Normalized Fourier-induced PINN method for solving the wave propagation equation in a non-unitized domain over an extended time range

TL;DR

This study introduces a normalized PINN (NPINN) framework to solve a class of wave propagation equations in non-unitized domains over extended time ranges through a normalization technique that involves either spatial or temporal variable normalization.

Abstract

Physics-Informed Neural Networks (PINNs) have gained significant attention for their simplicity and flexibility in engineering and scientific computing. In this study, we introduce a normalized PINN (NPINN) framework to solve a class of wave propagation equations in non-unitized domains over extended time ranges. This is achieved through a normalization technique that involves either spatial or temporal variable normalization. To enhance the capability of NPINN in solving wave equations, we integrate a Fourier-induced deep neural network as the solver, leading to a novel architecture termed NFPINN. Furthermore, we explore different normalization strategies for spatial and temporal variables and identify the optimal normalization approach for our method. To assess the effectiveness and robustness of the proposed NFPINN, we present numerical experiments in both two-dimensional and three-dimensional Euclidean spaces, considering regular and irregular domains. The results confirm the accuracy and stability of our approach.

Figures (9)

• Figure 1: Schematic diagram of FFM-based DNN with $n$ subnetworks, $\sigma$ stands for the activation function
• Figure 2: FPINN framework to solve wave propagation
• Figure 3: Numerical results of PINN and FPINN methods for test example in domain $\Omega_1$.
• Figure 4: Numerical results of PINN and FPINN methods for test example in domain $\Omega_2$.
• Figure 5: Numerical results of NFPINN in different normalized methods for test example in domain $\Omega$.

Theorems & Definitions (1)

• remark 1