On the Interplay between Deep Learning, Partial Differential Equations and Inverse Problems

Abstract

The intersection of deep learning and mathematics has led to profound developments in both fields. On one front, researchers have sought a deeper understanding of deep learning's mathematical underpinnings, aiming to enhance its robustness. Simultaneously, deep learning has been harnessed to address mathematical challenges, paving the way for scientific machine learning. This interdisciplinary synergy has revolutionised scientific computing, particularly in the context of partial differential equations (PDEs). Innovative neural network architectures have emerged, tailor-made for solving specific classes of PDEs, capitalising on inherent PDE properties. These advancements have significantly impacted mathematical modelling, where parametric PDEs are pivotal in representing natural and physical processes in science and engineering.

This thesis looks at these specialised neural network methods and extends them for parametric studies and the solution of related inverse problems. The relevance of these approaches is showcased across various industrial applications; namely, in a continuum mechanics problem encountered in the automotive industry during vehicle development. Building upon this foundation, the research extends to more intricate PDEs encountered in scientific and engineering domains, including the Navier-Stokes equation, Helmholtz equation, advection equation, and Solid mechanics equation. Methodologies are rigorously examined, comparing neural operator-based techniques with classical finite element solvers and Tikhonov functional-based approaches. Extensive numerical experiments are conducted under varying noise levels, providing insights into the trade-offs and applicability of different methods for diverse PDE-based challenges.

Another aspect of this thesis is exploring Electrical Impedance Tomography (EIT), a powerful imaging technique with diverse applications, primarily focusing on solving its challenging (PDE-based) inverse problem. A comprehensive examination of deep learning-based and analytic-based strategies is conducted, emphasising their strengths and limitations. Novel variable conductivity scenarios are introduced to mimic real-world complexities, facilitating a nuanced assessment of the methods' robustness and adaptability.