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Volume 31, Issue 4
Deep Unfitted Nitsche Method for Elliptic Interface Problems

Hailong Guo & Xu Yang

DOI: 10.4208/cicp.OA-2021-0201

Commun. Comput. Phys., 31 (2022), pp. 1162-1179.

Published online: 2022-03

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  • Abstract

This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method.

  • Keywords

Deep learning, unfitted Nitsche method, interface problem, deep neural network.

  • AMS Subject Headings

78M10, 78A48, 47A70, 35P99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hailong.guo@unimelb.edu.au (Hailong Guo)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-31-1162, author = {Guo , Hailong and Yang , Xu}, title = {Deep Unfitted Nitsche Method for Elliptic Interface Problems}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {4}, pages = {1162--1179}, abstract = {

This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0201}, url = {http://global-sci.org/intro/article_detail/cicp/20380.html} }
TY - JOUR T1 - Deep Unfitted Nitsche Method for Elliptic Interface Problems AU - Guo , Hailong AU - Yang , Xu JO - Communications in Computational Physics VL - 4 SP - 1162 EP - 1179 PY - 2022 DA - 2022/03 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0201 UR - https://global-sci.org/intro/article_detail/cicp/20380.html KW - Deep learning, unfitted Nitsche method, interface problem, deep neural network. AB -

This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method.

Guo , Hailong and Yang , Xu. (2022). Deep Unfitted Nitsche Method for Elliptic Interface Problems. Communications in Computational Physics. 31 (4). 1162-1179. doi:10.4208/cicp.OA-2021-0201
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