Adaptive methods for the neural network approximation of PDEs: Integration and Dual norm computations

This thesis aims to utilize neural networks for solving partial differential equations. To this end, it proposes novel approaches that are efficient and enhance reliability and robustness across a wide range of problems in engineering and the sciences. The central theme of this work is the concept of adaptivity, which yields superior accuracy in approximating target solutions while ensuring computational efficiency. For practical usability, a novel, high-performance, and unified implementation of existing approaches in the literature is proposed, capable of utilizing modern high-performance computing hardware such as GPUs.