Solving parametric PDE problems with artificial neural networks
The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few fea...
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| Published in | European journal of applied mathematics Vol. 32; no. 3; pp. 421 - 435 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.06.2021
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| Subjects | |
| Online Access | Get full text |
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| Abstract | The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of the coefficient fields. Based on such observation, we propose using neural network to parameterise the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural network can be justified by viewing the neural network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics. |
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| AbstractList | The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of the coefficient fields. Based on such observation, we propose using neural network to parameterise the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural network can be justified by viewing the neural network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics. |
| Author | KHOO, YUEHAW YING, LEXING LU, JIANFENG |
| Author_xml | – sequence: 1 givenname: YUEHAW orcidid: 0000-0002-8472-8984 surname: KHOO fullname: KHOO, YUEHAW email: ykhoo@uchicago.edu organization: 1Department of Statistics, University of Chicago, IL 60615, USA, email: ykhoo@uchicago.edu – sequence: 2 givenname: JIANFENG surname: LU fullname: LU, JIANFENG email: jianfeng@math.duke.edu organization: 2Department of Mathematics, Department of Chemistry and Department of Physics, Duke University, Durham, NC 27708, USA, email: jianfeng@math.duke.edu – sequence: 3 givenname: LEXING surname: YING fullname: YING, LEXING email: lexing@stanford.edu organization: 3Department of Mathematics and ICME, Stanford University, Stanford, CA 94305, USA, email: lexing@stanford.edu |
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| Cites_doi | 10.1016/j.cma.2004.05.027 10.1017/S0962492917000083 10.1137/S1064827501387826 10.1109/72.712178 10.1137/130913249 10.1109/CVPR.2016.90 10.2307/2371268 10.1126/science.aag2302 10.1007/s40687-018-0160-2 10.1016/j.neucom.2014.11.058 10.1016/j.cma.2008.11.007 10.1038/nature14539 10.1016/j.neunet.2014.09.003 10.1126/science.1127647 10.1103/PhysRevB.94.165134 |
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| SubjectTerms | Applied mathematics Artificial neural networks Coefficients Neural networks Partial differential equations |
| Title | Solving parametric PDE problems with artificial neural networks |
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