Physics-informed KAN-coupled FEM for deformation analysis of complex shells

Abstract

This study addresses the challenges of weak geometric adaptability and low computational accuracy in Physics-Informed Neural Networks (PINNs) for predicting static mechanical responses of complex shell structures. Despite growing applications of PINNs in shell static analysis, their use for complex geometries like stiffened shells remains hindered by limited network capacity, subpar training accuracy, and ineffective physical constraint enforcement. No reliable PINN solutions currently exist for such complex geometries. We propose an innovative computational framework that deeply integrates the Finite Element Method (FEM) with Physics-Informed Kolmogorov–Arnold Networks (PIKANs). The framework spatially discretizes shell structures using FEM, solves shell governing equations at each grid node, and establishes the neural network’s loss function based on these equations. Building on this foundation, an improved High-Order ReLU-KAN (HRKAN) is employed to construct the FEM-PIKAN computational model, which innovatively introduces sigmoid activation functions at the hidden layer inputs to significantly enhance the network’s nonlinear mapping capability. To validate the method’s effectiveness, systematic studies are conducted on four typical shell structures — flat plate, cylindrical shell, stiffened plate, and stiffened cylindrical shell — under body force, surface pressure, and concentrated loads. Additionally, the static response of a simplified submarine pressure hull subjected to complex loading scenarios was predicted. The results demonstrate that the proposed FEM-PIKAN method accurately predicts the static responses of various complex shell structures, providing a new paradigm for efficient and high-precision analysis of such structures. The source code is available at https://github.com/whatuphmz/FEM-PIKAN.