polynomial representations with a domain scaling approach that transforms spatial variables in PDEs into the standardized domain \([-1,1]^d\), as intrinsically required by Chebyshev polynomials. By combining the flexibility of Chebyshev-based KANs (cKANs) with the physics-driven principles of PINNs, and the spatial domain transformation, Scaled-cPIKAN enables efficient representation of oscillatory dynamics across extended spatial domains while improving computational performance. We demonstrate Scaled-cPIKAN efficacy using four benchmark problems: the diffusion equation, the Helmholtz equation, the Allen-Cahn equation, as well as both forward and inverse formulations of the reaction-diffusion equation (with and without noisy data). Our results show that Scaled-cPIKAN significantly outperforms existing methods in all test cases. In particular, it achieves several orders of magnitude higher accuracy and faster convergence rate, making it a highly efficient tool for approximating PDE solutions that feature oscillatory behavior over large spatial domains.

Figures

Figures reproduced from arXiv: 2501.02762 by the authors.

Figure 1
Figure 1. Comparison of the solutions predicted for the diffusion equation (Example 4.1) in [ [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the solutions predicted for the diffusion equation (Example 4.1) in [ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Comparison of the solutions predicted for the diffusion equation (Example 4.1) in [ [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Prediction results for the Helmholtz equation (Example 4.2) in the first scenario with parameters ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Prediction results for the Helmholtz equation (Example 4.2) in the second scenario with parameters ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the solutions predicted for the Allen-Cahn equation (Example 4.3) in a supervised version on [ [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the solutions predicted for the Allen-Cahn equation (Example 4.3) in a supervised version on [ [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Comparison of the solutions predicted for the Allen-Cahn equation (Example 4.3) in an unsupervised version on [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Comparison of the solutions predicted for the Allen-Cahn equation (Example 4.3) in an unsupervised version on [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Comparing the performance of Scaled-cPIKAN and cPIKAN on solving the Allen-Cahn equation (Example 4.3) in [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Training loss histories for the forward version of the reaction-diffusion equation (Example 4.4) across three domain [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Forward solutions of the reaction-diffusion equation (Example 4.4) for [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Training loss histories for the inverse version of the reaction-diffusion equation (Example 4.4) across different noise [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Inverse problem of the reaction-diffusion equation (Example 4.4) results for [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Inverse problem of the reaction-diffusion equation (Example 4.4) results for [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Overall comparison of the performance of Scaled-cPIKAN, Scaled-PINN, cPIKAN, and PINN on challenging examples [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.