Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities

Area of Science:

• Artificial Intelligence
• Numerical Analysis
• Computational Science

Background:

• Kolmogorov-Arnold Networks (KANs) are emerging as powerful alternatives to traditional Multilayer Perceptrons, featuring learnable activation functions.
• Various function representations have been explored for KANs, but a universally effective method for diverse function types remains an active research area.
• Physics-informed neural networks (PINNs) are increasingly used for solving differential equations, requiring robust function approximation capabilities.

Purpose of the Study:

• To investigate the efficacy of Sinc interpolation as a function representation within Kolmogorov-Arnold Networks.
• To evaluate the performance of SincKANs in function approximation tasks, particularly those involving smooth functions and singularities.
• To assess the applicability and benefits of SincKANs in the context of physics-informed neural networks for solving partial differential equations.

Main Methods:

• Implementation of Sinc interpolation for defining the learnable activation functions in Kolmogorov-Arnold Networks, creating SincKANs.
• Experimental validation through a series of comparative tests against existing function representation methods in KANs.
• Application of SincKANs to problems requiring function approximation and the solution of partial differential equations using a PINN framework.

Main Results:

• Sinc interpolation proves to be a viable and effective method for representing functions within KANs, handling both smooth and singular behaviors.
• SincKANs demonstrated superior performance across nearly all experimental cases considered, outperforming other function representation techniques.
• The integration of Sinc interpolation into PINNs shows promise for enhanced accuracy and robustness in solving differential equations.

Conclusions:

• Sinc interpolation offers a significant advancement for Kolmogorov-Arnold Networks, enhancing their capability to approximate complex functions.
• SincKANs represent a promising new architecture for machine learning tasks, particularly in scientific computing and differential equation solving.
• The study provides strong empirical evidence for the superiority of SincKANs, with open-source code available for reproducibility and further research.