Are KANs trained from data?

Yes. Their learnable functions are adjusted to fit an objective, subject to the chosen architecture and training method.

Do all variants use splines?

The original formulation uses spline-based functions, while later variants may use other parameterizations.

AI Glossary / AI and machine learning / Advanced

Kolmogorov - Arnold Networks

Definition

Kolmogorov - Arnold networks, usually called KANs, are neural-network architectures that learn univariate functions on connections between nodes. In the original formulation, these functions are represented with splines, and nodes sum incoming transformed values.

What it means

A familiar multilayer perceptron combines weighted inputs and then applies a fixed activation function. The original KAN formulation shifts much of the learnable nonlinearity onto the connections themselves. Each connection can learn a curve describing how one incoming value contributes to the next layer.

The architecture is inspired by the Kolmogorov - Arnold representation theorem, but the theorem is not a guarantee that practical KAN training will be easy or that KANs will outperform other networks on every task. Finite data, optimization, model size and implementation still matter.

KANs have been explored for scientific modeling and interpretable function learning. Learned curves can sometimes be inspected or simplified into useful expressions. However, a network containing many interacting curves can still be difficult to understand, and an attractive symbolic approximation needs validation beyond the examples used to fit it.

How it works

A layer takes an input vector and applies a separate learned one-dimensional function along each relevant connection. A receiving node sums the results. Stacking layers creates a more flexible multivariate mapping.

In the original work, spline parameterizations allow the connection functions to change shape during training. Grid choices affect flexibility and computation. Regularization and pruning can encourage a smaller structure, and researchers may explore symbolic approximations to learned functions.

Training optimizes a loss using data, much as other neural-network training does, but the parameterization and computational trade-offs differ. A fair comparison controls data splits, tuning effort and relevant resource budgets. Interpretability should be assessed from the resulting model and explanation process, not inferred solely from the architecture's name. Later variants can use different function families, so implementation details should be stated.

Three practical examples

1. A scientific modeler fits a small nonlinear relationship and inspects learned connection functions for a compact explanatory pattern.

2. An engineering team compares a KAN with an MLP on a tabular regression task, measuring both prediction error and training cost.

3. A teaching exercise contrasts fixed activation functions with learnable connection curves to show how two networks distribute their modeling flexibility differently.

Think of it this way

Think of a network of adjustable pipes where each connection has its own shaped response, and junctions add the incoming contributions. The limit is that these are mathematical functions, not physical flows, and seeing individual connection curves does not automatically explain the behavior of the whole network.

Worked example

Consider a toy KAN-like node with two learned connection functions: phi1(x) = x squared and phi2(y) = 2y. The node outputs phi1(x) + phi2(y).

For x = 3 and y = 4, the output is 9 + 8 = 17. If x changes to 4 while y stays 4, the output becomes 16 + 8 = 24. The contribution from x is nonlinear even before any additional layer is added.

This deliberately simple example illustrates learned univariate contributions. It does not reproduce a training procedure, establish a symbolic discovery or show that a KAN is the best way to represent this function.

Illustrative case study

A fictional materials team experiments with a KAN on a small laboratory dataset. Several connection curves appear smooth and easy to describe, so the team considers presenting them as discovered physical laws.

Before doing so, it checks performance on new experiments and compares the proposed expressions with known measurement limits. One apparent pattern changes when a few observations are removed. The team reports it as a model-dependent relationship needing further evidence.

The case illustrates that interpretability can support investigation without turning a fitted curve into a proven mechanism. Scientific meaning requires domain evidence and robustness, not merely a readable formula.

Common misconceptions and mistakes

The representation theorem guarantees superior practical performance. It does not settle optimization, data or resource trade-offs.
Every KAN is automatically interpretable. Model size and interactions can make explanations difficult.
A fitted symbolic expression is a discovered law. It may be an approximation specific to the available data.

Business use

KANs are a modeling option worth understanding in specialized research and development, particularly where function structure is of interest. Adoption should follow task-specific evaluation rather than broad claims about replacing established architectures.

Compare with simpler models and a well-tuned MLP, using comparable data and resources. Record the function parameterization, regularization and pruning choices. If interpretability is a goal, define what an explanation must help a person verify. A model's connection curves can be useful evidence while still leaving uncertainty about behavior outside the observed range.

Check your understanding

Why should a simple-looking KAN curve not immediately be described as a physical law?

Answer: It is a fitted model component. Its stability, domain validity and predictive behavior beyond the training examples still need evidence.

Related terms

Learn next

Read multilayer perceptrons and splines. Compare where fixed activations and learnable functions appear in the two architectures, then inspect a small fitted model without assuming every curve has a causal meaning.

Sources

Liu and colleagues, KAN: Kolmogorov–Arnold Networks (2024): https://arxiv.org/abs/2404.19756. Liu and colleagues, KAN 2.0: Kolmogorov–Arnold Networks Meet Science: https://arxiv.org/abs/2408.10205. The node calculation and materials-team scenario are invented. The source-list term contains a corrupted dash; the standard spelling is used in this explanation.

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