Abstract

The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by parameterizing a set of functions. Various approaches, such as the deep Ritz method and physics-informed neural networks, have been developed for numerical solutions. Training algorithms, including gradient descent and greedy algorithms, have been proposed to solve the resulting optimization problems. In this paper, we focus on the variational formulation of the problem and propose a Gauss–Newton method for computing the numerical solution. We provide a comprehensive analysis of the superlinear convergence properties of this method, along with a discussion on semi-regular zeros of the vanishing gradient. Numerical examples are presented to demonstrate the efficiency of the proposed Gauss–Newton method.

Department(s)

Mathematics and Statistics

Comments

Purdue University, Grant 1R35GM146894

Keywords and Phrases

Convergence analysis; Gauss–Newton method; Neural network discretization; Partial differential equations; Variational form

International Standard Serial Number (ISSN)

1573-7691; 0885-7474

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2024 Springer, All rights reserved.

Publication Date

01 Jul 2024

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