Generalization Error in the Deep Ritz Method with Smooth Activation Functions

Janne Siipola

doi:10.4208/cicp.OA-2023-0253

Generalization Error in the Deep Ritz Method with Smooth Activation Functions

Janne Siipola

Department of Mathematics and Statistics

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

Deep Ritz method is a deep learning paradigm to solve partial differential equations. In this article we study the generalization error of the Deep Ritz method. We focus on the quintessential problem which is the Poisson’s equation. We show that generalization error of the Deep Ritz method converges to zero with rate C√n , and we discuss about the constant C. Results are obtained for shallow and residual neural networks with smooth activation functions.

Original language	English
Journal	Communications in Computational Physics
Volume	35
Issue number	3
Pages (from-to)	761-815
Number of pages	55
ISSN	1815-2406
DOIs	https://doi.org/10.4208/cicp.OA-2023-0253
Publication status	Published - 12 Mar 2024
MoE publication type	A1 Journal article-refereed

Fields of Science

111 Mathematics

Access to Document

10.4208/cicp.OA-2023-0253Licence: CC BY

353_761Final published version, 336 KBLicence: CC BY

Cite this

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title = "Generalization Error in the Deep Ritz Method with Smooth Activation Functions",

abstract = "Deep Ritz method is a deep learning paradigm to solve partial differential equations. In this article we study the generalization error of the Deep Ritz method. We focus on the quintessential problem which is the Poisson{\textquoteright}s equation. We show that generalization error of the Deep Ritz method converges to zero with rate C√n , and we discuss about the constant C. Results are obtained for shallow and residual neural networks with smooth activation functions.",

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AB - Deep Ritz method is a deep learning paradigm to solve partial differential equations. In this article we study the generalization error of the Deep Ritz method. We focus on the quintessential problem which is the Poisson’s equation. We show that generalization error of the Deep Ritz method converges to zero with rate C√n , and we discuss about the constant C. Results are obtained for shallow and residual neural networks with smooth activation functions.

KW - 111 Mathematics

KW - deep learning

KW - partial differential equations

KW - error analysis

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