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.

Department(s)

Mathematics and Statistics

Comments

Alexander von Humboldt-Stiftung, Grant 22341302

Keywords and Phrases

Deep learning; Greedy algorithm; Indefinite elliptic problems

International Standard Serial Number (ISSN)

1573-7691; 0885-7474

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2025 Springer, All rights reserved.

Publication Date

01 Sep 2025

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