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

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.

💡 Research Summary

The paper introduces a novel physics‑informed neural network (PINN) framework, termed Physics‑Informed Kolmogorov‑Arnold Networks (PIKAN), for solving elasticity problems in multi‑material electronic packaging structures. Traditional numerical methods such as finite element analysis (FEA) face challenges when dealing with heterogeneous materials, complex interfaces, and the need for mesh generation, especially in high‑density electronic packages. While PINNs have emerged as a mesh‑free alternative, most implementations rely on multilayer perceptrons (MLPs) that suffer from spectral bias, large parameter counts, and difficulty capturing sharp material discontinuities.

To address these issues, the authors replace the conventional MLP with a Kolmogorov‑Arnold Network (KAN). A KAN differs from an MLP by using trainable one‑dimensional B‑spline activation functions for each neuron, while the linear weights remain simple summations. B‑splines possess local support, enabling the network to represent piecewise‑smooth functions efficiently. This property is crucial for multi‑material problems where material properties change abruptly across interfaces. The spline‑based activations allow the network to adapt locally without requiring a globally deep or wide architecture, thereby reducing the total number of trainable parameters and mitigating the spectral bias that hampers high‑frequency solution components.

The authors embed the KAN within an energy‑based PINN, specifically the Deep Energy Method (DEM). Instead of enforcing the governing PDE residuals directly (strong form), DEM constructs a loss function equal to the total potential energy of the elastic system: \