A Physics-Informed Neural Network Approach for Solving Structural Eigenvalue Problem

Research on physics-informed neural networks (PINNs), which incorporate physics into the learning process, is being actively conducted across many research groups. PINNs are utilized in various applications, such as in the forward problem, the inverse problem, and in deep operator networks. In this study, we focus on utilizing PINNs to address eigenvalue problems, which are of fundamental significance in engineering. The eigenvalue problem possesses characteristics of both the forward and inverse problems. It exhibits characteristics of an inverse problem in inferring eigenvalues and eigenfunctions simultaneously. Likewise, it shows features similar to the forward problem in its ability to learn without using data. However, in terms of convergence properties, the eigenvalue problem stands out in its ability to propose infinite number of global solutions. This is unlike the forward problem, which generally offers mostly unique solutions. Further, it does not converge to local solutions, which is a characteristic found in inverse problems. This study discusses points to consider when solving eigenvalue problems with PINNs and proposes a PINN-based method for solving eigenvalue problems. Finally, this research study validates the proposed method through case studies.