Resolución de la ecuación diferencial de una viga mediante la implementación de Physics-Informed Neural Networks

Abstract

This thesis focuses on the implementation and validation of the Physics-Informed Neural Networks (PINNs) method, using a variational approach in a common application of Solid Mechanics, specifically in the structural area that deals with deflection in beams under the Euler-Bernoulli theory. The mathematical formulation used in this method is based on the variational approach of the analyzed beam, which allows obtaining an approximate solution through the implementation of a neural network. Since the nonlinearity of the model may influence the resolution of this method, modifications were made to both the mathematical formulation and the hyperparameters of the neural network implementation to address this complexity. Although traditional methods have the rigor and validity necessary to solve most problems in science and engineering, new methods based on neural networks, such as the one presented in this degree work, are promising due to the constant increase in the available computational capacity. The approximate solution obtained by the studied method shows consistent results in the three beam problems presented, as compared to the exact solution. This research contributes to the growth and development of novel approaches to tackle complex problems in the field of solid mechanics and demonstrates the potential of the integration of neural networks in the solution of differential equations in engineering.