Physics-Informed Extreme Learning Machine framework for solving linear elasticity mechanics problems

In neural network-based methods for elasticity mechanics such as Physics-Informed Neural Networks (PINNs), the training process is typically time-consuming due to multiple optimization iterations and reliance on automatic differentiation for gradient computation. This paper proposes a novel framework that combines Physics-Informed Extreme Learning Machines (PIELM) with linear elastic mechanics, focusing on the integration of discretization operators with the principles of linear elasticity mechanics. These discretization operators are constructed based on the form of higher-order derivatives in PIELM, along with governing equations and boundary conditions. Physical information is incorporated into the loss function through discretization operators applied at collocation points. The solution of the proposed method is calculated through the least squares method , which determines the output weights. The comparison with PINN under nonlinear loads and geometric defects demonstrates that PIELM achieves superior error control and prediction accuracy. Furthermore, PIELM achieves more precise in evaluated error norms compared to Finite Element Method (FEM) in a three-dimensional problem with an analytical solution. In conclusion, the proposed method offers an robust and efficient alternative method for linear elasticity.