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Volume 36, Issue 3
A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions

Jingrun Chen, Rui Du & Keke Wu

Commun. Math. Res., 36 (2020), pp. 354-376.

Published online: 2020-07

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  • Abstract

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions. Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.

  • Keywords

Partial differential equations, boundary conditions, deep Galerkin method, deep Ritz method, penalty method.

  • AMS Subject Headings

65K10, 65N06, 65N22, 65N99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-36-354, author = {Jingrun and Chen and and 8405 and and Jingrun Chen and Rui and Du and and 8406 and and Rui Du and Keke and Wu and and 8407 and and Keke Wu}, title = {A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {3}, pages = {354--376}, abstract = {

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions. Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0051}, url = {http://global-sci.org/intro/article_detail/cmr/17853.html} }
TY - JOUR T1 - A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions AU - Chen , Jingrun AU - Du , Rui AU - Wu , Keke JO - Communications in Mathematical Research VL - 3 SP - 354 EP - 376 PY - 2020 DA - 2020/07 SN - 36 DO - http://doi.org/10.4208/cmr.2020-0051 UR - https://global-sci.org/intro/article_detail/cmr/17853.html KW - Partial differential equations, boundary conditions, deep Galerkin method, deep Ritz method, penalty method. AB -

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions. Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.

Jingrun Chen, Rui Du & Keke Wu. (2020). A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions. Communications in Mathematical Research . 36 (3). 354-376. doi:10.4208/cmr.2020-0051
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