A Posteriori Error Control of Neural Network Solutions to Plane Problems of Linear Elasticity Theory

This article investigates the computational properties of a functional‑type a posteriori error estimate for analyzing the accuracy of solutions to plane linear elasticity problems obtained by neural networks. Problems with boundary conditions in displacements and boundary tractions are considered. Special attention is paid to a comparison of solutions calculated using the classical finite element method (FEM) and the physics-informed neural networks (PINN). The functional a posteriori error estimate obtained by S.I. Repin is used as the error analysis tool. In order to ensure exact satisfaction of the Dirichlet boundary condition, the neural network solution is constructed based on bubble functions. In addition to analytical specification of the bubble functions, a method for constructing them by solving Poisson’s equation on the same computational domain is employed. Numerical experiments conducted using adaptive algorithms based on the first-order Raviart–Thomas element showed that the efficiency index of the estimate tends to an optimal value with local mesh refinement in areas of maximum error. The obtained results confirm that functional a posteriori estimates provide reliable and efficient error control for solutions obtained by neural networks.