A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic Response
Abstract
This paper introduces a framework that integrates the dynamic stiffness matrix (DSM) with physics-informed neural networks (PINN). The DSM-PINN embeds physical constraints within the model and demonstrates robustness, particularly when addressing limited datasets across diverse investigations. In this approach, deep neural network outputs approximate the displacement fields of element nodes. Unlike the finite element method (FEM), the element shape functions are homogeneous solutions to the governing partial differential equation, forming the basis of the exact dynamic stiffness matrix, thereby avoiding high-order derivative terms. This matrix also serves as a frequency-domain spectral element, resulting in a strong-form PINN. The loss function is produced by connecting neural networks with dynamic stiffness matrices. We focus on utilising PINNs to resolve eigenvalue problems by employing the Wittrick-Williams algorithm, which overcomes the challenge of neural networks failing to converge to higher-order eigenvalues. Additionally, the frequency domain-PINN method is used to analyse structural dynamic responses under moving and impulsive loads, addressing the limitation of neural networks in handling complex numbers. Theoretical convergence stability of the suggested approach is also analysed even DSM is an indefinite matrix after implementing the boundary condition. The numerical results validate the practicality and efficacy of the recommended approach.
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BibTeX
@misc{an2026framework,
title = {{A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic Response}},
author = {Yi An and Zhijiang Chen and Zhiqiang Feng and Qian Cheng and Jack C. P. Cheng and Haijiang Li and Dalei Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28683},
eprint = {2608.28683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28683},
}
RIS
TY - UNPB
TI - A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic Response
AU - Yi An
AU - Zhijiang Chen
AU - Zhiqiang Feng
AU - Qian Cheng
AU - Jack C. P. Cheng
AU - Haijiang Li
AU - Dalei Wang
PY - 2026
DA - 2026/08/26
PB - arXiv
DO - 10.48550/arxiv.2608.28683
UR - https://arxiv.org/abs/2608.28683
AB - This paper introduces a framework that integrates the dynamic stiffness matrix (DSM) with physics-informed neural networks (PINN). The DSM-PINN embeds physical constraints within the model and demonstrates robustness, particularly when addressing limited datasets across diverse investigations. In this approach, deep neural network outputs approximate the displacement fields of element nodes. Unlike the finite element method (FEM), the element shape functions are homogeneous solutions to the governing partial differential equation, forming the basis of the exact dynamic stiffness matrix, thereby avoiding high-order derivative terms. This matrix also serves as a frequency-domain spectral element, resulting in a strong-form PINN. The loss function is produced by connecting neural networks with dynamic stiffness matrices. We focus on utilising PINNs to resolve eigenvalue problems by employing the Wittrick-Williams algorithm, which overcomes the challenge of neural networks failing to converge to higher-order eigenvalues. Additionally, the frequency domain-PINN method is used to analyse structural dynamic responses under moving and impulsive loads, addressing the limitation of neural networks in handling complex numbers. Theoretical convergence stability of the suggested approach is also analysed even DSM is an indefinite matrix after implementing the boundary condition. The numerical results validate the practicality and efficacy of the recommended approach.
ER -