De Rosa, Mariapia (2025) Computational learning methodologies for solving PDEs: The Physics-Informed Neural Network framework. [Tesi di dottorato]

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Abstract

This thesis presents a comprehensive study on the application of Physics-Informed Neural Networks (PINNs) for solving Partial Differential Equations (PDEs), integrating the realms of Deep Learning (DL) and computational science. The work begins by establishing a robust theoretical foundation for computational learning, emphasizing the principles of error decomposition, expressivity, and optimization within Neural Network (NN) frameworks. By examining the interplay between DL architectures and mathematical modeling, this research provides insights into enhancing the accuracy and efficiency of solutions to complex PDEs. The core of the thesis is dedicated to advancing PINN methodologies, showcasing innovations such as adaptive loss functions, multi-output architectures, and hyperparameter optimization strategies. These methodologies are rigorously tested through diverse case studies, demonstrating their applicability in both industrial and environmental contexts. Ultimately, this research contributes to the evolution of Scientific Machine Learning (SciML) by offering a powerful alternative to traditional numerical methods, reinforcing the potential of PINNs to transform how one can approach complex scientific problems.

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