Greedy Algorithm for Neural Networks for Indefinite Elliptic Problems

TL;DR

This paper develops a neural network approach with a specialized greedy algorithm to effectively solve indefinite elliptic problems, demonstrating superior performance over traditional methods through rigorous analysis and extensive numerical experiments.

Contribution

It introduces a novel application of the Orthogonal Greedy Algorithm to neural networks for indefinite elliptic equations, with comprehensive error analysis and practical implementation.

Findings

Neural networks can approximate indefinite elliptic problems effectively.

OGA-based methods outperform traditional finite element methods in experiments.

Theoretical error bounds are validated by numerical results.

Abstract

The paper presents a priori error analysis of the shallow neural network approximation to the solution to the indefinite elliptic equation and and 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 nontraditional difficulties due to the lack of coerciveness. A rigorous a priori error analysis that shows the neural networks ability to approximate indefinite problems is confirmed numerically by OGA methods. We also present a discretization error analysis of the relevant numerical quadrature. In particular, massive numerical implementations are conducted to justify the theory, some of which showcase the OGAs 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.