asKAN: Active subspace embedded Kolmogorov-Arnold network.

The Kolmogorov-Arnold Network (KAN) has emerged as a promising neural network architecture for small-scale AI Science applications. We explore the capability of KAN in representing ridge functions through the theoretical framework of the Kolmogorov-Arnold theorem, which starts the representation of multivariate functions from constructing the univariate components rather than combining the independent variables. Our analysis reveals that leveraging linear combinations of input variables can lead to a simplification of network architectures when representing ridge functions. Inspired by this finding, we propose active subspace embedded KAN (asKAN), a hierarchical framework that integrates KAN's expressive function representation with the active subspace methodology. asKAN embeds active subspace detection between KANs, identifying dominant ridge directions and adaptively projecting independent variables onto these directions to obtain new linear input combinations. asKAN is implemented in an iterative way without increasing the number of neurons in the original KAN. The proposed method is validated through fitting function, solving the Poisson equation, and reconstructing sound field. Compared with KAN, asKAN significantly reduces the error using the same network architecture.