Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

Two-dimensional (plane) elasticity equations in solid mechanics are solved numerically with the use of an ensemble of physics-informed neural networks (PINNs). The system of equations consists of the kinematic definitions, i.e. the strain-displacement relations, the equilibrium equations connecting a stress tensor with external loading forces and the isotropic constitutive relations for stress and strain tensors. Different boundary conditions for the strain tensor and displacements are considered. The proposed computational approach is based on principles of artificial intelligence and uses a developed open-source machine learning platform, scientific software Tensorflow, written in Python and Keras library, an application programming interface, intended for a deep learning. A deep learning is performed through training the physics-informed neural network model in order to fit the plain elasticity equations and given boundary conditions at collocation points. The numerical technique is tested on an example, where the exact solution is given. Two examples with plane stress problems are calculated with the proposed multi-PINN model. The numerical solution is compared with results obtained after using commercial finite element software. The numerical results have shown that an application of a multi-network approach is more beneficial in comparison with using a single PINN with many outputs. The derived results confirmed the efficiency of the introduced methodology. The proposed technique can be extended and applied to the structures with nonlinear material properties.