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On the study of frequency control and spectral bias in Wavelet-Based Kolmogorov Arnold networks: A path to physics-informed KANs

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that the frequency of the mother wavelet in a Wav-KAN controls the decay rate of the network's neural tangent kernel eigenvalues, giving a tunable mechanism to counteract spectral bias in function approximation and in…

desk verdict A plausible empirical trend wrapped in an invalid NTK argument; the frequency-control claim is not supported by the math. read the letter →

arxiv 2502.00280 v1 pith:5WDPN3DW submitted 2025-02-01 cs.LG cs.NAmath.NA

Juan Daniel Meshir , Abel Palafox , Edgar Alejandro Guerrero This is my paper · ORCID
classification cs.LGcs.NAmath.NA MSC 68T0742C40
keywords spectralbiasneuraltangentkernelWaveletKolmogorov-Arnoldnetworksphysics-informedfrequencycontrolMorletFourierfeaturesmulti-scalePDEs
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

This paper claims that spectral bias in Wavelet Kolmogorov-Arnold Networks (Wav-KANs) — the tendency to learn low-frequency function components before high-frequency ones — can be controlled by tuning the frequency of the mother wavelet. Analyzing the network's neural tangent kernel (the kernel that governs training dynamics under gradient flow), the authors derive an eigenvalue lower bound for a Morlet-wavelet network showing that eigenvalue decay slows as the wavelet frequency parameter $b$ grows. They argue this gives a direct mechanism to accelerate learning of high-frequency components, and they verify it numerically on a multiscale target function and on Poisson, heat, wave, and Helmholtz equations solved by physics-informed versions of the network (Wav-KINNs). The payoff, if the analysis transfers to full networks, is a parameter-efficient way to mitigate spectral bias without the domain partitioning or extra hyperparameters that Fourier-feature methods require.

What carries the argument

The load-bearing object is the neural tangent kernel of a single-layer Wav-KAN, $K(x_r, x_s) = \sum_i \psi_i(x_r) \psi_i(x_s)$, obtained by fixing the translation and scale parameters $T$ and $S$ so that only the weights $W$ train (equation 10). In one dimension this becomes the rank-one kernel $\psi_1(x_r)\psi_1(x_s)$; Proposition 2 shows the eigenfunctions take the form $g(x_r) = C \psi_1^{1/S}(x_r)$, and Proposition 3 derives the eigenvalue lower bound $\lambda \geq \tfrac{1}{4} e^{-4b^2(x_r - T)^2}$ for the Morlet wavelet with $S = 1$. This bound is the whole mechanism: it expresses how the wavelet frequency $b$ controls the kernel's spectral decay, which through the gradient-flow solution $f(X; t) = (I - e^{-tK_{\mathrm{ntk}}})Y$ determines how fast each frequency component of the target is learned. The same NTK equations (4)-(7) connect larger eigenvalues to faster convergence, so slowing the decay transfers directly into faster high-frequency learning.

What would settle it

Compute the neural tangent kernel eigenvalues of the full two-layer $[1, 35, 1]$ Wav-KAN at initialization over the 100 training points for $b = 1, 5, 10, 15, 25$ on repeated random initializations, and check whether the spectral decay rate is monotonically decreasing in $b$. Because Proposition 3 is proven only for the rank-one kernel with fixed $S$ and $T$, any failure of that monotonicity in the full kernel — or evidence that hidden-unit count, not $b$, dominates the decay — would falsify the claimed frequency-control mechanism.

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Extended reading notes

Core claim

The paper's central claim is that the decay rate of a Wav-KAN's neural tangent kernel eigenvalues is controlled by the frequency of the mother wavelet, giving a handle on spectral bias. Working in the single-layer case with translation and scale parameters fixed so that only the weights train, the kernel becomes rank-one, $K(x_r, x_s) = \psi_1(x_r)\psi_1(x_s)$ (equation 10), and for the Morlet wavelet $\psi(x) = e^{-x^2/2}\cos(bx)$ with $S = 1$ and $T \in [0,1]$ the authors prove the eigenvalue satisfies $\lambda \geq \tfrac{1}{4} e^{-4b^2(x_r - T)^2}$ (Proposition 3). Larger $b$ keeps the bound from dropping as fast, which they interpret as slower eigenvalue decay and therefore faster convergence on high-frequency components. Numerically, on the target $u(x) = \sin(2\pi x) + 0.1\sin(50\pi x)$, raising $b$ from 1 to 15 lets a two-layer $[1, 35, 1]$ Wav-KAN capture the high-frequency detail, while $b = 25$ overfits; increasing the hidden-unit count achieves a similar effect at fixed $b$. The paper then embeds Wav-KANs in physics-informed networks (Wav-KINNs) and reports that they approximate solutions to Poisson, heat, and wave equations with high-frequency content without domain separation, and that the Helmholtz and wave equations require balancing the domain and boundary terms in the loss.

Load-bearing premise

The proof of frequency control covers only a single wavelet neuron with fixed translation and scale parameters, and the paper assumes the same behavior carries over to full multi-layer Wav-KANs with all parameters trainable.

Editorial extensions

If this is right

  • Practitioners can treat the mother wavelet's frequency $b$ as a tunable control over learning speed across frequency bands: raising $b$ accelerates convergence on high-frequency components, at the cost of overfitting when $b$ is too large (the paper observes this at $b = 25$).
  • Wav-KINNs approximate solutions of elliptic, parabolic, and hyperbolic PDEs with high-frequency content without partitioning the domain or adding Fourier-feature hyperparameters, and the paper reports they do so with fewer parameters than multiple-Fourier-feature methods.
  • Increasing the number of hidden units at fixed wavelet frequency flattens the NTK eigenvalue spectrum in the same direction as raising $b$, giving an architecture-based alternative to frequency tuning that the paper finds does not overfit.
  • Loss terms for Wav-KINNs must be balanced (the paper uses weights $\lambda_D$ and $\lambda_{\partial D}$) to reach accurate solutions on Helmholtz and wave equations; without balancing, training stalls on one term.
  • Fixing the translation and scale parameters ($T \in [0,1]$, $S = 1$) leaves only the weights trainable, which the paper shows reproduces the accuracy of the full model and reduces the parameter count.

Reading between the lines

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

  • The rank-one argument proves the eigenvalue bound for a single wavelet neuron, so the transfer to deep, wide networks with trainable $T$ and $S$ is an empirical question that the paper's own experiments only partially cover; a direct test would be measuring the full network's NTK spectrum at initialization for several $b$ values.
  • If the mechanism holds generally, the wavelet frequency behaves like a bandwidth knob, which suggests an automatic tuning scheme: choose $b$ so that the NTK eigenvalue decay rate matches the target function's known or estimated frequency content, something the paper does not develop.
  • The overfitting observed at $b = 25$ implies a bias-variance trade-off, so for targets with unknown frequency ranges, cross-validating $b$ against the NTK spectrum or the validation loss may be a cheaper alternative to grid-searching Fourier-feature scales.
  • The authors note that an NTK analysis of Wav-KINNs (where the loss includes PDE residuals) is computationally prohibitive; an approximate or empirical kernel for the residual loss could turn the observed loss-imbalance problem into a principled weighting rule instead of manual tuning.
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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 / 5 minor

Summary. The paper studies spectral bias in Wavelet-based Kolmogorov-Arnold networks (Wav-KANs) through the lens of the neural tangent kernel (NTK). It claims that the decay rate of the NTK eigenvalues of a Wav-KAN can be controlled by the frequency of the mother wavelet, thereby mitigating spectral bias and accelerating learning of high-frequency components. The theoretical analysis is carried out in Section 3.2 for a simplified rank-one kernel and is then used to motivate experiments in which the mother-wavelet frequency b is varied while approximating u(x)=sin(2πx)+0.1 sin(50πx). The paper also replaces the neural network inside PINNs by Wav-KANs, forming 'Wav-KINNs', and reports experiments on Poisson, heat, Helmholtz, and wave equations with high-frequency components.

Significance. If the central claim were established, the paper would provide a useful design principle for Wav-KANs: choose the mother-wavelet frequency to reshape the NTK spectrum and thereby control the convergence of high-frequency components in both function approximation and physics-informed settings. The empirical trend reported in Section 3.3—larger b slows eigenvalue decay and improves fitting of the 50π component—is plausible and visible in the figures, and the paper is commendable for testing the approach on several PDEs and for explicitly acknowledging limitations such as loss-term imbalance and the computational difficulty of Wav-KINN NTKs. However, the theoretical propositions that are supposed to explain this trend are invalid, and the experiments are not designed as an independent test of the theory. As it stands, the manuscript does not provide a sound theoretical foundation for the claimed frequency-control mechanism, so its significance is currently limited to an empirical observation.

major comments (4)
  1. [Sec. 3.2, Eq. (10), Proposition 2] The rank-one kernel K(x_r,x_s)=ψ1(x_r)ψ1(x_s) has exactly one nonzero eigenvalue, λ=∫_C ψ1^2(x_s) dx_s, with eigenfunction ψ1; every function orthogonal to ψ1 has eigenvalue zero. Consequently there is no eigenvalue-decay spectrum whose rate could be controlled by b. Proposition 2's assertion that g=C1 ψ1^{1/S} is an eigenfunction is also algebraically wrong for S≠1: substituting it into the integral equation gives ψ1(x_r) times a constant on the left-hand side, not λ ψ1^{1/S}(x_r). The many-eigenvalue spectra plotted in Figures 3(b), 4, 11(b), 12(b), and 13(b) therefore cannot be produced by the kernel analyzed in Proposition 2, and the theoretical mechanism asserted in the abstract is not supported.
  2. [Sec. 3.2, Proposition 3, Eqs. (36)-(39)] Equation (39) is not an eigenvalue equation: it expresses λ as a function of x_r, whereas an eigenvalue of the rank-one kernel is a constant independent of x_r. For S=1 the actual nonzero eigenvalue is ∫_0^1 e^{-(x_s-T)^2} cos^2(b(x_s-T)) dx_s, which does not contain the factor e^{-2(x_r-T)^2} cos^2(b(x_r-T)) appearing in Eq. (39). The lower bounds (14) and (15) therefore do not bound any eigenvalue, and the conclusion that 'the decay rate of the eigenvalues can be modulated by adjusting the frequency of the selected mother wave function' is not established by this proposition.
  3. [Sec. 3.3] The empirical validation is partially circular. The text states that 'the frequency parameter b can be empirically selected based on learning curves to optimize performance,' and the experiments then show that b=15 works well for target frequency 50π. The theoretical bound in Proposition 3 does not predict which b should be used, does not involve the target frequency, and does not predict the observed transition near b≈15. Thus the experiments demonstrate a fitted observation rather than a successful prediction of the proposed spectral-bias mechanism.
  4. [Sec. 3.2, Eqs. (4)-(5); Sec. 4] The NTK dynamics in Eqs. (4)-(5) require the infinite-width gradient-flow limit in which the kernel remains constant during training, but the experiments use finite-width networks (e.g., [1,35,1], [2,15,15,1], [2,20,20,20,1]) trained with Adam and LBFGS. The paper explicitly assumes that 'similar behavior will extend to the general architecture of Wav-KANs,' but this is the load-bearing transfer step from the rank-one toy model to the actual networks whose spectra are plotted. No argument, convergence check, or width-scaling study is provided to justify this transfer, so the connection between the theory and the measured finite-width eigenvalue spectra is not established.
minor comments (5)
  1. [Fig. 3 caption vs. Eq. (38)] The Morlet wavelet is written as ψ(x)=e^{-x^2} cos(bx) in the Figure 3 caption but as e^{-x^2/2} cos(bx) earlier in the text and as e^{-(x-T)^2/(2S^2)} cos(b(x-T)/S) in Eq. (38); the normalization should be made consistent throughout.
  2. [Proposition 3, Eqs. (14) and (15)] Equation (14) writes the exponent as -4(bx_r - T/S)^2, while the proof's Eq. (40) derives -4b^2(x_r-T)^2; these are not equivalent. Eq. (15) is also not implied for arbitrary T∈[0,1], since for T<1/2 the minimum over x_r∈[0,1] of the exponent occurs at x_r=0, not at x_r=1.
  3. [Sec. 4, Eq. (21)] The Helmholtz equation and the forcing q(x,y) contain the parameter k, but k is never specified. Since the solution and the forcing depend on k through the k^2 u term, the problem is incompletely defined as stated.
  4. [Sec. 3.4] The claim that increasing the number of hidden units avoids overfitting while still learning high-frequency components is supported only by training curves and eigenvalue plots; no test error or other generalization metric is reported for the hidden-unit experiment.
  5. [Throughout] There are numerous typos and minor wording issues, including 'scalars' for 'scales' in Section 3.3, 'hiden units' in Figure 6(b), 'freq' in Figure 3(c), and '100 data points x ∈ [0,1] that are spaced' in Section 3.3, which should be 'evenly spaced'.

Circularity Check

2 steps flagged · score 6.0 of 10

Frequency-control claim is partly constructed: Proposition 3's bound is an ansatz-dependent expression, and the reported best b is chosen from the same learning curves it is said to explain.

  1. other [Section 3.2, Proposition 3 and Appendix C, Eq. (40)]
    "since cos^2(b(x − T)) provides a controlling factor, λ is bounded from below by: 1/4 e^{-4b^2(x_r−T)^2} ≤ λ."

    The bound is not an eigenvalue statement: λ in the integral equation (11) is a number independent of x_r, while the RHS retains the sample point x_r. It is obtained by writing the Morlet wavelet with frequency b and reading off the b-dependent factor, not by solving the eigenproblem. The subsequent conclusion that the eigenvalue decay rate can be modulated by b therefore follows from the definition of the wavelet used in the kernel, not from the spectral analysis; the claimed control is equivalent to the input ansatz.

  2. fitted input called prediction [Section 3.3, paragraph after Fig. 3]
    "The frequency parameter b can be empirically selected based on learning curves to optimize performance and prevent overfitting. Thus, b is a valuable hyperparameter in balancing the trade-off between learning speed for high frequencies and generalization to avoid overfitting."

    This admits that the specific b values are chosen from the learning curves on the same target u(x)=sin(2πx)+0.1 sin(50πx) used for validation. The paper presents b as the discovered mechanism controlling NTK eigenvalue decay and high-frequency convergence, but b=15 (and later b=10, b=5) is selected post hoc from the loss curves; the theory does not predict these values. The demonstrated effect is therefore partly a fitted hyperparameter choice rather than an independent prediction.

full rationale

The central theoretical step, Proposition 3, is constructed rather than derived: its x_r-dependent lower bound is not a valid eigenvalue bound, and the claimed frequency dependence is already present in the chosen Morlet wavelet. A correct rank-one kernel has only one nonzero eigenvalue, so the plotted decay spectra cannot come from the given theory. I do not count the algebraic errors in Propositions 2-3 as circularity by themselves; they are correctness risks. The paper also explicitly says the extension to general Wav-KANs is anticipated, not proven. The empirical sweeps over b, hidden units, and alternative wavelets supply some independent evidence that spectral bias can be influenced, but they do not validate the constructed bound, and the best b is selected on the same data it is claimed to explain. No load-bearing self-citation chain exists here; cited works on KANs, Wav-KANs, and NTK are external. Because the main theoretical 'prediction' reduces in part to the input ansatz and the key hyperparameter is fitted post hoc, the circularity score is 6.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a sequence of modeling choices: treating the kernel as rank-one with fixed T,S; invoking the infinite-width NTK limit for finite networks; assuming the frequency-control result extends to multi-layer trainable Wav-KANs; and selecting b, loss weights, and widths by hand. None of these is externally benchmarked, and the eigenvalue bound itself contains an algebraic error.

free parameters (4)
  • Morlet wavelet frequency b = 1, 5, 10, 15, 25 in function experiments; 10 for Poisson; 5 for Heat, Helmholtz, Wave
    Chosen by hand from learning curves to make the high-frequency target learnable; b=25 overfits. This parameter is the central control claimed by the paper.
  • Loss-balancing weights λD, λ∂D, λbc, λic, λnbc = not reported
    Manually selected in Section 4 to make Helmholtz and Wave equation approximations succeed; no values or selection procedure are given.
  • Number of hidden units n = 10, 50, 100, 150, 200 in Section 3.4; 35 and 64 in other experiments
    Architectural choice shown to affect NTK eigenvalue decay; the paper offers no theory linking width to spectral control.
  • Translation and scale parameters T, S when fixed = T in [0,1], S=1
    Assumed fixed in the theoretical analysis for tractability; in most experiments T and S are trainable, so the theoretical setting does not match the experiments.
assumptions (6)
  • standard math Kolmogorov-Arnold representation theorem (Theorem 1)
    Used as the architectural justification for KANs and Wav-KANs in Section 3.
  • domain assumption NTK gradient-flow and infinite-width limits apply (Eqs. 4-5)
    The paper solves the gradient-flow dynamics with a constant NTK, but the experiments use finite-width networks trained with Adam and LBFGS.
  • ad hoc to paper Rank-one kernel with fixed T and S is representative of full Wav-KANs
    Section 3.2 reduces the kernel to K=ψψ and says 'we anticipate that similar behavior will extend to the general architecture', an extrapolation not proven.
  • ad hoc to paper Eigenfunctions have the form g = C ψ^(1/S) (Proposition 2)
    Derived in Appendix B under the fixed-S differential equation; the subsequent integral-eigenvalue computation in Appendix C uses this form and produces an x_r-dependent λ.
  • ad hoc to paper Setting S=1 and T in [0,1] preserves the spectral conclusion
    Appendix C states 'Let us assume that the weights can adapt the mother function ψ and set S=1', which is needed to obtain the lower bound.
  • standard math Mercer/integral-equation eigenvalue framework on compact domains
    Equation (11) uses the standard integral equation for kernel eigenfunctions on compact C; this background is standard.

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Cite this review

Pith. "Pith review of On the study of frequency control and spectral bias in Wavelet-Based Kolmogorov Arnold networks: A path to physics-informed KANs." pith.science (2026). https://pith.science/paper/5WDPN3DW

@misc{pith2026250200280,
  author       = {Pith},
  title        = {Pith review of: On the study of frequency control and spectral bias in Wavelet-Based Kolmogorov Arnold networks: A path to physics-informed KANs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5WDPN3DW}},
  note         = {Machine review of arXiv:2502.00280}
}
read the original abstract

Spectral bias, the tendency of neural networks to prioritize learning low-frequency components of functions during the initial training stages, poses a significant challenge when approximating solutions with high-frequency details. This issue is particularly pronounced in physics-informed neural networks (PINNs), widely used to solve differential equations that describe physical phenomena. In the literature, contributions such as Wavelet Kolmogorov Arnold Networks (Wav-KANs) have demonstrated promising results in capturing both low- and high-frequency components. Similarly, Fourier features (FF) are often employed to address this challenge. However, the theoretical foundations of Wav-KANs, particularly the relationship between the frequency of the mother wavelet and spectral bias, remain underexplored. A more in-depth understanding of how Wav-KANs manage high-frequency terms could offer valuable insights for addressing oscillatory phenomena encountered in parabolic, elliptic, and hyperbolic differential equations. In this work, we analyze the eigenvalues of the neural tangent kernel (NTK) of Wav-KANs to enhance their ability to converge on high-frequency components, effectively mitigating spectral bias. Our theoretical findings are validated through numerical experiments, where we also discuss the limitations of traditional approaches, such as standard PINNs and Fourier features, in addressing multi-frequency problems.

Figures

Figures reproduced from arXiv: 2502.00280 by the authors.

Figure 1
Figure 1. Kolmogorov-Arnold representation as KAN of [2,5,1] layers, where each is a learnable [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Example of a Wav-KAN used to approximate the function [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Analyzing Wav-KAN behavior through the NTK. a) Approximation of function ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: NTK eigenvalues in descending order for b = 15 and b = 25. similar accuracy to that of the Wav-KAN with non-fixed parameters shown in [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: a) Function approximation for (16) with b = 15. b) Function approximation for (16) with b = 25. c) Loss evolution for b = 15. d) Loss evolution for b = 25. patterns in data while maintaining robust generalization. 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.5 0.0 0.5 1.0 Actual Pred…
Figure 6
Figure 6. Figure 6: Approximation of function 16 using a Wav-KAN with a fixed frequency Morlet mother wavelet. a) Approximation by a two-layer Wav-KAN with structure [1, n, 1], where n = 10, 50, 100, 150, 200, trained for 1000 epochs. b) NTK eigenvalues in descending order for each value …
Figure 7
Figure 7. Figure 7: Approximation of the Poisson equation with high-frequency components using a Wav [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Approximation of the Heat equation with high-frequency components using a Wav [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: a) Solution approximation of the Helmholtz equation with high-frequency components [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: a) Solution approximation of the Wave equation with high-frequency components us [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: a) Approximation of function (16) using a Wav-KAN with two layers [1,35,1] and the real Shannon Wavelet as mother wavelet for different ω1 values: 1, 3, 5 y 7, trained for 1000 epochs. b) NTK eigenvalues in descending order for each value of ω1. 27 [PITH_FULL_IMAGE:f…
Figure 12
Figure 12. Figure 12: a) Approximation of function (16) using a Wav-KAN with two layers [1,35,1] and the Mexican Hat Wavelet as the mother wavelet for different σ values: 0.1, 0.2, 5 y 10, trained for 1000 epochs. b) NTK eigenvalues in descending order for each value of σ. Derivative of Ga…
Figure 13
Figure 13. Figure 13: a) Approximation of function (16) using a Wav-KAN with two layers [1,35,1] and the Derivative of Gaussian Wavelet as the mother wavelet for different σ values: 0.1, 0.2, 5 y 10, trained for 1000 epochs. b) NTK eigenvalues in descending order for each value of σ. 29 […

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Pith tools

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