Use of Galerkin Physics-Informed Neural Network for Solving the Wave Equation

Several physical phenomena in science and engineering are modeled by partial differential equations. The solution of these equations can be challenging because few simplified cases allow analytical solutions, while most of them require numerical techniques to generate approximated solutions. Therefore, when accurate results are needed or geometrically complex domains are considered, a very fine mesh must be used, leading to a time-consuming process for high-dimensional problems or sensitivity analysis, where different configurations of the same problem are necessary. An alternative approach for such cases is the use of artificial intelligence (AI) techniques, which can easily handle several variables and generate continuous results across the problem domain. For this purpose, it is possible to replace the data requirements of traditional AI algorithms with a mathematical formulation, leading to the recently introduced physics-informed machine learning approaches. For these cases, neural networks are trained to generate a continuous solution for partial/ordinary differential equations. However, standard physics-informed machine learning methods still have limitations and generally require large networks with several hidden layers and neurons to generate acceptable solutions. To overcome this limitation, a global Galerkin formulation was developed by using feedforward neural networks as the kernel to generate the basis functions of the approximation and an improved two-step training process, which allows to calculate the network internal parameters and Galerkin coefficients simultaneously, reducing the number of iterations. The proposed method was successfully employed to solve the wave equation, which is a second order partial differential equation that arises in different areas, such as acoustics, fluid dynamics, and geophysics, among others. It was possible to solve the PDE for different domains in 1D and 2D by using smaller models with improved accuracy when compared to the standard physics-informed techniques.