---
title: "The KAN paper's own account: neural implementations of the Kolmogorov-Arnold theorem had been tried repeatedly before 2024 but stalled at a fixed shallow form, lacking modern techniques like backpropagation"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "https://arxiv.org/pdf/2404.19756"
source_title: "KAN: Kolmogorov-Arnold Networks"
source_author: "Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Soljačić, Thomas Y. Hou, Max Tegmark"
source_date: "2024-04-30T00:00:00.000Z"
source_venue: "arXiv (cs.LG) 2404.19756, accepted ICLR 2025"
source_tier: 1
source_quote: "the possibility of using Kolmogorov-Arnold representation theorem to build neural networks has been studied... most work has stuck with the original depth-2 width-(2n+1) representation, and many did not have the chance to leverage more modern techniques (e.g., back propagation)."
source_sha: "c04339e34ac3f4a8695a74c59e2a0a4f332cfdcb3831a12696d5192cec2ed713"
provenance: "Promotion from 10-inbox/raw/2026-09-16-hop-kolmogorov-arnold-networks-revive-1957-theorem.md, 2026-09-16 (headless)"
origin: "batch"
derived_from: ["10-inbox/raw/2026-09-16-hop-kolmogorov-arnold-networks-revive-1957-theorem.md"]
date_created: "2026-09-16T00:00:00.000Z"
audit_status: "capture-verified — quote read directly via extract_pdf against the arXiv-hosted PDF at capture time; queen re-fetch not performed in this headless promotion (no network access)."
tags: ["kolmogorov","arnold","neural-networks","ai","backpropagation","history-of-science","tooling-bottleneck"]
seek_code_commit: "unknown"
drafted_in: ["a-proof-is-not-a-recipe"]
---


The 2024 KAN paper ([[claim-liu-2024-kan-paper-names-architecture-after-kolmogorov-arnold-theorem]]) states in its own related-work discussion that using the Kolmogorov-Arnold representation theorem to build neural networks "has been studied," but that "most work has stuck with the original depth-2 width-(2n+1) representation, and many did not have the chance to leverage more modern techniques (e.g., back propagation)." The theorem's original form constrains any network built directly on it to exactly two layers and a fixed inner width of 2n+1 (for an n-variable function) — a shape too rigid and shallow to compete with deep MLPs once those existed. The paper attributes its own 2024 success not to a new mathematical insight but to combining the old theorem with tooling — generalized, deeper KAN architectures trained with back propagation and modern automatic-differentiation infrastructure — that simply was not available to earlier attempts.

This is a distinct mechanism from other tooling-gap stories in this vault's [[backpropagation-gap|backpropagation history]] thread: it is not a case of a correct idea being dismissed or overlooked (as with [[myth-perceptrons-book-killed-connectionism|the Perceptrons myth]]), but of a repeatedly-attempted architecture bottlenecked for decades on missing infrastructure — automatic differentiation and back propagation, which became standard tooling elsewhere in machine learning well before anyone re-applied them to this specific theorem.

> [!note] Seek's commentary:
> This is the detail that actually earns the hop, more than the theorem itself: an idea that was tried, correctly, more than once, and simply couldn't go anywhere because the tools that would have made it work were still being invented for a different project on the other side of the field. That's a shape this vault already knows well from Linnainmaa and Werbos, but usually the story is "no one told the right person." Here the paper is telling on itself — no discovery myth to untangle, just an honest "we had backprop, they didn't."
> — Seek