Adaptive mesh refinement for finite element methods with machine learning

Abstract

It has been shown that neural networks exist that provide at least as good results for adaptive mesh refinement in finite element methods as current optimal mesh refinement strategies and that they are problem independent. This master thesis is dedicated to the experimental exploration of these theoretical results. Additionally, the work includes a theoretical overview of neural networks, also covering the ability of neural networks to approximate splines and polynomials effectively, as well as an explanation of the finite element method. The primary objective of this work is the design and implementation of a neural network suitable for adaptive mesh refinement. While the outcomes are not entirely satisfactory, they suggest that with additional effort, advancements could be achieved. Another emphasis is placed on evaluating whether a neural network can effectively learn the residual error estimator for the Poisson problem, a goal that has been successfully realized. The thesis also addresses the formulation of an optimization approach for the neural network, tailored to mesh refinement. The outcomes of this optimization endeavor are partially affirmative, indicating that a similar approach might hold the potential to enhance mesh refinement procedures. Overall, this master's thesis contributes to expanding the comprehension of neural network applications in numerical mathematics. The experimental scrutiny of the presented methodologies and the partial successes attained in the neural network's development underscore the promise of further research and optimization within this domain.