Generalized Kolmogorov–Arnold theorem for deeper KAN architectures

Develop a generalized Kolmogorov–Arnold representation theorem that corresponds to network architectures with more than two non-linear layers, thereby formalizing deeper Kolmogorov–Arnold Network compositions and their properties.

Background

The classical Kolmogorov–Arnold representation theorem underpins two-layer KANs, but the paper extends KANs to arbitrary depth without an accompanying generalized theorem.

The authors explicitly note the absence of a generalized version corresponding to deeper KANs, highlighting a foundational mathematical gap.

References

To the best of our knowledge, there is not yet a 'generalized' version of the theorem that corresponds to deeper KANs.

— KAN: Kolmogorov-Arnold Networks  (2404.19756 - Liu et al., 2024) in Subsection 2.2, KAN architecture

Whether universal approximation is preserved under these modifications remains an open question.

— Kolmogorov-Arnold Classifier Systems as Universal Approximators  (2609.37958 - Shiraishi et al., 29 Sep 2026) in Section Conclusion

Their theoretical grounding remains debated \citep{girosi1989representation,lin1993realization,koppen2002training,sprecher2002space,montanelli2020error,he2023optimal,lai2021kolmogorov}, yet KANs perform strongly across scientific domains \citep{scidirectkan2025,oxfordgenomicskan2025,naturegraphkan2025,coxkan2025}, with a growing set of architectural refinements \citep{Bagrow2025,Koenig2025,xkan2024}; in dynamical systems, KAN-based ODEs learn dynamics and hidden physics \citep{Koenig2024}.

— Higher-Order Kolmogorov-Arnold Networks for Dynamics  (2610.02637 - Slote et al., 2 Oct 2026) in Introduction, paragraph beginning “Separately, Kolmogorov--Arnold Networks”