A Gentle Introduction to Physics-Informed Neural Networks, with Applications in Static Rod and Beam Problems

Abstract

A modern approach to solving mathematical models involving differential equations, the so-called Physics-Informed Neural Network (PINN), is based on the techniques which include the use of artificial neural networks and the method of fitting the governing differential equations at collocation points. In this paper, training of the PINN with an application of optimization techniques is performed on simple one-dimensional mechanical problems of elasticity, namely rods and beams. Different boundary conditions are considered. Required computer algorithms are implemented using Python programming packages with the intention of creating neural networks. Numerical results are presented, and the efficiency of the proposed technique is investigated through numerical experiments with different numbers of epochs, batches, hidden layers, neurons, and collocation points. The paper provides useful skills for using a PINN for different problems of solid mechanics. The proposed methodology is a continuation of our intention of using PINNs for problems of the theory of elasticity. The objectives are to present simply the main steps of constructing PINN and an implementation of it. A detailed explanation of the Python programming code, based on the scientific software Tensorflow, built in the Keras library and optimizers, may help compose an effective code for complicated models in mechanics. PINNs are proposed in many recent publications to solve complicated direct and inverse problems. It seems to be a promising method that will play a central role in the development of computational mechanics in the near future. Nevertheless, the lack of educational material does not help new users to enter this scientific area. The present contribution describes the method for the solution of elementary rod and beam problems and gives computer codes that may help the reader to understand the method and to apply it to other problems.