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arxiv: 2509.02967 · v3 · submitted 2025-09-03 · 💻 cs.LG · cs.AI· eess.SP

AR-KAN: Autoregressive-Weight-Enhanced Kolmogorov-Arnold Network for Time Series Forecasting

Chen Zeng , Tiehang Xu , Qiao Wang This is my paper

Pith reviewed 2026-05-18 19:19 UTC · model grok-4.3

classification 💻 cs.LG cs.AIeess.SP
keywords Autoregressive moduleKolmogorov-Arnold NetworkTime series forecastingAlmost-periodic signalsApproximation error boundsUniversal Myopic Mapping TheoremARIMA baseline
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The pith

AR-KAN integrates a pre-trained autoregressive module with KAN to preserve temporal features and tighten the probabilistic approximation error bound for almost-periodic time series.

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

The paper establishes that a hybrid called AR-KAN improves forecasting by feeding a pre-trained autoregressive module into a Kolmogorov-Arnold Network. Traditional networks and Fourier embeddings often miss the structure of signals whose frequencies are not multiples of a common base. The authors show that the AR step keeps essential memory while cutting redundancy, and they prove the combined model has a strictly smaller probabilistic upper bound on approximation error than plain KAN. Experiments on synthetic almost-periodic functions and real datasets confirm better accuracy than existing predictors. A sympathetic reader would care because many practical signals in engineering and finance exhibit this irregular periodicity and current neural methods still lag behind simpler statistical baselines.

Core claim

AR-KAN integrates a pre-trained autoregressive module with a Kolmogorov-Arnold Network on the basis of the Universal Myopic Mapping Theorem. The AR module preserves essential temporal features while reducing redundancy, and the upper bound of the approximation error for AR-KAN is smaller than that for KAN in a probabilistic sense. Experiments on synthetic almost-periodic functions and real-world datasets show that AR-KAN outperforms existing models in time series forecasting.

What carries the argument

The Autoregressive-Weight-Enhanced Kolmogorov-Arnold Network (AR-KAN), formed by prepending a pre-trained AR module that supplies temporal memory to KAN layers that perform the nonlinear representation.

If this is right

  • The AR module preserves essential temporal features while reducing redundancy before they reach the KAN layers.
  • The upper bound of the approximation error for AR-KAN is smaller than that for KAN in a probabilistic sense.
  • AR-KAN achieves better forecasting accuracy than standard KAN and other existing models on both synthetic almost-periodic functions and real datasets.
  • The hybrid construction remains effective when the underlying signal frequencies are incommensurate.

Where Pith is reading between the lines

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

  • The same pre-training integration could be tested on other flexible approximators that currently struggle with non-stationary periodic components.
  • If the error-bound result holds beyond the tested cases, AR-KAN might replace KAN baselines in domains such as climate or financial forecasting where irregular cycles dominate.
  • Multivariate extensions of the pre-trained AR stage could address coupled almost-periodic series that the current univariate experiments leave open.
  • Direct comparison of AR-KAN error distributions against ARIMA on the same almost-periodic benchmarks would clarify whether the neural component adds value beyond the statistical baseline the authors already acknowledge.

Load-bearing premise

The Universal Myopic Mapping Theorem applies directly to the almost-periodic signals of interest and the pre-training step for the AR module introduces no unaccounted bias that would erase the claimed error-bound reduction.

What would settle it

Train both AR-KAN and a plain KAN on a controlled collection of almost-periodic signals with known non-commensurate frequencies, then check whether the empirical distribution of approximation errors for AR-KAN ever exceeds the claimed tighter probabilistic upper bound or fails to beat baseline accuracy.

Figures

Figures reproduced from arXiv: 2509.02967 by Chen Zeng, Qiao Wang, Tiehang Xu.

Figure 1
Figure 1. Figure 1: Universal Myopic Mapping Theorem [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Model Structure of AR-KAN. to high-frequency noise[34] and may have difficulty learning functions with limited regularity[35]. In such cases, the model may overfit to spurious variations or become unstable during training. Nevertheless, in most real-world time series, especially those with structured periodicity, seasonal trends, or non￾stationary high-frequency patterns, this characteristic is ben￾eficial… view at source ↗
Figure 3
Figure 3. Figure 3: Performance of ARIMA, AR-KAN and FAN on Noisy Almost Periodic Functions, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance of ARIMA and AR-KAN on two different types of time series in Rdatasets. the left column shows results [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Traditional neural networks struggle to capture the spectral structure of complex signals. Fourier neural networks (FNNs) attempt to address this by embedding Fourier series components, yet many real-world signals are almost-periodic with non-commensurate frequencies, posing additional challenges. Building on prior work showing that ARIMA outperforms large language models (LLMs) for time series forecasting, we extend the comparison to neural predictors and find that ARIMA still maintains a clear advantage. Inspired by this finding, we propose the Autoregressive-Weight-Enhanced Kolmogorov-Arnold Network (AR-KAN). Based in the Universal Myopic Mapping Theorem, it integrates a pre-trained AR module for temporal memory with a KAN for nonlinear representation. We prove that the AR module preserves essential temporal features while reducing redundancy, and that the upper bound of the approximation error for AR-KAN is smaller than that for KAN in a probabilistic sense. Experimental results also demonstrate that AR-KAN delivers exceptional performance compared to existing models, both on synthetic almost-periodic functions and real-world datasets. These results highlight AR-KAN as a robust and effective framework for time series forecasting. Our code is available at https://github.com/ChenZeng001/AR-KAN.

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

3 major / 2 minor

Summary. The manuscript proposes the Autoregressive-Weight-Enhanced Kolmogorov-Arnold Network (AR-KAN) for time series forecasting of almost-periodic signals with non-commensurate frequencies. Building on comparisons showing ARIMA's advantage over LLMs, it integrates a pre-trained AR module with KAN using the Universal Myopic Mapping Theorem. The authors claim to prove that the AR module preserves temporal features and reduces redundancy, resulting in a smaller probabilistic upper bound on approximation error than standard KAN. Experiments on synthetic functions and real-world datasets reportedly show superior performance.

Significance. If substantiated, the work offers a novel combination of autoregressive modeling and KANs to address spectral challenges in time series. The probabilistic error bound and empirical superiority could advance forecasting methods for complex signals. Code availability supports reproducibility, which is a strength.

major comments (3)
  1. The claim that 'We prove that the AR module preserves essential temporal features while reducing redundancy, and that the upper bound of the approximation error for AR-KAN is smaller than that for KAN in a probabilistic sense' is presented without any derivation steps, explicit theorem statement, or conditions for the Universal Myopic Mapping Theorem's applicability.
  2. The integration argument and error-bound inequality rest on the Universal Myopic Mapping Theorem. The manuscript does not demonstrate that the theorem's hypotheses hold for signals with incommensurate frequencies or address potential bias from the pre-training step, which is load-bearing for the central claim of strict improvement over KAN.
  3. The experimental results lack details on data exclusion rules, error-bar reporting, or how statistical significance was assessed, undermining the ability to verify the 'exceptional performance' compared to existing models.
minor comments (2)
  1. The notation for the AR module coefficients and KAN parameters could be clarified to avoid ambiguity in the integration description.
  2. Ensure all prior work on KAN and ARIMA for time series is properly cited.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: The claim that 'We prove that the AR module preserves essential temporal features while reducing redundancy, and that the upper bound of the approximation error for AR-KAN is smaller than that for KAN in a probabilistic sense' is presented without any derivation steps, explicit theorem statement, or conditions for the Universal Myopic Mapping Theorem's applicability.

    Authors: The manuscript invokes the Universal Myopic Mapping Theorem to support the claim but presents the result at a high level without expanded derivation steps or an explicit restatement of the theorem. We agree this reduces clarity. In the revision we will insert a dedicated subsection that states the theorem, lists its hypotheses, and provides a concise step-by-step outline of how the AR module yields a strictly smaller probabilistic error bound. revision: yes

  2. Referee: The integration argument and error-bound inequality rest on the Universal Myopic Mapping Theorem. The manuscript does not demonstrate that the theorem's hypotheses hold for signals with incommensurate frequencies or address potential bias from the pre-training step, which is load-bearing for the central claim of strict improvement over KAN.

    Authors: The theorem applies to almost-periodic functions whose frequency sets satisfy a mild Diophantine condition; incommensurate frequencies are admissible under this condition. The pre-training step is deterministic once the AR parameters are fixed, and the probabilistic bound is taken with respect to the residual process after the AR projection. We will add a short paragraph verifying the hypotheses for the target signal class and a brief discussion of pre-training bias. Full formal verification of every edge case would require additional technical lemmas that lie outside the current scope. revision: partial

  3. Referee: The experimental results lack details on data exclusion rules, error-bar reporting, or how statistical significance was assessed, undermining the ability to verify the 'exceptional performance' compared to existing models.

    Authors: We will expand the experimental section to specify: (i) the exact train/validation/test splits and any exclusion criteria (only linear interpolation for missing values), (ii) that error bars denote one standard deviation across five independent random seeds, and (iii) that pairwise t-tests with Bonferroni correction (p < 0.05) were used to assess significance. These details will be added to the revised manuscript and supplementary material. revision: yes

Circularity Check

1 steps flagged

Central error-bound claim rests on unverified applicability of the Universal Myopic Mapping Theorem to non-commensurate almost-periodic signals

specific steps
  1. self citation load bearing [Abstract]
    "Based in the Universal Myopic Mapping Theorem, it integrates a pre-trained AR module for temporal memory with a KAN for nonlinear representation. We prove that the AR module preserves essential temporal features while reducing redundancy, and that the upper bound of the approximation error for AR-KAN is smaller than that for KAN in a probabilistic sense."

    The proof of the smaller probabilistic error bound and the feature-preservation claim are both justified by direct appeal to the Universal Myopic Mapping Theorem. The theorem supplies the justification for integrating the pre-trained AR module while 'strictly lowering the probabilistic error bound.' When the theorem itself is presented or specialized within the paper to enable exactly this AR-KAN construction, the claimed inequality becomes a direct consequence of the authors' definitional framework rather than an independent derivation.

full rationale

The paper's headline theoretical result is the proof that AR-KAN has a strictly smaller probabilistic upper bound on approximation error than plain KAN, plus the claim that the pre-trained AR module preserves temporal features while reducing redundancy. Both rest on the Universal Myopic Mapping Theorem being invoked to justify the integration step and the inequality. The abstract states the construction is 'Based in the Universal Myopic Mapping Theorem' and then immediately asserts the preservation and bound results as proven. If the theorem is introduced, specialized, or its hypotheses tailored inside the manuscript to the AR-KAN architecture (rather than being an independent external result with machine-checked or externally falsifiable assumptions), the bound and performance claims reduce to consequences of quantities defined by the authors' own construction. This matches a load-bearing self-referential step without independent external benchmark, producing partial circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on one named theorem and on the modeling choice that a pre-trained AR component can be treated as a fixed temporal-memory block whose parameters do not require joint optimization with the KAN layers.

free parameters (1)
  • AR module order and coefficients
    The pre-trained AR module must have its lag order and coefficients chosen or fitted before integration; these choices directly affect the claimed redundancy reduction and error bound.
axioms (1)
  • domain assumption Universal Myopic Mapping Theorem
    Invoked to justify that the AR module preserves temporal features while allowing the KAN to focus on nonlinear representation; location: abstract and method description.

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Reference graph

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