Kolmogorov-Arnold Networks (KANs) for Time Series Analysis
(2405.08790)Abstract
This paper introduces a novel application of Kolmogorov-Arnold Networks (KANs) to time series forecasting, leveraging their adaptive activation functions for enhanced predictive modeling. Inspired by the Kolmogorov-Arnold representation theorem, KANs replace traditional linear weights with spline-parametrized univariate functions, allowing them to learn activation patterns dynamically. We demonstrate that KANs outperforms conventional Multi-Layer Perceptrons (MLPs) in a real-world satellite traffic forecasting task, providing more accurate results with considerably fewer number of learnable parameters. We also provide an ablation study of KAN-specific parameters impact on performance. The proposed approach opens new avenues for adaptive forecasting models, emphasizing the potential of KANs as a powerful tool in predictive analytics.
Overview
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The paper introduces Kolmogorov-Arnold Networks (KANs) for time series forecasting, positioning them as an innovative approach superior to traditional methods like ARIMA and modern machine learning models such as MLPs, LSTMs, and CNNs.
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KANs leverage the Kolmogorov-Arnold representation theorem and use spline-parametrized univariate functions as adaptive activation functions, enhancing interpretability and efficiency.
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Experimental results on satellite traffic data demonstrate that KANs outperform MLPs in forecasting accuracy, using fewer parameters, and show promise for real-world applications and future research directions.
Kolmogorov-Arnold Networks for Time Series Forecasting
Introduction
Time series forecasting is crucial for many fields, from finance to meteorology. Traditionally, predicting future data points based on past observations relied on statistical methods like ARIMA or exponential smoothing. These methods are well-established but sometimes struggle with complex, nonlinear relationships in the data. Enter Machine Learning (ML) and, more recently, Deep Learning (DL), with models like Multi-Layer Perceptrons (MLPs), Long Short-Term Memory (LSTM) networks, and Convolutional Neural Networks (CNNs), which have revolutionized the forecasting landscape.
However, these modern methods have their challenges, particularly in scaling and interpretability. This paper investigates Kolmogorov-Arnold Networks (KANs) as an innovative approach promising enhanced performance and efficiency in time series forecasting.
Understanding Kolmogorov-Arnold Networks
Kolmogorov-Arnold Representation Theorem
KANs are rooted in the Kolmogorov-Arnold representation theorem. This theorem states that any multivariate continuous function can be expressed as a finite sum of continuous univariate functions. This breaks down the daunting task of learning a high-dimensional function into learning several simpler one-dimensional functions.
What Makes KANs Different?
Instead of linear weights, KANs use spline-parametrized univariate functions as activation functions. These splines (often B-splines) dynamically adapt during training, enhancing both the interpretability and the efficiency of the network. Here's a comparison of KAN features with typical MLPs:
- Learnable Splines: Unlike fixed activation functions (like ReLUs in MLPs), KANs use splines that adapt during training.
- Layer Configuration: KANs follow a 2-layer architecture based on the theorem, but deeper and broader KANs can be constructed for more complex tasks.
- Parametrization: Adjusting nodes and grid sizes in KANs can significantly affect model performance, offering fine-grained control over the network's capacity.
Experimental Setup and Model Configurations
The study investigates KANs' forecasting capabilities using real-world satellite traffic data. Satellite traffic data is particularly challenging due to its dynamic nature, making it an excellent benchmark.
Model Configurations
Four different models were compared:
- MLP (3-depth): Traditional MLP with three hidden layers.
- MLP (4-depth): Traditional MLP with four hidden layers.
- KAN (3-depth): KAN with three layers, using B-splines with specific configurations.
- KAN (4-depth): KAN with four layers, similarly configured.
All models were trained using the Adam optimizer for 500 epochs with a mean absolute error optimization goal.
Results and Performance Analysis
Comparison of KANs and MLPs
The study reveals that KANs outperform MLPs in forecasting accuracy while needing considerably fewer parameters. Here's a summary of key findings:
- Error Metrics: KANs, particularly the 4-depth model, showed lower Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE) compared to MLPs.
- Parameter Efficiency: KANs achieved better performance with a significantly reduced number of parameters (e.g., 109k vs. 329k for KAN 4-depth vs. MLP 4-depth).
Ablation Study of KAN Parameters
The study explored the impact of varying the number of nodes and grid sizes within KAN configurations:
- Increasing the number of nodes generally improved performance.
- Larger grid sizes within reason enhanced the network's ability to capture complex patterns, especially when paired with a high number of nodes.
Practical and Theoretical Implications
Real-World Applications
For practical forecasting tasks like traffic prediction in satellite networks, KANs offer superior accuracy and efficiency. Their ability to quickly adapt to rapid changes in data makes them well-suited for dynamic environments.
Theoretical Insights
KANs represent an interesting melding of MLPs and splines, blending the strengths of both approaches. This dual-level flexibility (node-based and spline-based) allows KANs to handle complex, nonlinear data more effectively than traditional methods.
Future Directions
Given their promising performance, future research could focus on:
- Robustness Studies: Further testing KANs on diverse datasets.
- Hybrid Architectures: Exploring combinations of KANs with other deep learning architectures like CNNs or LSTMs.
Conclusion
Kolmogorov-Arnold Networks present a compelling alternative for time series forecasting, offering robust performance with fewer parameters. Their innovative use of adaptive splines provides unique advantages in modeling complex temporal data, potentially transforming forecasting tasks across various domains.