Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging

TL;DR

PIKAN solves multi-material elasticity by minimizing the total potential energy $\mathcal{L} = \Psi_{in} - \Psi_{ex}$ using a single Kolmogorov-Arnold Network (KAN) within the Deep Energy Method, avoiding domain decomposition. It uses admissible displacement fields that automatically satisfy essential boundary conditions and trainable B-spline activations to efficiently represent discontinuities at material interfaces. Through 2D numerical experiments on cantilever beams, plates with central holes, DBC substrates, and TGV-Cu structures, PIKAN achieves high accuracy relative to FEM and robust performance across complex interface geometries, while avoiding penalty-based interface constraints. The work provides open-source code and outlines future extensions to 3D, adaptive sampling, and potential hybrid MLP-KAN architectures to balance efficiency and accuracy.

Abstract

This paper proposes a Physics-Informed Kolmogorov-Arnold Network for analyzing elasticity problems in multi-material electronic packaging structures. The method replaces traditional Multi-Layer Perceptrons with Kolmogorov-Arnold Networks within an energy-based Physics-Informed Neural Network framework. By constructing admissible displacement fields satisfying essential boundary conditions and optimizing network parameters through numerical integration, the proposed method effectively handles material property discontinuities. Unlike traditional methods that require domain decomposition and interface constraints for multi-material problems, Kolmogorov-Arnold Networks' trainable B-spline activation functions provide inherent piecewise characteristics. This capability stems from B-splines' local support, which enables effective approximation of discontinuities despite their individual smoothness. Consequently, this approach enables accurate approximation across the entire domain using a single network and simplifying the computational framework. Numerical experiments demonstrate that the proposed method achieves excellent accuracy and robustness in multi-material elasticity problems, validating its practical potential for electronic packaging analysis. Source codes are available at https://github.com/yanpeng-gong/PIKAN-MultiMaterial.

Figures (35)

• Figure 1: Kolmogorov-Arnold Network schematic (adapted and redrawn from the original KAN architecture liu2024kan).
• Figure 2: PIKAN schematic: Deep Energy Method using KAN instead of MLP.
• Figure 3: Triangular integration scheme: sample points and control areas.
• Figure 4: Delaunay integration scheme: sample points and control areas.
• Figure 5: PIKAN flowchart for multi-material elasticity problems.