Utilizing curriculum learning for high-frequency eigenfunction discovery through a physics-informed neural network

A physics-informed neural network (PINN) is a type of learning model that incorporates the governing physical laws (i.e., ordinary/partial differential equations and initial/boundary conditions) directly into the training process of neural network architecture for approximating the solution function. The focus of this study is to apply the PINN to various eigenvalue problems found in engineering. A notable issue when solving the eigenvalue problem with the PINN is its convergence ability to obtain the solution to high-frequency eigenmodes due to spectral bias. To address this, we propose a curriculum learning-based PINN for not only high-frequency eigenfunctions (forward problem) but also corresponding eigenvalues (inverse problem). The proposed method is composed of two sequentially executed training processes. In the first stage (easier training), the eigenvalue is fixed to a certain value for learning the eigenfunction, aligning it not exactly but closely with the target eigenfunction. In the second stage (harder training), the eigenfunction learned in the first stage serves as the initial model. Then both the eigenvalue and eigenfunction are trained simultaneously to refine the solution toward the target eigenfunction. The effectiveness of the proposed method was verified through two case studies (i.e., Schrödinger equation and L-shaped membrane). The results demonstrate that the proposed method helps to mitigate spectral bias and improves convergence robustness for selected higher eigenmodes.