PINN-Based Kolmogorov-Arnold Networks with RAR-D Adaptive Sampling for Solving Elliptic Interface Problems

TL;DR

This paper introduces a novel PINN architecture based on Kolmogorov-Arnold Networks with adaptive sampling, achieving higher accuracy and faster convergence for elliptic interface problems with smaller networks.

Contribution

It proposes a dual KANs structure for PINNs and integrates RAR-D adaptive sampling to improve training efficiency and accuracy for interface PDE problems.

Findings

Achieves more uniform error distribution across the domain.

Requires smaller network sizes for high accuracy.

Demonstrates faster convergence compared to standard PINNs.

Abstract

Physics-Informed Neural Networks (PINNs) have become a popular and powerful framework for solving partial differential equations (PDEs), leveraging neural networks to approximate solutions while embedding PDE constraints, boundary conditions, and interface jump conditions directly into the loss function. However, most existing PINN approaches are based on multilayer perceptrons (MLPs), which may require large network sizes and extensive training to achieve high accuracy, especially for complex interface problems. In this work, we propose a novel PINN architecture based on Kolmogorov-Arnold Networks (KANs), which offer greater flexibility in choosing activation functions and can represent functions with fewer parameters. Specifically, we introduce a dual KANs structure that couples two KANs across subdomains and explicitly enforces interface conditions. To further boost training efficiency and convergence, we integrate the RAR-D adaptive sampling strategy to dynamically refine training points. Numerical experiments on the elliptic interface problems yield more uniform error distributions across the computational domain, which demonstrates that our PINN-based KANs achieve superior accuracy with significantly smaller network sizes and faster convergence compared to standard PINNs.