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.

Perspectives of Physics-Based Machine Learning for Geoscientific Applications Governed by Partial Differential Equations

“…They have a better performance for the small data regime since the physical constraints regularize the system and they constrain the size of admissible solutions (Raissi et al, 2019). Within the field of physics-guided loss functions various different approaches are available (e.g., Bar-Sinai et al, 2019;De Bézenac et al, 2019;Dwivedi and Srinivasan, 2020;Geneva and Zabaras, 2020;Karumuri et al, 2020;Pang et al, 2019Pang et al, , 2020Peng et al, 2020;Raissi et al, 2019;Shah et al, 2019;Sharma et al, 2018;Wu et al, 2020;Yang and Perdikaris, 2018;Yang et al, 2020;Zhu et al, 2019). For the following, we as an example focus on the method of physics-informed neural networks (PINNs).…”

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

“…In many recent studies [16][17][18]62], the implications of imposing essential boundary conditions via the loss function in PINN have been studied, and numerical experiments have affirmed that the presence of the boundary residual terms compromises the convergence of the backpropagation algorithm and the accuracy of the method. To address this problem, remedies have been introduced in the PINN literature, such as using two neural networks, one for the PDE and the other to satisfy the essential boundary condition [3,7,[61][62][63], introduction of a penalty parameter via an augmented variational formulation to weakly impose the essential boundary conditions [9], and Nitsche's method to impose the essential boundary condition [64]. Some of these approaches mirror those previously pursued in meshfree and particle methods to satisfy essential boundary conditions [19,20,65].…”