Volume 2, Issue 4
Enforcing Exact Boundary and Initial Conditions in the Deep Mixed Residual Method

Liyao Lyu, Keke Wu, Rui Du & Jingrun Chen

CSIAM Trans. Appl. Math., 2 (2021), pp. 748-775.

Published online: 2021-11

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  • Abstract

Boundary and initial conditions are important for the well-posedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years, we have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically by using distance functions and explicit constructions. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.

  • Keywords

Machine learning, deep neural networks, enforcement of boundary/initial conditions, penalty.

  • AMS Subject Headings

65M99, 65N99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-748, author = { and Liyao Lyu and and 21506 and and Liyao Lyu and and Keke Wu and and 21507 and and Keke Wu and and Rui Du and and 21509 and and Rui Du and Jingrun and Chen and and 21510 and and Jingrun Chen}, title = {Enforcing Exact Boundary and Initial Conditions in the Deep Mixed Residual Method}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {4}, pages = {748--775}, abstract = {

Boundary and initial conditions are important for the well-posedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years, we have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically by using distance functions and explicit constructions. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0011}, url = {http://global-sci.org/intro/article_detail/csiam-am/19991.html} }
TY - JOUR T1 - Enforcing Exact Boundary and Initial Conditions in the Deep Mixed Residual Method AU - Liyao Lyu , AU - Keke Wu , AU - Rui Du , AU - Chen , Jingrun JO - CSIAM Transactions on Applied Mathematics VL - 4 SP - 748 EP - 775 PY - 2021 DA - 2021/11 SN - 2 DO - http://doi.org/10.4208/csiam-am.SO-2021-0011 UR - https://global-sci.org/intro/article_detail/csiam-am/19991.html KW - Machine learning, deep neural networks, enforcement of boundary/initial conditions, penalty. AB -

Boundary and initial conditions are important for the well-posedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years, we have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically by using distance functions and explicit constructions. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.

Liyao Lyu, Keke Wu, Rui Du & Jingrun Chen. (2021). Enforcing Exact Boundary and Initial Conditions in the Deep Mixed Residual Method. CSIAM Transactions on Applied Mathematics. 2 (4). 748-775. doi:10.4208/csiam-am.SO-2021-0011
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