Developing of neural network computing methods for solving inverse elasticity problems

This paper examines the use of neural network methods to solve inverse problems in the mechanics of elastic materials. The aim is to design physics-informed neural networks that can predict the parameters of structural components, and the physical properties of materials based on a specified displacement distribution. A key feature of the specified neural networks is the integration of differential equations and boundary conditions into the loss function calculation. This approach ensures that the error in approximating unknown functions has a direct impact on optimizing the network's weights. As a result, the resulting neural network approximations of unknown functions comply with the differential equations and boundary conditions. To test the capabilities of the designed neural networks, inverse problems involving the bending of plates and beams have been solved, focusing on determining one or two unknown parameters. Comparison of predicted and exact values demonstrates high accuracy of the constructed neural network models, with a relative prediction error of less than 3 % across all cases. Unlike analytical methods for solving inverse problems, the primary advantage of physics-informed neural networks is their flexibility when addressing both linear and nonlinear problems. For instance, the same network architecture can be employed to solve various boundary-value problems without modification. Compared to classical numerical methods, the parallelization capability of neural networks is inherently supported by modern software libraries. Therefore, the application of physics-informed neural networks for solving inverse elasticity problems of plates and beams is effective, as evidenced by the achieved relative errors and the computational robustness of the method. In practice, the proposed solution can be used for relevant calculations during the design of structural elements. The developed software code can also be integrated into automated design systems or computer algebra systems