asKAN: Active subspace embedded Kolmogorov-Arnold network.

Area of Science:

• Artificial Intelligence
• Scientific Machine Learning
• Numerical Analysis

Background:

• Kolmogorov-Arnold Network (KAN) shows promise for AI science but has limitations in representing complex functions.
• The Kolmogorov-Arnold theorem provides a theoretical basis for function representation using univariate components.
• Representing ridge functions efficiently is crucial for simplifying neural network architectures.

Purpose of the Study:

• To investigate the capability of KANs in representing ridge functions.
• To develop an improved KAN architecture that leverages insights from the Kolmogorov-Arnold theorem and active subspace methods.
• To reduce the error and improve the efficiency of neural networks in AI science applications.

Main Methods:

• Theoretical analysis based on the Kolmogorov-Arnold theorem to understand KAN's function representation.
• Development of active subspace embedded KAN (asKAN), a hierarchical framework integrating KAN with active subspace methodology.
• Iterative implementation of asKAN, identifying dominant ridge directions and projecting input variables onto these directions without increasing neuron count.

Main Results:

• Linear combinations of input variables can simplify network architectures for ridge function representation.
• asKAN significantly reduces approximation errors compared to standard KAN across various tasks.
• Validation on function fitting, Poisson equation solving, and sound field reconstruction demonstrates asKAN's effectiveness.

Conclusions:

• asKAN offers a more efficient and accurate approach to function representation in AI science compared to standard KAN.
• The integration of active subspace methodology enhances KAN's ability to capture essential data variations.
• asKAN represents a significant advancement for small-scale AI science applications requiring high-fidelity function approximation.