Some regards on using physics-informed neural networks for solving two-dimensional elasticity problems

Abstract

This paper investigates stress analysis in two-dimensional solid structures using Physics-Informed Neural Networks (PINNs) as an alternative to the Finite Element Method (FEM). Unlike FEM, which depends on fine meshing and substantial computational resources, PINNs incorporate physical laws and boundary conditions directly into the neural network. Although training a PINN can be computationally demanding, the trained model can be reused at a significantly lower cost for similar analyses, reducing the need for repeated meshing and re-computation. The study focuses on three benchmark cases, in which stress distributions are modeled using three neural networks: two plate problems and the classical Kirsch problem, which involves stress concentration around a central hole. The networks take random coordinates as input and output the corresponding stress tensor components. By minimizing a loss function based on the mean squared error (MSE) of the governing equations and boundary conditions, PINNs generate physically consistent stress distributions that are validated against analytical solutions. A key advantage of PINNs is their ability to generalize across different geometries and boundary conditions, making them a powerful tool for solid mechanics analysis. A detailed explanation of the numerical implementation and corresponding codes is also provided to ensure reproducibility and to facilitate further research.