full
search.png

Search

close.png

Cancel

arrow
Volume 35, Issue 3
Generalization Error in the Deep Ritz Method with Smooth Activation Functions

Janne Siipola

DOI: 10.4208/cicp.OA-2023-0253

Commun. Comput. Phys., 35 (2024), pp. 761-815.

Published online: 2024-04

[An open-access article; the PDF is free to any online user.]

Preview Full PDF 1488 15056

Cited by

google scholar semantic scholar
Export citation
  • 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 $\frac{C}{\sqrt{n}},$ and we discuss about the constant $C.$ Results are obtained for shallow and residual neural networks with smooth activation functions.

  • Keywords

Deep learning, Deep Ritz method, Poisson’s equation, residual neural networks, shallow neural networks, generalization.

  • AMS Subject Headings

35A35, 65C05, 92B20, 68T10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-35-761, author = {Siipola , Janne}, title = {Generalization Error in the Deep Ritz Method with Smooth Activation Functions}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {3}, pages = {761--815}, 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 $\frac{C}{\sqrt{n}},$ and we discuss about the constant $C.$ Results are obtained for shallow and residual neural networks with smooth activation functions.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0253}, url = {http://global-sci.org/intro/article_detail/cicp/23059.html} }
TY - JOUR T1 - Generalization Error in the Deep Ritz Method with Smooth Activation Functions AU - Siipola , Janne JO - Communications in Computational Physics VL - 3 SP - 761 EP - 815 PY - 2024 DA - 2024/04 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0253 UR - https://global-sci.org/intro/article_detail/cicp/23059.html KW - Deep learning, Deep Ritz method, Poisson’s equation, residual neural networks, shallow neural networks, generalization. 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 $\frac{C}{\sqrt{n}},$ and we discuss about the constant $C.$ Results are obtained for shallow and residual neural networks with smooth activation functions.

Siipola , Janne. (2024). Generalization Error in the Deep Ritz Method with Smooth Activation Functions. Communications in Computational Physics. 35 (3). 761-815. doi:10.4208/cicp.OA-2023-0253
Copy to clipboard
BibteX RIS TXT
The citation has been copied to your clipboard