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Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Conformal prediction gives Kolmogorov-Arnold network ensembles prediction intervals with a distribution-free coverage guarantee.

desk verdict Competent application of split conformal prediction to KAN ensembles; theory is sound, but the empirical claims need repeated-seed statistics to support them. read the letter →

arxiv 2504.15240 v1 pith:L5C6M6YW submitted 2025-04-21 cs.LG

Amirhossein Mollaali , Christian Bolivar Moya , Amanda A. Howard , Alexander Heinlein , Panos Stinis , Guang Lin This is my paper · ORCID
classification cs.LG MSC 68T0762G15
keywords conformalpredictionKolmogorov-Arnoldnetworksuncertaintyquantificationensemblemethodscoverageguaranteesscientificmachinelearningmulti-fidelitymodelsdomaindecomposition
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 claims that wrapping a KAN ensemble in split conformal prediction turns the ensemble spread into prediction intervals with a distribution-free coverage guarantee: for any test input, the true output lands in the interval with probability at least $1-\alpha$, provided the calibration data are exchangeable with the test data. The method is demonstrated on a 1-D function, a 2-D function, a multi-fidelity problem, and a wave-equation PDE, where the conformalized intervals reach roughly 95% coverage while raw ensemble intervals often under-cover. A secondary finding is that domain-decomposed FBKANs produce considerably narrower conformal intervals than standard KANs at the same coverage. This matters for scientific machine learning because it attaches a rigorous, user-chosen reliability statement to KAN predictions in data-sparse settings.

What carries the argument

The load-bearing object is split conformal prediction applied to a KAN ensemble. Conformal prediction is a distribution-free wrapper: it uses a held-out calibration set to estimate how badly the model's own uncertainty measure misses, then widens the ensemble interval by the calibration quantile. The ensemble standard deviation serves as the denominator of the nonconformity score, so the final interval has the shape $\mu_M \pm \hat{q}_\alpha \sigma_M$, and the conformal theorem converts the empirical quantile into the coverage guarantee. For FBKANs the ensemble is built from domain-decomposed sub-KANs stitched by a partition of unity; for MFKANs a frozen low-fidelity KAN feeds an ensemble of linear and nonlinear high-fidelity KANs.

What would settle it

Rerun the conformalized ensemble on exchangeable calibration and test data with a different seed or base model and check whether the empirical coverage on a large test set stays between $1-\alpha$ and $1-\alpha+1/(n+1)$. The claim would be falsified if coverage fell systematically below $1-\alpha$; the paper's own experiments are this check for its four test problems.

Watch

Extended reading notes

Core claim

The central claim is that ensembles of KAN-based models, including FBKANs and MFKANs, can be conformalized to yield calibrated prediction intervals with guaranteed coverage without assuming a particular error distribution. Concretely, after training $M$ models from different random initializations, the ensemble mean $\mu_M(x)$ and standard deviation $\sigma_M(x)$ define a heuristic uncertainty. The paper's procedure computes nonconformity scores $s_j = |f(x_j)-\mu_M(x_j)|/\sigma(x_j)$ on held-out calibration data and rescales the ensemble spread by the empirical quantile $\hat{q}_\alpha = \lceil(n+1)(1-\alpha)\rceil/n$ of these scores, producing $C_\alpha(x) = [\mu_M(x)-\hat{q}_\alpha\sigma_M(x),\ \mu_M(x)+\hat{q}_\alpha\sigma_M(x)]$. The paper states that exchangeability of calibration and test data gives $P(f(x_{\mathrm{test}}) \in C_\alpha(x_{\mathrm{test}})) \ge 1-\alpha$, and its four experiments report empirical coverage near the target 95%.

Load-bearing premise

The coverage guarantee rests entirely on the calibration data being exchangeable with the test data; the paper assumes i.i.d. draws, so if the test inputs come from a different distribution, are correlated with the calibration set, or arrive in a non-exchangeable order, the stated coverage can fail.

Editorial extensions

If this is right

  • At miscoverage level $\alpha = 0.05$, conformalized KAN, FBKAN, and MFKAN intervals reach roughly 95% empirical coverage across all four test problems, correcting the under-coverage of raw ensemble intervals.
  • With $n$ exchangeable calibration points, coverage is at least $1-\alpha$ and at most $1-\alpha + 1/(n+1)$, independent of the data distribution.
  • Because the ensemble standard deviation is the only model-specific ingredient, the same wrapper can be applied to other KAN variants beyond FBKANs and MFKANs.
  • Domain decomposition improves interval sharpness: conformalized FBKANs give narrower intervals than standard conformalized KANs at the same coverage.
  • Larger calibration sets reduce coverage fluctuations, while ensemble size mainly influences interval width rather than the coverage guarantee.

Reading between the lines

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

  • The wrapper should transfer to any KAN variant that can be ensembled, such as physics-informed or wavelet KANs, as long as the ensemble standard deviation remains a meaningful uncertainty heuristic; a direct test would be to rerun the same conformal procedure with a different base architecture.
  • The guarantee is marginal over exchangeability, not conditional on the input; users who need per-input reliability in heterogeneous domains would have to calibrate locally or conditionally, which the paper does not develop.
  • A natural extension is local or adaptive calibration: binning calibration points by input region and computing region-specific quantiles could shrink the widest intervals while preserving coverage, especially where FBKAN subdomains localize error.
  • For multi-fidelity problems, conformalized MFKAN interval width could serve as an acquisition signal for choosing where to collect new high-fidelity data, since the intervals reflect where the surrogate is least certain.
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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

3 major / 5 minor

Summary. The paper proposes Conformalized-KANs, a UQ framework that combines an ensemble of Kolmogorov-Arnold Networks (KANs) with split conformal prediction. The ensemble mean is used as the point predictor and the ensemble standard deviation as a heuristic uncertainty scale; conformal calibration then yields intervals of the form [mu_M(x) - qhat*sigma_M(x), mu_M(x) + qhat*sigma_M(x)] with a claimed coverage guarantee P(f(x_test) in C_alpha(x_test)) >= 1-alpha. The method is applied to standard KANs and to two KAN variants, FBKANs and MFKANs, in four experiments: a 1-D function, a 2-D function, a multi-fidelity problem, and a wave-equation PDE. The reported empirical coverage values are close to 95%, and ablations study the effect of ensemble size, number of subdomains, and calibration-set size. The theoretical content is a correct application of standard split-conformal theory; the main weaknesses are empirical reporting (single-run coverage statistics) and some algorithmic details in the definition of the nonconformity score.

Significance. If the empirical claims are made rigorous, this would be a useful and practical application of established conformal-prediction theory to a popular class of SciML models. The paper's strength is that it correctly identifies the standard split-conformal theorem and applies it in the right way: the ensemble is trained before calibration, the nonconformity scores are computed on held-out calibration data, and the intervals have the correct conformal form. The application to FBKANs and MFKANs broadens the scope of conformal UQ to recent KAN variants. The coverage guarantee itself is not new, so the contribution is primarily empirical and architectural. The current evidence, however, is weakened by the absence of repeated-seed or multi-split statistics: all reported coverage values are single realizations, and under the nominal 95% level with test sets of size 1000 the sampling noise is roughly +/-1.4 percentage points at two standard deviations, which is comparable to the observed differences between methods.

major comments (3)
  1. [Section 3.2, Algorithm 1] The nonconformity score is defined as s_j = |f(x_j) - mu_M(x_j)| / sigma(x_j). For the conformal guarantee to hold, this score must be a well-defined, fixed function on all calibration and test points. If the ensemble standard deviation is zero or numerically negligible for any point, the score is undefined or extremely large, which can make the conformal quantile and the resulting intervals unstable. The paper should state a safeguard such as sigma_epsilon(x) = max(sigma_M(x), epsilon) and discuss how this affects the coverage guarantee. In addition, Algorithm 1 uses inconsistent notation: line 4 divides by sigma(x_j) and line 8 uses sigma(xtest), while Section 3.1 defines sigma_M; this should be corrected.
  2. [Section 4, Tables 1-4] Each reported coverage value (e.g., 96.10%, 95.50%, 96.01%, 95.44%) comes from a single data split and a single trained ensemble. For a test set of size 1000 and nominal coverage 0.95, the standard error of the empirical coverage is about 0.7 percentage points, so two-standard-deviation intervals span roughly 1.4 percentage points. The differences between methods and the deviations from 95% in Tables 1-4 are therefore all within ordinary sampling noise. The claims that Conformalized-KANs 'consistently achieve the target 95% coverage' and that FBKANs 'consistently outperform KANs' are not substantiated by single-run statistics. Please report coverage means and standard deviations across multiple independent data splits and model initializations.
  3. [Appendix A, Figures 6-8] The ablation studies visualize coverage and width curves as functions of ensemble size M and the number of subdomains L, but the paper does not state whether each plotted curve is a single realization or an average over repeated runs. If these are single runs, the same sampling-noise critique applies to the ablation conclusions, including the claim that the conformalized variants 'consistently' maintain coverage near the target. The figure captions and text should clarify the number of independent realizations and include error bars or confidence bands.
minor comments (5)
  1. [Abstract and Section 2.2] There are typographical errors: 'multifideilty' should be 'multi-fidelity' in the abstract, and 'parition' should be 'partition' in Section 2.2.
  2. [Section 2.3, Eq. (3)] The penalty term involving lambda_alpha is garbled in the displayed equation ('+ lambda_alpha alpha^n + w' appears as '+ λααn +w'). Please rewrite the equation so that the exponent on alpha is clear.
  3. [Section 3.1] The text says the 1.96-sigma interval gives 'approximately 95% confidence, assuming the predictions follow a normal distribution.' This is only valid for Gaussian predictive distributions, and the paper correctly notes this limitation; however, the point is stated somewhat informally and could be clarified.
  4. [Section 4.5] For the PDE experiment, the calibration and test sets should be drawn from the same distribution over (x,t) for the exchangeability assumption to hold. The paper should state explicitly how the 1200 calibration points and 10000 test points are sampled, especially because the training uses physics-informed losses with initial, boundary, and residual data.
  5. [Section 3.2] The sentence 'Given that the data is exchangeable, the method assures coverage' would be clearer if it specified that the guarantee holds over the randomness in the calibration data for a fixed test point, rather than suggesting a deterministic statement for any test set.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the conformal coverage guarantee is a direct application of the external split-conformal theorem, not a restatement of fitted inputs.

full rationale

The claimed derivation chain is: (i) train an ensemble of KANs on training data; (ii) compute nonconformity scores s_j = |f(x_j)-mu_M(x_j)|/sigma(x_j) on a held-out calibration set; (iii) take q_hat as the empirical ceil((n+1)(1-alpha))/n quantile; and (iv) form intervals C_alpha(x_test) = [mu_M(x_test) - q_hat * sigma_M(x_test), mu_M(x_test) + q_hat * sigma_M(x_test)]. The coverage statement in Section 3.2, 'Given that the data is exchangeable, the method assures coverage: P(f(x_test) in C_alpha(x_test)) >= 1-alpha,' is the standard split-conformal theorem imported from external references [40,42]. The quantile is estimated from calibration data, not from the test set, and the model is fixed before calibration, so the guarantee is not a fitted quantity in disguise. Self-citations [28,29] describe the FBKAN and MFKAN architectures, and [43,44] describe previous applications of conformal prediction to DeepONets; neither is used as the source of the coverage theorem. The empirical coverage values in Tables 1-4 are held-out checks of the theorem under synthetic i.i.d. data, not consequences of any fitted parameter. The acknowledged exchangeability assumption is a validity condition, not a circular step. The fact that each experiment reports a single coverage realization is a statistical precision concern, but it does not make the derivation circular.

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

The central claim, that conformalized KAN ensembles achieve target coverage, rests on the conformal prediction theorem and the exchangeability assumption, not on fitted constants. Hyperparameters such as ensemble size, subdomain count, and loss weights affect interval width and model accuracy but do not enter the coverage guarantee.

free parameters (4)
  • Ensemble size M = 4 (1D, 2D), 5 (MF), 10 (PDE)
    Chosen per experiment; affects the heuristic sigma and interval width, but not the conformal coverage guarantee.
  • Number of FBKAN subdomains L = 10 (1D), 4 (2D and PDE)
    Chosen by hand; affects accuracy and interval width, but not the conformal validity.
  • PDE residual weighting lambda_res = 0.01
    Hand-set to balance the physics loss terms; not fit to the coverage claim.
  • MFKAN exponent n = 4
    Hyperparameter controlling the linear and nonlinear blend alpha, taken from prior work [29].
assumptions (4)
  • standard math Kolmogorov-Arnold representation theorem
    Justifies the KAN architecture; not central to the UQ claim.
  • domain assumption Calibration and test data are exchangeable (i.i.d. in experiments)
    Required for the conformal coverage guarantee; stated in Section 3.2.
  • domain assumption The trained ensemble model is fixed before calibration
    Split conformal prediction requires that the predictive model is not updated with calibration data.
  • standard math Conformal quantile theorem of Vovk et al. and Angelopoulos and Bates
    Imported from refs [40,42] to convert score quantiles into coverage bounds.

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Cite this review

Pith. "Pith review of Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning." pith.science (2026). https://pith.science/paper/L5C6M6YW

@misc{pith2026250415240,
  author       = {Pith},
  title        = {Pith review of: Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5C6M6YW}},
  note         = {Machine review of arXiv:2504.15240}
}
read the original abstract

This paper explores uncertainty quantification (UQ) methods in the context of Kolmogorov-Arnold Networks (KANs). We apply an ensemble approach to KANs to obtain a heuristic measure of UQ, enhancing interpretability and robustness in modeling complex functions. Building on this, we introduce Conformalized-KANs, which integrate conformal prediction, a distribution-free UQ technique, with KAN ensembles to generate calibrated prediction intervals with guaranteed coverage. Extensive numerical experiments are conducted to evaluate the effectiveness of these methods, focusing particularly on the robustness and accuracy of the prediction intervals under various hyperparameter settings. We show that the conformal KAN predictions can be applied to recent extensions of KANs, including Finite Basis KANs (FBKANs) and multifideilty KANs (MFKANs). The results demonstrate the potential of our approaches to improve the reliability and applicability of KANs in scientific machine learning.

Figures

Figures reproduced from arXiv: 2504.15240 by the authors.

Figure 1
Figure 1. Schematic representation of the Conformalized-KANs framework. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Conformalized prediction intervals generated on 1-D function test dataset, computed with an ensemble size of 4 and a [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Conformalized prediction intervals generated by Conformalized-KANs and Conformalized-FBKANs on the 2-D [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Conformalized prediction intervals generated by ensemble MFKANs and Conformalized-MFKANs on the high-fidelity [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Conformalized prediction intervals generated by Conformalized-KANs and Conformalized-FBKANs on the wave [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Experiment 1: 1-D Function. (a) Coverage performance of ensemble KANs and ensemble FBKANs, along with their [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Experiment 3: Multi-Fidelity Problem. (a) Coverage performance of ensemble MFKANs, along with its conformalized [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Experiment 4: PDE Problem. (a) Coverage performance of ensemble KANs and ensemble FBKANs, along with [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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Forward citations

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.