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

TL;DR

The paper addresses elliptic interface problems with discontinuous coefficients by replacing standard MLP backbones in PINNs with Kolmogorov-Arnol'd Networks (KANs) in a dual-network, domain-decomposition setup that explicitly enforces interface jump conditions. It introduces Residual-based Adaptive Refinement with Diversity (RAR-D) to dynamically refine collocation points where the PDE residual is largest, improving convergence near interfaces. The key contributions are (i) a dual-KANs architecture that can capture non-smooth interface features with fewer parameters, (ii) a rigorous coupling across subdomains through interface losses, and (iii) the integration of RAR-D to produce more uniform error distributions and faster training. Numerical results on continuous and discontinuous interfaces show that PINN-based KANs achieve higher accuracy with smaller networks and faster convergence than baseline dual-PINNs, highlighting the practical impact of KANs in meshfree, physics-informed PDE solvers for complex interface geometries.

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.

Figures (11)

• Figure 1: A dual PINNs structure for solving elliptic interface problem.
• Figure 2: Schematic illustration of a dual KAN with structure [2,3,3,3,1].
• Figure 3: Contour plots of the Example \ref{['E1']}: exact solution(a), approximation solutions by PINN(b) and KAN(c), and the absolute error by PINN(d) and KAN(e).
• Figure 4: The evolution of loss error for Example \ref{['E1']}.
• Figure 5: Visualization of the activation functions of a dual KAN after training for Example \ref{['E1']}.

Theorems & Definitions (4)

• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4