# LLM.txt - How KANs Work: Why Learnable Edges Survived the MLP-Killer Hype
## Article Metadata
- **Title**: How KANs Work: Why Learnable Edges Survived the MLP-Killer Hype
- **URL**: https://www.llmrumors.com/news/how-kans-work-learnable-edges-after-mlp-hype
- **Publication Date**: July 11, 2026
- **Reading Time**: 15 min read
- **Tags**: Kolmogorov-Arnold Networks, KAN, Neural Networks, Scientific AI, Symbolic Regression, B-Splines, Transformers, AI Infrastructure
- **Slug**: how-kans-work-learnable-edges-after-mlp-hype
## Summary
Kolmogorov-Arnold Networks replace scalar weights with learned functions. Two years of evidence show where KANs work, where they fail, and why the idea survived.
## Key Topics
- Kolmogorov-Arnold Networks
- KAN
- Neural Networks
- Scientific AI
- Symbolic Regression
- B-Splines
- Transformers
- AI Infrastructure
## Content Structure
This article from LLM Rumors covers:
- Industry comparison and competitive analysis
- Data acquisition and training methodologies
- Financial analysis and cost breakdown
- Human oversight and quality control processes
- Comprehensive source documentation and references
## Full Content Preview
TL;DR: Kolmogorov-Arnold Networks replace scalar edge weights with learnable one-dimensional functions, usually splines in the original design, so each connection learns a curve instead of a number.[1] One Poisson PDE experiment reported 100 times lower squared error with 100 times fewer parameters, but the same paper disclosed a typical 10-times training slowdown at equal parameter count.[1] By 2026, accepted work had extended KANs into symbolic neural operators, hardware-aware transformers, multivariate lookup layers, and restricted training theory, but every major success changed or constrained the original spline design.[8][10][12][13]
KANs arrived in April 2024 with the kind of pitch the AI industry cannot resist. Change one foundational assumption, then make decades of neural-network design look unnecessarily rigid. Multi-layer perceptrons, or MLPs, learn scalar weights between neurons and apply fixed nonlinear activations on the neurons. KANs put learnable nonlinear functions on the connections instead.
That sounds cosmetic. It is not. A scalar weight can only amplify, suppress, or reverse a signal. A learned function can bend, saturate, oscillate, flatten, or sharpen the signal differently at every input value. The connection stops being a volume knob and becomes a tiny curve-fitting machine.
The real story isn't that KANs killed the MLP. They did not. The real story is that KANs exposed a valuable design axis: a network can learn the shape of a connection, not merely its strength. The research since 2024 has been a two-year stress test of when that extra freedom earns its cost.
The follow-up record is no longer empty. Accepted 2025-2026 work now spans KANO at ICLR, FlashKAT at AAAI, KAN optimization and privacy theory at ICML, lmKAN scaling at ICLR, ReLU convertibility at AISTATS, and a peer-reviewed warning that scientific model discovery can be nonunique.[4][5][8][10][12][13] The real 2026 verdict is not that KANs replaced MLPs. It is that learnable functional connections became a serious research program.
The Architecture: Every Connection Learns A Curve
A conventional MLP layer performs two familiar operations. It multiplies an input vector by a matrix of learned scalar weights, adds biases, then applies a fixed activation such as ReLU, GELU, or tanh at each node.
A KAN layer reorganizes that bargain. For every connection from input coordinate i to output coordinate j, the network learns a univariate function phi[j,i]. Each output coordinate is the sum of those transformed inputs:
x_next[j] = sum_i phij,i
There is no single shared activation curve sitting on the destination node. Every edge can learn its own curve. In the original PyKAN design, that curve combines a residual base function with a trainable B-spline expansion. The model adjusts multiple coefficients over a grid, allowing one edge to behave differently across different input regions.[1][1]
Here is the genius. B-spline basis functions have local support. Adjusting one coefficient changes a limited region of a curve instead of rewriting its entire shape. PyKAN can also extend a trained spline grid from coarse to fine, preserving the learned function while adding local resolution.[1]
The cost appears immediately. One MLP edge stores one number. One original KAN edge stores multiple spline coefficients and scales. For width N, depth L, grid size G, and spline order k, the original paper gives KAN parameter scaling of O(N²L(G+k)), versus O(N²L) for an MLP of comparable width and depth.[1]