Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities

Ivan Oseledets; Jiang Yang; Jingwei Qiu; Tianchi Yu

arxiv: 2410.04096 · v2 · pith:VXEFVHB2new · submitted 2024-10-05 · 💻 cs.LG · cs.AI· cs.NA· cs.NE· math.NA· physics.comp-ph

Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities

Tianchi Yu , Jingwei Qiu , Jiang Yang , Ivan Oseledets This is my paper

Pith reviewed 2026-05-23 19:40 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.NAcs.NEmath.NAphysics.comp-ph

keywords Sinc interpolationKolmogorov-Arnold NetworksPhysics-informed neural networksPartial differential equationsFunction approximationLearnable activationsSingular functions

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The pith

Sinc interpolation serves as an effective learnable activation basis in Kolmogorov-Arnold Networks for function approximation and PDE solving.

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

The paper introduces SincKAN by substituting Sinc interpolation into the learnable activation functions of Kolmogorov-Arnold Networks. This choice draws on the established ability of Sinc functions to represent both smooth behaviors and functions with singularities. The substitution matters for physics-informed neural networks because many differential equations produce solutions with sharp features. Experiments across multiple examples indicate that SincKANs achieve lower errors than earlier KAN variants in nearly all cases tested.

Core claim

We propose to use Sinc interpolation in the context of Kolmogorov-Arnold Networks, neural networks with learnable activation functions. Many different function representations have already been tried, but Sinc interpolation proposes a viable alternative since it is known in numerical analysis to represent well both smooth functions and functions with singularities. This is important not only for function approximation but also for the solutions of partial differential equations with physics-informed neural networks. Through a series of experiments, we show that SincKANs provide better results in almost all of the examples we have considered.

What carries the argument

Sinc interpolation employed as the basis for learnable activation functions inside the Kolmogorov-Arnold Network architecture.

If this is right

SincKANs can approximate functions containing singularities more accurately than prior KAN versions.
Physics-informed neural networks that employ SincKANs solve partial differential equations with reduced error.
The KAN framework gains applicability to problems whose solutions exhibit non-smooth or singular behavior.
Replacement of the activation basis with Sinc functions maintains compatibility with existing KAN optimization routines.

Where Pith is reading between the lines

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

SincKAN layers could be hybridized with other bases to handle mixed smooth and singular regions within one network.
The method may prove useful for time-dependent PDEs where singularities move or appear dynamically.
Scalability questions arise for very high-dimensional inputs not covered in the reported experiments.

Load-bearing premise

Sinc interpolation can be substituted directly for other bases inside the KAN framework while preserving training dynamics, and the chosen examples represent the intended applications.

What would settle it

A benchmark function approximation or PDE task where SincKAN produces higher error than a standard KAN or MLP under identical training conditions and comparable parameter counts.

Figures

Figures reproduced from arXiv: 2410.04096 by Ivan Oseledets, Jiang Yang, Jingwei Qiu, Tianchi Yu.

**Figure 2.** Figure 2: Fig. 2(a) depicts the function of ’piece-wise’ in Table 1 and compares the performance [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Fig. 3(a) shows the inverse decay approach on sin-low function; Fig. 3(b) shows the inverse [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Demonstration of boundary layer problems, Fig. 4(a) depicts the exact solution of Eq. (15) [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Fig. 5(a) depicts the exact solution of Eq. (34), Fig. 5(b) shows the solution predicted by [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Fig. 6(a) used high [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: Figures of different Nponits with increasing degree. 1 2 3 M 2.0 6.0 10.0 h0 7.80e-04 7.06e-04 6.33e-04 1.60e-04 5.40e-04 8.02e-04 2.91e-04 1.03e-03 3.47e-03 1.49e-04 1.29e-03 3.30e-03 (a) 1 2 3 M 2.0 6.0 10.0 h0 7.07e-01 7.07e-01 7.02e-01 7.07e-01 6.89e-01 2.29e-02 6.98e-01 1.20e-02 2.17e-02 6.58e-01 7.22e-02 3.48e-02 (b) [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗

**Figure 8.** Figure 8: Fig. 8(a) shows the exponential decay approach on sin-low function; Fig. 8(b) shows the [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 9.** Figure 9: Fig. 9(a) solve Eq. (32) accurately. However, SincKANs have oscillations after either [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗

read the original abstract

In this paper, we propose to use Sinc interpolation in the context of Kolmogorov-Arnold Networks, neural networks with learnable activation functions, which recently gained attention as alternatives to Multilayer Perceptron. Many different function representations have already been tried, but we show that Sinc interpolation proposes a viable alternative, since it is known in numerical analysis to effectively represent both smooth functions and functions with singularities. This is important not only for function approximation but also for solving the partial differential equations with physics-informed neural networks. Through a series of experiments, we show that SincKANs provide better results in almost all of the examples we have considered.

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.

Desk Editor's Note private letter to a colleague

SincKAN adds Sinc interpolation as an activation option inside KANs for better handling of singular functions in PINNs, with the main uncertainty being whether the claimed experimental gains are large enough to matter.

read the letter

The paper proposes replacing the usual activation functions in Kolmogorov-Arnold Networks with Sinc interpolation. The authors note that Sinc bases are already known in numerical analysis to approximate both smooth functions and those with singularities, which is a direct motivation for using them in physics-informed neural networks for PDEs. That pairing itself does not appear in the prior KAN literature they cite, so the technical step is new at this level of specificity. The experiments are presented as showing better results than alternatives in almost all cases tested, and the claim is framed as coming from those runs rather than from any internal fitting inside the model equations. This keeps the circularity burden low. The substitution of Sinc is treated as preserving the overall KAN training setup, and the paper's experiments are meant to check whether that holds in practice. The main limitation visible from the description is that no error values, baseline details, or statistical comparisons are supplied, which makes it impossible to judge the size or reliability of the reported advantage. If the full manuscript contains controlled, reproducible runs with clear metrics, that would address the gap; otherwise the advantage stays hard to evaluate. The work sits inside the narrow area of KAN variants for scientific computing and PDE approximation. Readers already experimenting with different bases inside KANs or looking for activation choices that cope with singularities could find the concrete proposal worth testing. I would send the paper to peer review because the rationale is grounded in existing numerical analysis and the change is small enough that referees can focus on whether the experiments actually support the performance claim.

Referee Report

0 major / 3 minor

Summary. The paper proposes Sinc Kolmogorov-Arnold Networks (SincKANs) by using Sinc interpolation as the learnable univariate activation functions within the KAN framework. It motivates the choice via the established ability of Sinc bases in numerical analysis to approximate both smooth functions and functions with singularities, and applies the resulting architecture to function approximation tasks as well as physics-informed neural networks. A series of experiments is reported to show that SincKANs yield better results than alternative KAN variants in almost all examples considered.

Significance. If the reported empirical gains hold under scrutiny, the work supplies a theoretically grounded alternative basis for KAN activations that may be advantageous for PINN problems involving singularities. The explicit link to classical approximation theory is a positive feature, and the focus on practical performance in both approximation and PDE settings adds relevance.

minor comments (3)

[Abstract] Abstract: the claim of superior performance would be more informative if accompanied by at least one concrete quantitative improvement (e.g., error reduction on a named benchmark) rather than the qualitative statement alone.
The manuscript should clarify the precise discretization and truncation parameters used for the Sinc basis (e.g., number of nodes, cutoff radius) and whether these are held fixed or adapted during training.
Ensure that all comparison baselines (other KAN variants) are described with identical hyper-parameter regimes and that any differences in computational cost are reported.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the link to classical approximation theory, and recommendation of minor revision. The assessment that SincKANs may be advantageous for PINN problems with singularities aligns with our motivation. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; empirical proposal with external grounding

full rationale

The paper proposes SincKAN by direct substitution of the known Sinc interpolation basis into the KAN framework, justified by pre-existing numerical-analysis facts about its ability to represent both smooth and singular functions. The performance claim is presented strictly as an outcome of experiments rather than any derivation, fitted parameter, or self-referential definition. No load-bearing self-citations, uniqueness theorems, or ansatzes appear in the text; the chain is self-contained and externally falsifiable via the reported benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no concrete free parameters, axioms, or invented entities; a full-text audit is required.

pith-pipeline@v0.9.0 · 5654 in / 955 out tokens · 28847 ms · 2026-05-23T19:40:19.228103+00:00 · methodology

discussion (0)

Forward citations

Cited by 3 Pith papers

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