The rapid advancements in algorithms, hardware, and graphical processors, along with the increasing availability of data across various fields, have led to widespread interest in modern artificial intelligence approaches, particularly deep learning, for solving a diverse range of problems. Physics-informed neural networks (PINNs) are a type of scientific machine learning technique used to solve problems involving ordinary and partial differential equations. In this type of neural network, the solutions to differential equations are approximated during the learning process by minimizing a loss function that incorporates the effects of boundary conditions, initial conditions, and the residuals of the differential equations at selected points. Essentially, these networks transform the process of directly solving governing equations into an optimization problem by incorporating the physical knowledge of the problem. This approach is particularly effective for both forward and inverse problems. In this thesis, the PINN approach—an innovative class of scientific machine learning techniques and part of the mesh-free methods category—has been employed for the first time to analyze both forward and inverse problems related to nanobeams and plates. In the first part, the governing equations for the behavior of nanobeams resting on a nonlinear elastic foundation were derived using Hamilton’s principle, based on Euler-Bernoulli beam theory and Eringen’s nonlocal theory. Subsequently, the static bending problem was analyzed under various boundary conditions, loadings, and elastic foundations for different values of nonlocal parameters. Additionally, the impact of network components on the performance of the PINN approach was examined. Inverse problems involving the identification of the nonlocal parameter, transverse loading intensity, critical buckling load, and free vibration frequency were also eva‎luated. In the second part, the bending analysis of plates was conducted based on classical plate theory and first-order shear deformation theory under various loadings and material compositions. Furthermore, inverse problems related to the identification of transverse loading intensity, critical buckling load, and free vibration frequency were assessed. In summary, it was observed that any increase in stiffness originating from boundary conditions, the elastic foundation, or higher modes leads to an increase in the number of computational iterations (epochs) required for both forward and inverse problems. Moreover, the location of the extrema in the deformation and bending moment diagrams is influenced by asymmetry in loading and boundary conditions, which is significantly related to the nonlocal parameter and foundation stiffness. It is worth noting that this approach demonstrates favorable performance even in the presence of noise and with varying initial guesses. Finally, the use of nondimensionalized reduced order differential equations and boundary conditions, along with adaptive weighting methods, significantly improves the performance of the utilized approach.

سعيد صرامي , بشير موحديان عطار

مجتبي ازهري

نسرين جعفري , حسين عموشاهي , مهدي زندي آتشبار