Deep Domain Decomposition Method: Elliptic Problems

Wuyang Li, Xueshuang Xiang, Yingxiang Xu
Proceedings of The First Mathematical and Scientific Machine Learning Conference, PMLR 107:269-286, 2020.

Abstract

This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential equations (PDE). Using DNN to solve PDE is a physics-informed learning problem with the objective involving two terms, domain term and boundary term, which respectively make the desired solution satisfy the PDE and corresponding boundary conditions. DeepDDM will exchange the subproblem information across the interface in DDM by adjusting the boundary term for solving each subproblem by DNN. Benefiting from the simple implementation and mesh-free strategy of using DNN for PDE, DeepDDM will simplify the implementation of DDM and make DDM more flexible for complex PDE, e.g., those with complex interfaces in the computational domain. This paper will firstly investigate the performance of using DeepDDM for elliptic problems, including a model problem and an interface problem. The numerical examples demonstrate that DeepDDM exhibits behaviors consistent with conventional DDM: the number of iterations by DeepDDM is independent of network architecture and decreases with increasing overlapping size. The performance of DeepDDM on elliptic problems will encourage us to further investigate its performance for other kinds of PDE and may provide new insights for improving the PDE solver by deep learning.

Cite this Paper

BibTeX
@InProceedings{pmlr-v107-li20a, title = {Deep Domain Decomposition Method: {E}lliptic Problems}, author = {Li, Wuyang and Xiang, Xueshuang and Xu, Yingxiang}, booktitle = {Proceedings of The First Mathematical and Scientific Machine Learning Conference}, pages = {269--286}, year = {2020}, editor = {Lu, Jianfeng and Ward, Rachel}, volume = {107}, series = {Proceedings of Machine Learning Research}, month = {20--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v107/li20a/li20a.pdf}, url = {https://proceedings.mlr.press/v107/li20a.html}, abstract = {This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential equations (PDE). Using DNN to solve PDE is a physics-informed learning problem with the objective involving two terms, domain term and boundary term, which respectively make the desired solution satisfy the PDE and corresponding boundary conditions. DeepDDM will exchange the subproblem information across the interface in DDM by adjusting the boundary term for solving each subproblem by DNN. Benefiting from the simple implementation and mesh-free strategy of using DNN for PDE, DeepDDM will simplify the implementation of DDM and make DDM more flexible for complex PDE, e.g., those with complex interfaces in the computational domain. This paper will firstly investigate the performance of using DeepDDM for elliptic problems, including a model problem and an interface problem. The numerical examples demonstrate that DeepDDM exhibits behaviors consistent with conventional DDM: the number of iterations by DeepDDM is independent of network architecture and decreases with increasing overlapping size. The performance of DeepDDM on elliptic problems will encourage us to further investigate its performance for other kinds of PDE and may provide new insights for improving the PDE solver by deep learning. } }
Endnote
%0 Conference Paper %T Deep Domain Decomposition Method: Elliptic Problems %A Wuyang Li %A Xueshuang Xiang %A Yingxiang Xu %B Proceedings of The First Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2020 %E Jianfeng Lu %E Rachel Ward %F pmlr-v107-li20a %I PMLR %P 269--286 %U https://proceedings.mlr.press/v107/li20a.html %V 107 %X This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential equations (PDE). Using DNN to solve PDE is a physics-informed learning problem with the objective involving two terms, domain term and boundary term, which respectively make the desired solution satisfy the PDE and corresponding boundary conditions. DeepDDM will exchange the subproblem information across the interface in DDM by adjusting the boundary term for solving each subproblem by DNN. Benefiting from the simple implementation and mesh-free strategy of using DNN for PDE, DeepDDM will simplify the implementation of DDM and make DDM more flexible for complex PDE, e.g., those with complex interfaces in the computational domain. This paper will firstly investigate the performance of using DeepDDM for elliptic problems, including a model problem and an interface problem. The numerical examples demonstrate that DeepDDM exhibits behaviors consistent with conventional DDM: the number of iterations by DeepDDM is independent of network architecture and decreases with increasing overlapping size. The performance of DeepDDM on elliptic problems will encourage us to further investigate its performance for other kinds of PDE and may provide new insights for improving the PDE solver by deep learning.
APA
Li, W., Xiang, X. & Xu, Y.. (2020). Deep Domain Decomposition Method: Elliptic Problems. Proceedings of The First Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 107:269-286 Available from https://proceedings.mlr.press/v107/li20a.html.

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