{"title":"Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations","authors":"Shi Jin,Zheng Ma, Keke Wu","doi":"10.4208/cicp.oa-2023-0211","DOIUrl":null,"url":null,"abstract":"In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation in all\nranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the small\nKnudsen number regime. The first APNN for linear transport equation is based on\neven-odd decomposition, which relaxes the stringent conservation prerequisites while\nconcurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundary\nconditions is crucial for this network. For the Boltzmann-BGK equation, the APNN\nincorporates the conservation of mass, momentum, and total energy into the APNN\nframework as well as exact boundary conditions. A notable finding of this study is that\napproximating the zeroth, first, and second moments—which govern the conservation\nof density, momentum, and energy for the Boltzmann-BGK equation, is simpler than\nthe distribution itself. Another interesting phenomenon observed in the training process is that the convergence of density is swifter than that of momentum and energy.\nFinally, we investigate several benchmark problems to demonstrate the efficacy of our\nproposed APNN methods.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"15 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Computational Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.4208/cicp.oa-2023-0211","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation in all
ranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the small
Knudsen number regime. The first APNN for linear transport equation is based on
even-odd decomposition, which relaxes the stringent conservation prerequisites while
concurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundary
conditions is crucial for this network. For the Boltzmann-BGK equation, the APNN
incorporates the conservation of mass, momentum, and total energy into the APNN
framework as well as exact boundary conditions. A notable finding of this study is that
approximating the zeroth, first, and second moments—which govern the conservation
of density, momentum, and energy for the Boltzmann-BGK equation, is simpler than
the distribution itself. Another interesting phenomenon observed in the training process is that the convergence of density is swifter than that of momentum and energy.
Finally, we investigate several benchmark problems to demonstrate the efficacy of our
proposed APNN methods.
期刊介绍:
Communications in Computational Physics (CiCP) publishes original research and survey papers of high scientific value in computational modeling of physical problems. Results in multi-physics and multi-scale innovative computational methods and modeling in all physical sciences will be featured.