A neural network-based approach for bending analysis of strain gradient nanoplates

Purpose of this paper is the presentation of a novel Machine Learning (ML) technique for nanoscopic study of thin nanoplates. The second-order strain gradient theory is used to derive the governing equations and account for size effects. The ML framework is based on Physics-Informed Neural Networks (PINNs), a new concept of Artificial Neural Networks (ANNs) enriched with the mathematical model of the problem. Training of PINNs is performed using a highly efficient learning algorithm, known as Extreme Learning Machine (ELM). Two applications of this ANNs-based method are illustrated: solution of the Partial Differential Equations (PDEs) modeling the flexural response of thin nanoplates (direct problem), and identification of the length scale parameter of the nanoplate mathematical model with the aid of measurement data (inverse problem). Comparison with analytical and Finite Element (FE) solutions demonstrate the accuracy and efficiency of this ML framework as meshfree solver of high-order PDEs. The stability and reliability of the present method are verified through parameter studies on hyperparameters, network architectures, data noise and training initializations. The results presented give evidence of the effectiveness and robustness of this new ML approach for solving both direct and inverse nanoplate problems.