Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions
Abstract In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whol...
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| Published in | IMA journal of numerical analysis Vol. 42; no. 3; pp. 2055 - 2082 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford University Press
22.07.2022
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0272-4979 1464-3642 |
| DOI | 10.1093/imanum/drab031 |
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| Abstract | Abstract
In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method. |
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| AbstractList | Abstract
In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method. In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method. |
| Author | Herrmann, Lukas Grohs, Philipp |
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| Cites_doi | 10.1109/TIT.2021.3062161 10.1088/1361-6420/abaf64 10.1093/imanum/drx042 10.1006/jcom.1999.0499 10.1214/aoms/1177728169 10.1016/j.jcp.2020.109409 10.1007/BF01831723 10.1090/S0002-9947-1961-0137148-5 10.1093/comjnl/1.3.142 10.1007/s40304-018-0127-z 10.1016/j.neucom.2018.06.003 10.1016/j.neunet.2017.07.002 10.1080/03605301003657843 10.1142/S0219530519410136 10.1016/j.jcp.2020.109792 10.1137/19M125649X 10.1016/j.jcp.2016.03.005 10.1007/BF03014033 10.1007/978-3-642-12245-3 10.1016/j.jcp.2020.109339 10.1137/100787842 10.1007/s10915-018-00903-0 10.1214/aop/1176994833 10.1007/s10208-015-9265-9 10.1016/j.jcp.2018.08.029 10.1007/978-1-4684-0302-2 10.1051/m2an:2004005 10.2478/cmam-2011-0020 10.1142/S0219530518500203 10.1137/18M118709X 10.1007/s00365-009-9064-0 |
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| Keywords | neural network approximation Monte Carlo methods high-dimensional approximation |
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In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial... In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial... |
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