Selected Applications of Least-Squares in FEM and Neural Networks

Abstract

This thesis studies the applications of the least-squares functional in the adaptive finite element method and neural network method. The least-squares functional with a first-order system provides a useful tool for mesh refinement, error control, and energy minimization.

In the first part, the non-intrusive least-squares functional a posteriori error estimator is studied. The a posteriori error estimator can be used for adaptive mesh refinement, and error control within the numerical approximations is obtained from different methods. An alternative view is proposed that the non-intrusive least-squares functional error estimator can be seen as an estimator for the combined solve-recover process. This view can simplify the reliability and efficiency analysis. The plain convergence is proved for the adaptive algorithms of the general second-order elliptic equations driven by the non-intrusive least-squares functional estimator. Furthermore, we extend the idea to a model nonlinear problem.

In the second part, for second-order general elliptic equation in one dimension, we study the neural network with the least-squares functional loss function and ReLU activation function. Unlike optimizing all parameters together, we divide the neural network's parameters into linear parameters and nonlinear parameters in each optimization step. The linear parameters can be solved directly from the view that the shallow ReLU neural network function set is the same as the continuous linear FKS in one dimension. The nonlinear parameters can use the Adam method to update, which means the breakpoints can move to make the least-squares energy smaller.

In the final part, we discuss the possible future works on the least-squares functional.

Date of Award	19 Aug 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Shun ZHANG (Supervisor)