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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.

Climate consists of many components, for example, atmosphere, hydrosphere, cryosphere, and biosphere. All the components act under mechanisms that relate them in a highly non-linear way, making the climate a complex system. This complexity is a challenge to study the climate and its implications at various spatiotemporal scales. However, the dependence of anthropogenic activities on the climate has encouraged its study in order, for example, to anticipate its periodic changes and, as far as possible, extreme events that may have adverse effects. As climate study is an intricate task, several approaches have been used to unravel the underlying processes that dominate its behavior. Those approaches range from linear correlation analysis to complex machine learning-based knowledge discovery analysis. This last approach has become more relevant after the introduction of sophisticated climate simulation models and high-tech equipment (e.g., satellite) that allow a climate record of greater coverage (spatial and temporal) and that, together, have generated ubiquitous large databases. One of the knowledge discovery approaches based on this big data is based on climate networks. Nevertheless, causal reasoning methods have also been used recently to infer and characterize these networks, which are called causal climate networks. Several studies have been carried out with climate networks; however, the recent introduction of causality methods makes the study of climate with causal climate networks an opportunity to explore and exploit them more widely. In addition, the particularities of the climate make it necessary to understand specific operational issues that must be taken into account when applying networks. This thesis aims to propose new methodologies and applications of causal climate networks following as a common thread the characterization of physical phenomena that manifest themselves at different spatial scales. For this, different case studies have been taken. T

Floods represent a severe cause of deaths and economic loss. In order to prevent, mitigate, and reduce flood risks and their consequences, hydraulic models allow analysing and mapping floods. The results of an appropriate model that works under local conditions are a valuable tool for local governments leading to sustainable management of floodplains. Around the world, high-mountain rivers have been poorly modelled; their orography and data scarcity present an extra research difficulty. Considering that all one-dimensional models assume that the river bed slope is small, this study evaluated two widely applied one-dimensional models: Mike11 and HEC-RAS, for modelling a high mountain river. Their best configuration under complex topographical conditions and their potential use was assessed by calibration and validation of the models. We found that the HEC-RAS model was not able to define a stable solution of the hydrodynamic modelling of the river, while Mike11 yielded stable results. Furthermore, the validation of the Mike11 model showed good performance. This study sets a precedent in the 1D modelling of high-mountain rivers with data scarcity., Las inundaciones son una causa severa de muertes y pérdidas económicas. Para prevenir, mitigar y reducir los riesgos por inundaciones y sus consecuencias, los modelos hidráulicos permiten el análisis y mapeo de dichas inundaciones. Los resultados de un modelo apropiado, que trabaje en base a condiciones locales, son herramientas valiosas para los gobiernos locales, conduciendo a un manejo sustentable de las llanuras de inundación. Alrededor del mundo, muy pocos ríos de alta montaña han sido modelados; y, debido a su orografía, la escasez de datos presenta una dificultad adicional en su investigación. Tomando en cuenta que todos los modelos unidimensionales asumen que el fondo del río tiene una pendiente pequeña, este estudio evalúa dos modelos unidimensionales ampliamente usados: Mike11 y HEC-RAS, para modelar un río de alta montaña. La mejor configuración del modelo, bajo condiciones topográficamente complejas, y su potencial uso fueron valorados mediante su calibración y validación. Al contrario de los resultados obtenidos para el modelo Mike11, tanto en calibración como en validación, hemos encontrado que el modelo HEC-RAS no es capaz de encontrar una solución estable durante el modelamiento hidrodinámico del río. Este estudio sienta un precedente en cuanto a modelación unidimensional en ríos de alta montaña con escasez de datos.