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...

Full description

Saved in:
Bibliographic Details
Published inEuropean journal of applied mathematics Vol. 32; no. 3; pp. 421 - 435
Main Authors KHOO, YUEHAW, LU, JIANFENG, YING, LEXING
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.06.2021
Subjects
Applied mathematics
Artificial neural networks
Coefficients
Neural networks
Partial differential equations
uncertainty quantification
parametric PDE
65Nxx
Neural-network
Online AccessGet full text

Cover

More Information
Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792520000182