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Solving parametric PDE problems with artificial neural networks

Published online by Cambridge University Press:  01 July 2020

YUEHAW KHOO Open the ORCID record for YUEHAW KHOO [Opens in a new window] ,
JIANFENG LU  and
LEXING YING
YUEHAW KHOO
Affiliation:
Department of Statistics, University of Chicago, IL60615, USA, email: ykhoo@uchicago.edu
JIANFENG LU
Affiliation:
Department of Mathematics, Department of Chemistry and Department of Physics, Duke University, Durham, NC27708, USA, email: jianfeng@math.duke.edu
LEXING YING
Affiliation:
Department of Mathematics and ICME, Stanford University, Stanford, CA94305, USA, email: lexing@stanford.edu
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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.

Keywords

Neural-networkparametric PDEuncertainty quantification
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 421 - 435
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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