Solution of Biharmonic Equation in Complicated Geometries With Physics Informed Extreme Learning Machine

Abstract

Recently, physics informed neural networks (PINNs) have produced excellent results in solving a series of linear and nonlinear partial differential equations (PDEs) without using any prior data. However, due to slow training speed, PINNs are not directly competitive with existing numerical methods. To overcome this issue, the authors developed Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINN, and tested it on a range of linear PDEs of first and second order. In this paper, we evaluate the effectiveness of PIELM on higher-order PDEs with practical engineering applications. Specifically, we demonstrate the efficacy of PIELM to the biharmonic equation. Biharmonic equations have numerous applications in solid and fluid mechanics, but they are hard to solve due to the presence of fourth-order derivative terms, especially in complicated geometries. Our numerical experiments show that PIELM is much faster than the original PINN on both regular and irregular domains. On irregular domains, it also offers an excellent alternative to traditional methods due to its meshless nature.