Cargando…
Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem
Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the...
Autores principales: | , , , , , , |
---|---|
Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Nature Publishing Group UK
2022
|
Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9085858/ https://www.ncbi.nlm.nih.gov/pubmed/35534639 http://dx.doi.org/10.1038/s41598-022-11058-2 |
_version_ | 1784703910321061888 |
---|---|
author | Rodriguez-Torrado, Ruben Ruiz, Pablo Cueto-Felgueroso, Luis Green, Michael Cerny Friesen, Tyler Matringe, Sebastien Togelius, Julian |
author_facet | Rodriguez-Torrado, Ruben Ruiz, Pablo Cueto-Felgueroso, Luis Green, Michael Cerny Friesen, Tyler Matringe, Sebastien Togelius, Julian |
author_sort | Rodriguez-Torrado, Ruben |
collection | PubMed |
description | Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set. |
format | Online Article Text |
id | pubmed-9085858 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2022 |
publisher | Nature Publishing Group UK |
record_format | MEDLINE/PubMed |
spelling | pubmed-90858582022-05-11 Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem Rodriguez-Torrado, Ruben Ruiz, Pablo Cueto-Felgueroso, Luis Green, Michael Cerny Friesen, Tyler Matringe, Sebastien Togelius, Julian Sci Rep Article Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set. Nature Publishing Group UK 2022-05-09 /pmc/articles/PMC9085858/ /pubmed/35534639 http://dx.doi.org/10.1038/s41598-022-11058-2 Text en © The Author(s) 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Rodriguez-Torrado, Ruben Ruiz, Pablo Cueto-Felgueroso, Luis Green, Michael Cerny Friesen, Tyler Matringe, Sebastien Togelius, Julian Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem |
title | Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem |
title_full | Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem |
title_fullStr | Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem |
title_full_unstemmed | Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem |
title_short | Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem |
title_sort | physics-informed attention-based neural network for hyperbolic partial differential equations: application to the buckley–leverett problem |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9085858/ https://www.ncbi.nlm.nih.gov/pubmed/35534639 http://dx.doi.org/10.1038/s41598-022-11058-2 |
work_keys_str_mv | AT rodrigueztorradoruben physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem AT ruizpablo physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem AT cuetofelguerosoluis physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem AT greenmichaelcerny physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem AT friesentyler physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem AT matringesebastien physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem AT togeliusjulian physicsinformedattentionbasedneuralnetworkforhyperbolicpartialdifferentialequationsapplicationtothebuckleyleverettproblem |