Greedy Algorithm for Neural Networks for Indefinite Elliptic Problems

Abstract

The paper presents a priori error analysis of the shallow neural network approximation to the solution to the indefinite elliptic equation and a cutting-edge implementation of the Orthogonal Greedy Algorithm (OGA) tailored to overcome the challenges of indefinite elliptic problems, which is a domain where conventional approaches often struggle due to the lack of coerciveness. A rigorous priori error analysis that shows the neural network’s ability to approximate the solution of indefinite problems is confirmed numerically by OGA. We also present the error analysis of the relevant numerical quadrature. In particular, massive numerical implementations are conducted to justify the theory, some of which showcase the OGA’s superior performance in comparison to the traditional finite element method. This advancement illustrates the potential of neural networks enhanced by OGA to solve intricate computational problems more efficiently, thereby marking a significant leap forward in the application of machine learning techniques to mathematical problem-solving.