A physics-informed neural network-based method for dispersion calculations

Abstract

The study of dispersion relations of periodic structures or elastic metamaterials is essential to understand and optimize their unique wave propagation characteristics. By integrating physical laws for the generation of physically consistent results without labeled data, Physics-Informed Neural Networks (PINNs) offer a new perspective on scientific computation, which is a potential machine learning method in advancing the analysis and design of advanced materials. In this study, we introduce a novel PINN-based numerical method for the calculation of dispersion calculations of periodic structures. First, coupling the physical information of the dispersion problem of periodic structures and the neural networks PDE solver, the framework of the proposed method is constructed. Unlike those existing PINNs, the proposed PINN is designed for the first time to handle the dispersion problem as well as the equivalent eigenvalue problem. In particular, a unified framework is proposed to solve both the real and complex eigenvalue problems, from which the real and complex dispersion curves of periodic structures are obtained. Second, comparing with the analytical results, the correctness of the proposed method is validated. And, dispersion properties for propagative waves in pass bands and evanescent waves in stop bands are analyzed. Third, a comprehensive analysis of the convergence of the proposed method is performed. The Neural Tangent Kernel (NTK)-based adaptive loss weighting scheme is integrated into the proposed PINN to achieve the balanced convergence across different loss terms. Meanwhile, the Random Fourier Feature Mapping is implemented into the proposed method to mitigate the eigenfrequency bias problem. Comparison results demonstrate that such enhancements allow for a more accurate convergence. For the considered dispersion problem, a coherent convergence is achieved for all eigenfrequencies in the desired frequency range. In summary, the proposed physics-informed machine learning method is a promising computational method for the dispersion problem of periodic structures.