Evaluation of Physics Informed Neural Networks in engineering simple structural analysis problems

Sammanfattning: Neural Networks have found many applications for a long time in Machine Learning in different disciplines, and have especially flourished in the last decade because of the ever-increasing processing power especially from GPUs. Because of their ability to operate as universal function approximators and model nonlinear processes, attempts have been made in recent years to be also used for modeling partial differential equation (PDE) solutions. PDEs govern almost any engineering problem and in the past decades numerical methods like finite elements dominate the solving of such problems. The idea behind using a neural network to approximate the solution of a PDE is to incorporate the equations into its training process and then use them to regulate it. This makes it a Physics Informed Neural Network (PINN).  In this paper we are using a PINN to model an elastodynamics problem and more specifically a plane stress problem. Both data produced by a traditional solver and the governing PDEs with boundary conditions (BC), are incorporated into the loss function of the PINN, which is then minimized to regulate the network. The data help the training process, especially in this “soft” manner of enforcing BC, but they can be omitted completely to have the PINN function as a mesh-free solver. Both those two methods of training have been implemented and they have been compared with a traditional data-only training method of a neural network. Multiple neural networks have been used to represent the inputs and both displacement and stress components have been taken as the output. Different neural network complexities have been used and compared. The PINNs have proven to reach acceptable errors faster than the traditional neural networks, with less complexity, less data and less overfitting. They have also proven to behave quite well as surrogate models by being used to generate thicker grids than those used for training. The soft manner of enforcing BC though did not work acceptably when removing the data completely from the training process. Results confirm PINNs being apromising method in computational applications, but require still more refinement and investigation.