Kolmogorov-Arnold Networks (KAN): A Novel Approach to Neural Network Flexibility and Efficiency

New perspective for neural network flexibility..

KAN illustration created by DALLE-3

The universal approximation capability of traditional artificial neural networks is achieved through learnable weights and fixed activation functions. However, the limitations of these architectures, especially in terms of efficiency and interpretability, are noticeable. Inspired by the Kolmogorov-Arnold Representation Theorem, Kolmogorov-Arnold Networks (KAN) bring a breath of fresh air to neural networks by addressing these limitations.

Designed as an alternative to Multi-Layer Perceptron (MLP) models, KAN employs learnable activation functions on edges and summation operations at nodes. This approach enhances the model’s flexibility, allowing for more accurate data analysis while also providing scientists with a more understandable and collaborative architecture. KAN, particularly in data fitting and partial differential equation (PDE) solving applications, achieves better results than MLPs.

The theoretical differences between Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) models stem from the theorems they are based on. MLPs are grounded in the Universal Approximation Theorem, providing a structure that can approximate any continuous function with a certain degree of accuracy. However, MLPs’ reliance on fixed activation functions can limit learning.

In contrast, KAN is based on the Kolmogorov-Arnold Representation Theorem and offers a more flexible approach with learnable activation functions on edges. Each edge is modeled with spline-based functions and can respond differently depending on the input data. This allows KAN to develop a learning structure that can adapt to the input.

Network Structure

MLP:

• Activation Functions: Fixed activation functions (e.g. ReLU or sigmoid) are used and implemented at each node.
• Weights and Bias: Each edge has a fixed weight (w) and bias (b) value. These values are optimized during the learning process.
• Layers: MLPs usually have several hidden layers and a certain number of nodes in each layer.

Formula:

MLP formula

• W₁, W₂, W₃: learnable weights and σ₁, σ₂: Fixed activation functions

KAN:

• Activation Functions: Learnable activation functions (spline (piecewise polynomial) based) assigned to edges are used.
• Aggregate Operation: The interaction between nodes is expressed with learned functions and each node performs an aggregate operation. Nodes collect the results from splines.
• Layers: KANs have a multi-layered structure that includes learnable spline-based activation functions in each layer.

Formula:

KAN formula

• Φ₁, Φ₂, Φ₃: They are learnable, non-linear activation functions (splines) on edges.

The differences between activation functions can be explained as follows. In KAN architecture, spline-based functions replace the fixed weights in traditional neural networks. Instead of assigning fixed weights, KAN assigns parametrically modeled functions to each edge. A spline is a type of piecewise polynomial function that typically creates curves with smooth transitions by combining multiple polynomials. This structure allows the network to work more flexibly with different data sets.

In traditional neural networks, weight (w) and bias (b) values are optimized during training to achieve the best results. In KAN, each weight is represented not by a fixed value but by a spline function, which is optimized through training. The model then focuses on learning different parameters of a particular spline function rather than a fixed weight value for each connection.

For example, a piecewise polynomial spline function can be defined as follows:

Spline example

This function structure contains multiple polynomials, each defined to operate within different intervals, creating a curve that flexibly fits data across varying regions, allowing for more adaptable modeling. The KAN architecture employs B-Spline structures, which form the backbone of its learning mechanism. The formulation of B-Splines is provided below:

B Spline formula

Here:

• spline(x): Represents the spline function.
• cᵢ: Coefficients optimized during training.
• Bᵢ(x): B-Spline basic functions are defined on a specific grid.

B Spline illustration

In the B-Spline structure, the intervals in which functions are active are called grids. Grids define the ranges where the basis functions have an effect. The Bi(x) functions are activated at grid points, significantly contributing to the shape of the spline curve. Grid points are a hyperparameter that affects the precision of the network. More grid points increase the control over the spline curve, leading to more parameters being learned, which ensures more accurate and precise learning.

Training Process in KAN

During training, the ci coefficients of these splines are optimized to minimize prediction error. Techniques such as gradient descent are often used for this. In each iteration, the spline parameters are updated to reduce prediction error, allowing the model to find the most suitable curves for the data.

KAN Integrated LLMs?

Kolmogorov-Arnold Networks (KAN) and spline-based activation functions can provide a more flexible, adaptable, and powerful modeling approach when integrated with large language models (LLMs). This combination opens new pathways for enhancing language modeling capabilities. Compared to traditional neural network models, KANs better adapt to input variations, enabling LLMs to capture even subtle differences in data distribution. As a result, significant advancements can be made in language analysis and text generation.

To harness this potential, several projects are underway, aiming to integrate the KAN architecture with GPT models. For instance, the KAN-GPT project is one of the first open-source models in this area, demonstrating how KANs can be incorporated into LLMs (https://github.com/AdityaNG/kan-gpt). Although these projects are still in the development stage, they offer valuable insights into how KANs can be used alongside larger models in natural language processing. These efforts suggest that the flexible and adaptive nature of KAN can further enhance the performance of LLMs.

Conclusion

In summary, Kolmogorov-Arnold Networks (KAN) offer an innovative approach that transcends the limitations of classical neural networks. The adaptive nature of the spline-based edge functions makes KANs highly effective in data fitting and modeling. With its advanced flexibility, high accuracy, and interpretability, KAN has the potential to play a significant role in the future of AI. Future research will further explore the benefits of these networks in more complex fields, broadening their performance and applications.

Sources

https://arxiv.org/abs/2404.19756

https://hesamsheikh.substack.com/p/understanding-kolmogorovarnold-networks