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arxiv: 2410.03801 · v5 · submitted 2024-10-04 · 💻 cs.LG · cs.NE· stat.ML

P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimization

Xavier Warin This is my paper

Pith reviewed 2026-05-23 19:53 UTC · model grok-4.3

classification 💻 cs.LG cs.NEstat.ML
keywords Kolmogorov-Arnold networkP1-KANfunction approximationerror boundsuniversal approximationhydraulic optimizationneural networksmachine learning
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The pith

P1-KAN approximates high-dimensional irregular functions with error bounds and outperforms other KANs and MLPs in accuracy and speed.

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

This paper proposes P1-KAN, a new Kolmogorov-Arnold network for approximating potentially irregular functions in high dimensions. It derives error bounds under the assumption that the Kolmogorov-Arnold expansion functions are sufficiently smooth and establishes universal approximation theorems when the target function is only continuous. The network shows better accuracy and faster convergence than multilayer perceptrons. On irregular functions it outperforms several other KAN variants while achieving similar accuracy to the original spline-based KAN on smooth functions. The approach is demonstrated on the optimization of a French hydraulic valley.

Core claim

P1-KAN is an effective Kolmogorov-Arnold network that provides approximation error bounds assuming smooth expansion functions and universal approximation theorems for continuous functions. It demonstrates superior accuracy and convergence speed over multilayer perceptrons, outperforms competing KAN networks on irregular functions, and matches the original KAN on smooth functions, with an application to hydraulic valley optimization.

What carries the argument

P1-KAN architecture, a Kolmogorov-Arnold network variant that uses specific basis expansions to handle irregular high-dimensional functions.

If this is right

  • Hydraulic valley optimization yields improved results through more accurate function approximation.
  • KANs become applicable to a wider range of non-smooth high-dimensional problems.
  • Training converges faster than multilayer perceptrons while reaching comparable or better accuracy.
  • Similar performance to the original spline KAN is retained on smooth target functions.

Where Pith is reading between the lines

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

  • The architecture may extend to other engineering optimization tasks that involve irregular objective landscapes.
  • Combining P1-KAN with existing networks could address datasets containing both smooth and irregular regions.
  • If the smoothness assumption is verified in applications, the bounds could support more reliable error estimates in deployed models.

Load-bearing premise

The error bounds assume that the Kolmogorov-Arnold expansion functions are sufficiently smooth.

What would settle it

An irregular test function where P1-KAN fails to exceed the accuracy of the compared KAN networks or where measured errors violate the stated bounds.

Figures

Figures reproduced from arXiv: 2410.03801 by Xavier Warin.

Figure 1
Figure 1. Figure 1: Uniform P1 basis functions on [0, 1] with P = 5. where (a l,k,j p )p=0,P are trainable variables and (Ψl,j p )p=0,P is the basis of the shape function Ψl,j p with compact support in each interval [ˆx l j,p−1 , xˆ l j,p+1] for p = 1, . . . , P − 1 and defined as: Ψ l,j p (x) =    x−xˆ l j,p−1 xˆ l j,p−xˆ l j,p−1 x ∈ [ˆx l j,p−1 , xˆ l j,p] xˆ l j,p+1−x xˆ l j,p+1−xˆ l j,p x ∈ [ˆx l j,p, xˆ l j,p+1] suc… view at source ↗
Figure 2
Figure 2. Figure 2: Reference function and approximation with or without adaptation. Vertices on the x-axis. P Error without adaptation Error with adaptation 5 0.1545 0.00492 10 0.0737 3.55E-4 20 4.7E-4 5.22E-5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Function A in 2D [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: To approximate a function f with a neural network κ, we use the classical quadratic loss function defined as: L = E[(f(X) − κ(X))2 ], where X is a uniform random variable on [0, 1]n. Using a stochastic gradient algorithm with the ADAM optimizer, a learning rate of 10−3 , and a batch size of 1000, we minimize the loss L. The MLPs use a ReLU activation function, with either 2 layers with 10, 20, 40 neurons f… view at source ↗
Figure 5
Figure 5. Figure 5: Results in dimension 6 for function A [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results in dimension 12 for function A. As these graphs are obtained with one run, we provide in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results in dimension 2 for function B [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results in dimension 5 for function B Results in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Structure of the valley [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
read the original abstract

A new Kolmogorov-Arnold network (KAN) is proposed to approximate potentially irregular functions in high dimensions. We provide error bounds for this approximation, assuming that the Kolmogorov-Arnold expansion functions are sufficiently smooth. When the function is only continuous, we also provide universal approximation theorems. We show that it outperforms multilayer perceptrons in terms of accuracy and convergence speed. We also compare it with several proposed KAN networks: it outperforms all networks for irregular functions and achieves similar accuracy to the original spline-based KAN network for smooth functions. Finally, we compare some of the KAN networks in optimizing a French hydraulic valley.

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

2 major / 1 minor

Summary. The manuscript proposes P1-KAN, a Kolmogorov-Arnold network variant for high-dimensional approximation of potentially irregular functions. It derives approximation error bounds under the assumption that the Kolmogorov-Arnold expansion functions are sufficiently smooth, provides universal approximation theorems when the target is merely continuous, and reports empirical results claiming outperformance over MLPs in accuracy and convergence speed, outperformance over other KAN variants on irregular functions, and comparable accuracy to the original spline-based KAN on smooth functions. The work concludes with an application to optimization of a French hydraulic valley.

Significance. If the empirical comparisons prove robust, P1-KAN could supply a practical architecture for high-dimensional function approximation in engineering contexts. The combination of explicit error bounds (under smoothness) and a UAT for the continuous case is a constructive element; the hydraulic-valley application supplies a concrete, falsifiable use case.

major comments (2)
  1. [Theory / error bounds derivation] Theory section on error bounds: the stated bounds are derived only under the explicit assumption that the Kolmogorov-Arnold expansion functions are sufficiently smooth. The central empirical claim, however, concerns superior performance on irregular (likely non-smooth) functions, for which the smoothness hypothesis is violated and only the rate-free UAT remains. This leaves the theoretical rationale for the headline performance advantage on irregular targets unsupported by quantitative rates.
  2. [Experiments / comparison tables] Experimental section (comparisons with other KANs and MLPs): the manuscript does not supply sufficient detail on parameter-count matching, optimizer hyper-parameters, or basis-function choices to allow independent verification that the reported accuracy and convergence advantages on irregular functions are not artifacts of unequal experimental conditions.
minor comments (1)
  1. [Application / hydraulic valley] In the hydraulic-valley optimization subsection, confirm that all reported objective values include the number of independent runs and standard deviations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Theory / error bounds derivation] Theory section on error bounds: the stated bounds are derived only under the explicit assumption that the Kolmogorov-Arnold expansion functions are sufficiently smooth. The central empirical claim, however, concerns superior performance on irregular (likely non-smooth) functions, for which the smoothness hypothesis is violated and only the rate-free UAT remains. This leaves the theoretical rationale for the headline performance advantage on irregular targets unsupported by quantitative rates.

    Authors: We agree that the quantitative error bounds require the smoothness assumption on the expansion functions and therefore do not apply to irregular targets. The manuscript provides a rate-free universal approximation theorem for the continuous case. The reported advantages on irregular functions are empirical. We will revise the theory and discussion sections to explicitly separate the smooth-case bounds from the continuous-case UAT and to avoid any implication that quantitative rates are available for non-smooth targets. revision: partial

  2. Referee: [Experiments / comparison tables] Experimental section (comparisons with other KANs and MLPs): the manuscript does not supply sufficient detail on parameter-count matching, optimizer hyper-parameters, or basis-function choices to allow independent verification that the reported accuracy and convergence advantages on irregular functions are not artifacts of unequal experimental conditions.

    Authors: We acknowledge that the current experimental description lacks the level of detail needed for independent verification. In the revised manuscript we will add explicit tables or text specifying how parameter counts were matched across architectures, the full set of optimizer hyperparameters and training protocols, and the precise basis-function choices used for each KAN variant. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no reductions to inputs by construction

full rationale

The paper states error bounds under an explicit smoothness assumption on Kolmogorov-Arnold expansion functions and supplies universal approximation theorems when the target is merely continuous. Performance claims rest on direct empirical comparisons against external baselines (MLPs and other published KAN variants) rather than any fitted parameter renamed as a prediction or any self-citation chain. No self-definitional, fitted-input, or load-bearing self-citation patterns appear; the derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review prevents exhaustive ledger; the central claims rest on standard approximation-theory assumptions plus the new network definition itself.

axioms (1)
  • domain assumption Kolmogorov-Arnold expansion functions are sufficiently smooth for the error bounds to hold
    Explicitly invoked in the abstract when stating the error bounds.
invented entities (1)
  • P1-KAN network architecture no independent evidence
    purpose: Approximate irregular high-dimensional functions with improved accuracy and speed
    New proposed architecture whose definition and performance are the paper's contribution; no independent evidence outside the paper is described.

pith-pipeline@v0.9.0 · 5628 in / 1249 out tokens · 15805 ms · 2026-05-23T19:53:49.500290+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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Reference graph

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