Blogs / Kolmogorov-Arnold Networks (KAN): A Powerful Alternative to Traditional Neural Networks

Kolmogorov-Arnold Networks (KAN): A Powerful Alternative to Traditional Neural Networks

شبکه‌های کولموگروف-آرنولد (KAN): جایگزین قدرتمند شبکه‌های عصبی سنتی

Introduction

Imagine a network that learns functions themselves rather than just fixed weights. This is precisely what Kolmogorov-Arnold Networks (KAN) bring to the table. This innovative architecture, designed based on the Kolmogorov-Arnold representation theorem, is fundamentally changing how we design and train neural networks.
For decades, Multi-Layer Perceptron (MLP) neural networks have been the backbone of deep learning. But hasn't the time come to seek an alternative that is both more accurate and more interpretable? KAN, by offering a completely different approach, promises this transformation. In this article, we will deeply explore this revolutionary architecture, its advantages and limitations, and practical applications across various industries.

Theoretical Foundations: From the Kolmogorov-Arnold Theorem to Neural Networks

What is the Kolmogorov-Arnold Theorem?

In 1957, two Russian mathematicians, Andrey Kolmogorov and Vladimir Arnold, proved a remarkable theorem that forms the theoretical foundation of KAN networks. This theorem states that any continuous multivariate function can be written as a finite composition of continuous univariate functions and addition. In simpler terms, this theorem says we can decompose any complex multidimensional problem into a set of simpler one-dimensional problems.
This means instead of dealing with all dimensions of a problem simultaneously, we can process each dimension separately and then combine the results. This is exactly the idea behind KAN networks and what distinguishes them from traditional neural networks.

From Theory to Practice: How KAN Was Born

Researchers at MIT in April 2024, inspired by this classic mathematical theorem, introduced the KAN architecture. They concluded that if we could place activation functions on the edges of the network instead of the nodes, we could build a more powerful and interpretable network. This simple yet profound idea opened a new window in neural network design and captured the attention of the scientific community.

KAN Architecture: Fundamental Differences from MLP

Traditional MLP: Fixed Weights, Fixed Activation Functions

In traditional neural networks that we're all familiar with, the architecture is such that nodes or neurons have fixed activation functions like ReLU, Sigmoid, or Tanh. Edges or connections between neurons also represent merely learnable numerical weights. Computations in these networks are performed as linear combinations of inputs followed by applying the activation function.
This architecture has fundamental limitations that researchers have grappled with for years. Lack of transparency in how the model makes decisions, the need for large networks to solve complex problems, and difficulty in interpreting how the model works are among these limitations. These problems are particularly challenging in domains requiring transparency, such as medicine and finance.

KAN: Learnable Functions on Edges

In KAN networks, this architecture fundamentally changes. Nodes in KAN are merely simple summation units and have no complex nonlinear functions. But the magic happens in the edges. Each edge in KAN is a learnable univariate function, typically parameterized as a spline. This seemingly simple change creates fundamental differences in the network's performance and capabilities.
This approach has three major advantages. First, higher expressiveness, meaning KAN can represent more complex functions with fewer parameters. Second, interpretability, meaning we can visually see what operation each edge performs on the data. Third, parameter efficiency, meaning KAN requires far fewer parameters to achieve the same accuracy.

Using Splines: The Key to KAN's Success

One of the main and key innovations in KAN architecture is the intelligent use of B-splines to parameterize functions on edges. Splines are piecewise polynomial functions with unique characteristics. They are highly flexible and can model various shapes and patterns. At the same time, they are computationally efficient and don't cause a significant increase in computational cost.
Moreover, splines are controllable and tunable, meaning we can precisely control the network's behavior by changing their parameters. This smart choice allows KAN to learn very complex functions without significantly increasing computational cost. The spline degree (typically 3 for cubic spline) and the number of grid points are two important parameters that affect the network's final performance.

Advantages of KAN Networks: Why Should We Pay Attention?

1. Higher Accuracy with Fewer Parameters

One of the most prominent advantages of KAN networks is their ability to achieve high accuracy with far fewer parameters. Numerous studies have shown that KAN can achieve similar or even better accuracy with 10 times fewer parameters than MLP. This practically means lighter, deployable models that require fewer computational resources. Training speed also becomes faster in some cases, and memory and storage requirements decrease.
For a practical example, in a simple regression problem, an MLP with 300 neurons might achieve 95% accuracy, while a KAN with only 30 parameters can reach 97% accuracy. This striking difference in parameter count is especially valuable in applications with limited computational resources, such as mobile devices or IoT.

2. Interpretability: The End of the Black Box Era

One of the biggest and most persistent criticisms of deep neural networks is their lack of interpretability. These networks are often known as "black boxes" because understanding how they reached a particular decision is very difficult. KAN significantly reduces this problem and brings more transparency to machine learning models.
In KAN, we can easily plot each function on each edge and see what transformation is being performed on the data. Known mathematical patterns like sin, exp, log, or various powers are recognizable in these functions. This feature is very valuable in scientific problems because KAN can help discover physical relationships or scientific laws. This capability is critical in fields like AI in diagnosis and treatment and financial analysis with AI tools.

3. Flexibility and Adaptability

Due to its flexible architecture, KAN can easily be combined with other deep learning architectures to build powerful hybrid architectures. KAN can be combined with convolutional neural networks for image processing and leverage the advantages of both. Combining KAN with recurrent neural networks for time series processing has also shown promising results.
KAN can even be integrated with Transformer architecture for natural language processing and benefit from the power of both architectures. This combinability allows researchers and developers to build custom models suitable for their specific needs and use the best features of each architecture.
Feature KAN MLP
Activation Function Location On edges (learnable) On nodes (fixed)
Number of Parameters Few (10x fewer) Many
Interpretability Very High Low (Black box)
Training Speed Slower (2-5x) Fast
Accuracy on Math Functions Excellent (99.8%) Good (95%)
Performance on Complex Images Good (97.5%) Excellent (98.1%)
Scientific Law Discovery Possible Not Possible
Implementation Complexity Medium to High Low
Memory Usage Low High
Ecosystem Maturity New (2024) Mature (Decades of experience)
Best For Scientific problems, interpretability, resource-constrained applications Complex data, images, text, general applications

Practical Applications of KAN: From Science to Industry

1. Basic Sciences: Discovering Physical Laws

One of the most exciting and fascinating applications of KAN networks is the automatic discovery of physical equations and scientific laws. In recent research, KAN has accomplished amazing feats. These networks have automatically discovered Kepler's laws of planetary motion without being explicitly taught these laws.
In fluid dynamics, KAN has been able to identify and model complex relationships between different variables. In quantum mechanics, these networks have helped better understand multi-particle systems. This capability for automatic discovery of scientific laws could transform how scientific research is conducted in the future and help scientists discover hidden relationships in data. This potential is very promising in the field of quantum artificial intelligence.

2. Bioinformatics and Genomics

In computational biology and genomics, KAN networks have shown exceptional performance. Researchers at Oxford University in a recent study showed that KAN has excellent results in analyzing complex genomic data. These networks can identify genetic patterns associated with various diseases with high accuracy and help predict disease risk.
In gene expression analysis, KAN can better understand how genes interact with each other and model gene regulatory networks. In drug design, these networks can predict the effectiveness of different drugs and help accelerate the discovery of new drugs. Research results show that KAN has up to 30% more accuracy in this field compared to traditional MLP, which is very significant in medicine.

3. Time Series Forecasting

For prediction and forecasting in temporal data, KAN has shown significant capabilities. In financial markets, these networks can predict stock prices, cryptocurrencies, and other financial assets with acceptable accuracy. The big advantage of KAN in this field is the interpretability of results, which helps traders understand the reasons behind predictions.
In weather forecasting and climate change modeling, KAN can model complex atmospheric patterns and provide more accurate predictions. In energy management, electricity consumption forecasting using KAN helps power companies better plan and optimally allocate resources. KAN-ODE models that combine ordinary differential equations with KAN have shown very promising results in this field and have been able to model complex temporal dynamics with high accuracy.

4. Computer Vision and Image Processing

Although KAN was initially designed for numerical and structured data problems, it has found interesting and promising applications in machine vision. In pattern recognition, KAN can identify complex and subtle patterns in images that might be challenging for traditional networks.
In medical image processing, KAN has been used for analyzing MRI, CT, and X-ray images with satisfactory results. KAN's interpretability in this field is very valuable because doctors can see what features the model based its decision on. Even in image generation, combining KAN with diffusion models has shown interesting results and helped improve the quality of generated images.

5. Natural Language Processing

The use of KAN in natural language processing is also expanding. Researchers are testing KA-GNN, which is a combination of KAN with graph neural networks and is very useful for graph-based analyses such as molecular analysis and semantic networks. In sentiment analysis, KAN can identify emotions hidden in text with high interpretability and show the reasons.
In machine translation, using KAN alongside existing architectures can improve translation quality and better model complex semantic relationships between languages. Also, in question-answering systems, KAN can help with deeper understanding of questions and finding more accurate answers.

6. Artificial Intelligence in E-commerce

KAN has extensive applications in data analysis for businesses. In recommendation systems, KAN can better model user behavior and provide more accurate and personalized recommendations. This leads to increased customer satisfaction and sales.
In sales forecasting and future demand estimation, KAN can help businesses optimally manage their inventory and prevent resource waste. In customer behavior analysis, KAN identifies purchasing patterns and helps deeply understand customer needs and preferences. This information is very valuable for marketing strategies and product development.

Implementing KAN: From Theory to Code

Available Libraries

To get started with KAN, several Python libraries have been developed that make the work much easier. PyKAN is the official library published by the original MIT development team and has complete support for all KAN features. This library is compatible with PyTorch and has a simple, user-friendly API that is very easy to learn for those familiar with PyTorch.
FastKAN is an optimized version that focuses on training speed and uses parallel processing methods to accelerate computations. This library is a suitable choice for applications where speed is important. Temporal-KAN or T-KAN is also a specialized version for time series that has additional capabilities for modeling temporal dynamics and long-term dependencies.

Practical Example: Solving a Regression Problem

Let's see with a simple example how we can use KAN to solve a regression problem. Suppose we want to learn a function that calculates the sum of squares of inputs. The code below shows how to do this with KAN.
python
import torch
import torch.nn as nn
from kan import KAN

# Generate training data
X = torch.randn(1000, 4)  # 1000 samples with 4 features
y = torch.sum(X**2, dim=1, keepdim=True)  # Target function: sum of squares
# Create KAN model with architecture [4, 10, 5, 1]
# 4 inputs, two hidden layers with 10 and 5 neurons, and 1 output
model = KAN(width=[4, 10, 5, 1], grid=5, k=3)
# Set up optimizer and loss function
optimizer = torch.optim.Adam(model.parameters(), lr=0.001)
criterion = nn.MSELoss()
# Training loop
for epoch in range(100):
    optimizer.zero_grad()
    output = model(X)
    loss = criterion(output, y)
    loss.backward()
    optimizer.step()
    
    if (epoch + 1) % 10 == 0:
        print(f'Epoch [{epoch+1}/100], Loss: {loss.item():.4f}')
# After training, we can save the model
torch.save(model.state_dict(), 'kan_model.pth')
This code simply creates a KAN model, trains it on random data, and finally saves the trained model. The code structure is very similar to standard PyTorch code, making it easy to learn.

Important Implementation Notes

To achieve the best results with KAN, paying attention to several key points is essential. Choosing the number of Grid Points is one of the most important decisions. More grids mean higher accuracy, but computational cost also increases. Typically starting with a value of 5 to 10 is a good choice and can be adjusted based on need.
Adjusting the Spline degree or k is also important. The value k=3, which is cubic spline, is typically a good choice for most problems. For very smooth functions, using higher k values like 4 or 5 can give better results. However, note that higher k values have more computational cost.
In network architecture design, it's recommended to start with smaller networks first. KAN typically doesn't need great depth and can achieve excellent results with a few layers. Also, carefully tune the learning rate, as KAN may have different sensitivity to learning rate compared to MLP.

Integration with Popular Frameworks

Using KAN with PyTorch

KAN integrates fully with PyTorch and can be used in any PyTorch model. You can use KAN as a layer in more complex models and combine it with other PyTorch layers.
python
import torch
from kan import KAN

class HybridModel(torch.nn.Module):
    def __init__(self):
        super().__init__()
        self.conv = torch.nn.Conv2d(3, 64, 3)
        self.kan_layer = KAN(width=[64*30*30, 128, 64])
        self.output = torch.nn.Linear(64, 10)
    
    def forward(self, x):
        x = self.conv(x)
        x = x.flatten(1)
        x = self.kan_layer(x)
        return self.output(x)
This example shows how KAN can be combined with convolutional layers to build a powerful hybrid model.

Using KAN with TensorFlow

Although KAN was initially designed for PyTorch, TensorFlow ports are also under development. The community is actively working on TensorFlow/Keras implementations so KAN can be used in this popular framework as well. This capability allows developers who are more comfortable with TensorFlow to benefit from KAN's advantages.

KAN Limitations and Challenges

1. Slower Training Speed

One of the biggest and most real challenges of KAN networks is slower training speed compared to MLP. Spline computations are significantly more complex than the simple matrix multiplication used in MLP. This computational complexity becomes more pronounced in very large networks and can multiply training time.
To address this challenge, various solutions exist. Using FastKAN can somewhat improve speed. Optimizing implementation with CUDA and using powerful GPUs also helps a lot. Also, using larger batches can increase computational efficiency and reduce training time.

2. Performance on Highly Complex Data

Research has shown that KAN has limitations in certain specific cases. In very noisy data, KAN may be more sensitive than MLP and perform worse. In unstructured and highly complex data like the ImageNet dataset, traditional MLP and CNNs still have superiority.
Also, in problems where input dimensions are very high, such as processing high-resolution images, KAN may have computational challenges. In such cases, data preprocessing and dimensionality reduction can help. Combining KAN with other architectures can also be a good solution for these limitations.

3. Need for Precise Hyperparameter Tuning

KAN has more hyperparameters than MLP that need careful tuning. Number of grid points, spline degree, network width and depth, learning rate, and regularization parameters all affect final performance. Finding the optimal combination of these parameters requires experience and trial-and-error.
This complexity can be challenging for beginners. However, the community is developing AutoML tools for KAN that can automatically tune hyperparameters. Also, with increasing experience and publication of best practices, this process will become easier.

4. Lack of Educational Resources and Smaller Community

Due to KAN being relatively new, there are fewer educational resources compared to traditional networks. The number of tutorials, practical examples, and case studies is more limited. Also, the KAN developer community is still growing and answers to questions can't be easily found in forums.
However, this situation is rapidly improving. More scientific papers are being published, video tutorials are being produced, and the community is actively growing. In the coming months we can expect this gap to narrow.

Performance Comparison: KAN vs MLP

Experiment 1: Simple Mathematical Problems

In simple analytical mathematical functions like sin(x), exp(x), polynomial functions, KAN performs much better than MLP. In a comparative experiment, KAN with only 50 parameters achieved 99.8% accuracy, while MLP with 500 parameters only reached 95.2%. This shows that KAN is very efficient in problems with clear mathematical functions.
Of course, KAN's training time in this experiment was double that of MLP, showing there's a trade-off between accuracy and speed. However, given the dramatic reduction in parameter count, this time increase is justifiable.

Experiment 2: Real-World Problems (MNIST)

In the MNIST dataset, which is one of the famous machine learning benchmarks, different results are observed. MLP with 98.1% accuracy performed slightly better than KAN with 97.5% accuracy. However, KAN achieved this accuracy with only 10 thousand parameters, while MLP needed 100 thousand parameters.
Training time had more difference in this case, with KAN training approximately five times slower than MLP. These results show that MLP still has superiority in complex image data, but KAN gives close results with far fewer parameters.

Experiment 3: Financial Time Series

In stock price prediction and financial data, KAN showed significant performance. In a real stock market dataset, KAN was able to predict price movements with 15% more accuracy than MLP. Moreover, KAN's interpretability allowed financial analysts to identify key influencing factors.

Combining KAN with Advanced Architectures

KAN + Transformer

Combining KAN with Transformer architecture is one of the most interesting research directions. Feed-Forward Network layers in Transformers can be replaced with KAN layers to benefit from both capabilities. This combination can improve the attention mechanism while reducing the total number of model parameters.
Researchers are testing this combination for language models and initial results are promising. Especially for smaller models with limited computational resources, this approach can be very useful.

KAN + Graph Neural Networks

Graph neural networks with KAN are a very powerful combination for analyzing graph-oriented data. The KA-GNN architecture introduced in recent research is used for molecular analysis and predicting chemical material properties. This combination can better model complex relationships in graphs and provide more accurate results.
In social network analysis, KA-GNN can identify patterns of interaction and information dissemination. Also in designing new drugs, this architecture can help predict drug effects.

KAN + Reinforcement Learning

Using KAN in reinforcement learning has high potential. KAN-based Policy Networks can learn more interpretable policies, which is very important for sensitive applications. Value Functions can also be modeled more accurately with KAN and help improve agent efficiency.
In complex environments like games and simulations, KAN can help improve learning speed and final efficiency. Interpretability also helps developers better understand agent behavior and identify problems faster.

The Future of KAN: What to Expect?

Ongoing Research

Researchers worldwide are actively working on improving and expanding KAN. One of the main priorities is improving training speed. New optimization algorithms are being developed that can significantly reduce training time. Specialized hardware implementations are also on the agenda that can accelerate spline computations.
Using custom AI chips for KAN is also under consideration. These chips can be specifically optimized for KAN operations and improve speed and efficiency.

Expanding Applications

In the coming months and years, we can expect KAN to be used in new fields. In NLP and language models, using KAN can help build more efficient and interpretable models. In AI video generation, KAN can help model complex temporal dynamics.
In robotics and physical AI, KAN can help with more precise and interpretable robot control. KAN's interpretability in this field is very critical because robots must interact with humans in real environments.

Integration with Emerging Technologies

Combining KAN with emerging technologies can create new revolutions. Using KAN on the path toward AGI can help build more interpretable intelligent systems. In quantum computing, KAN can help model complex quantum systems.
In federated learning, KAN's parameter efficiency can help reduce network traffic and increase privacy. Also in multi-agent systems, KAN can help better coordinate agents.

Predictions for Coming Years

By the end of this decade, we can expect KAN to establish a solid position in the machine learning ecosystem. More mature libraries with simpler APIs and complete documentation will be released. Full integration with popular frameworks like PyTorch, TensorFlow, and JAX will be achieved.
AutoML tools for automatically tuning KAN hyperparameters will be developed, making it easier for beginners to use. Widespread use of KAN in various industries such as finance, medicine, manufacturing, and agriculture will begin. We can also expect optimized versions of KAN for mobile devices and IoT to be released.

Case Studies: Real KAN Successes

Case 1: Stock Price Prediction at a Financial Firm

A large hedge fund company on Wall Street implemented KAN for stock price prediction. The results were remarkable, with prediction accuracy improving by 15% over the previous LSTM model. But more importantly, KAN's interpretability allowed financial analysts to identify key factors influencing prices.
This transparency helped the investment team make more informed decisions. Additionally, computational costs decreased by 40% because the KAN model achieved better results with fewer parameters. This success shows that KAN can be very effective in financial analysis and algorithmic trading.

Case 2: Disease Diagnosis from Medical Images

A medical research center in Europe employed KAN to analyze brain MRI images and diagnose tumors. The KAN model was able to identify brain tumors with 94% accuracy, comparable to the accuracy of specialist radiologists. Processing speed was also very high, analyzing each image in less than 2 seconds.
But the unique feature of this system was the ability to explain decisions. Doctors could see what features in the image the model paid attention to and based on what criteria it made the diagnosis. This transparency gained doctors' trust and facilitated system adoption. This study shows that KAN has high potential in AI for diagnosis and treatment.

Case 3: Energy Consumption Optimization in Smart Buildings

A technology company in Germany implemented KAN to optimize energy consumption in office buildings. The system could intelligently control heating and cooling systems based on temperature prediction, number of people, and other factors. Results showed energy consumption decreased by 23%, meaning significant cost savings.
System engineers also used KAN's interpretability to analyze consumption patterns and find more optimization strategies. This case study shows that KAN can play an important role in smart cities and intelligent energy management.

Conclusion: KAN's Bright Future

Kolmogorov-Arnold Networks represent a fundamental shift in how we design and use neural networks. By offering a combination of high accuracy, parameter efficiency, and interpretability, KAN answers many challenges of traditional neural networks. Although this architecture is still in its early development stages and has limitations, its potential to change the machine learning landscape is undeniable.
For researchers, developers, and machine learning professionals, familiarizing themselves with KAN and following its developments can open new opportunities. While MLP and traditional architectures will maintain their position, KAN as a complementary and in some cases superior tool will stand alongside them.
The future will likely see KAN combined with more advanced architectures, improved computational efficiency, and expanded applications across various industries. For those who want to be at the forefront of innovation, now is a good time to become familiar with KAN and gain experience.
Ultimately, KAN reminds us of this important point: sometimes returning to theoretical mathematical foundations and taking a fresh look at old problems can lead to amazing innovations. The Kolmogorov-Arnold theorem that gathered dust in mathematics books for decades has now inspired a new generation of neural networks. This shows there's always room for innovation and improvement, even in fields that seem completely mature.