文档名：基于元学习的气体放电等离子体电子Boltzmann方程数值求解
摘要：在气体放电等离子体中,电子的输运行为可由Boltzmann方程精确描述,该方程的解是许多等离子体仿真模型的基础.物理信息神经网络作为一种求解Boltzmann方程的新型方法,虽克服了传统数值方法网格剖分和方程离散的缺陷,但其参数空间规模大,在求解多任务时训练效率较低.为此,该文构建了一种基于元学习的双循环物理信息神经网络,在内循环中对多个Boltzmann方程求解任务进行优化训练,得到各任务优化后的元损失函数,用于在外循环中进行网络参数更新,从而提高网络在求解新任务时的计算效率.计算结果表明,基于元学习的双循环物理信息神经网络在求解新的Boltzmann方程时,网络损失函数值和L2误差值的下降速度均显著快于普通的物理信息神经网络.此外,该文还研究了网络容量和内循环迭代次数对Boltzmann方程多任务求解效率的影响,结果显示计算效率并不随网络容量的增大而提高,且受内循环迭代次数影响较小.
Abstract：TheBoltzmannequationisapartialdifferentialequationthatdescribesthevariationofparticlesinanon-equilibriumthermodynamicsystem.Inthefieldofgasinsulationandplasmadischarge,theBoltzmannequationcanbeusedtoaccuratelydescribetheelectrontransportingasdischargeplasmas,anditssolutionisthebasisofmanyplasmasimulationmodels.However,thetraditionalnumericalmethodsforsolvingtheBoltzmannequationallrequiremeshingonthecomputationaldomain,andthesolutionaccuracysignificantlydependsonthequalityofthemeshing.Thephysics-informedneuralnetworks(PINNs),asanewmethodforsolvingBoltzmannequation,overcomestheshortcomingsoftraditionalnumericalmethodsinmeshgenerationandequationdiscretization,butitstrainingisinefficientwhendealingwithmulti-tasksbecauseofhugeparameterspaceofPINNs.Toaddressthisissue,thispaperproposesaMeta-PINNnetworkwithtwoloopsofPINNsbasedonmetalearning.Throughthetrainingininnerandouterloops,Meta-PINNsolvetheBoltzmannequationinmulti-tasksaccuratelyandefficiently.IntheMeta-PINN,therearetwotypesofnetworks,whicharethePINNnetworkandthemetanetwork.Intheinnerloop,thePINNnetworksolvetheBoltzmannequationinmulti-tasksbyminimizingthelossfunctionofPINN.Afterallmulti-tasksareoptimized,thesumofthePINNlossfunction,namelythemetalossfunction,isobtained.Then,themetanetworkupdatestheweightsbyminimizingthemetalossfunctionintheouterloop.Finally,theupdatedweightsareusedtoimprovethetrainingefficiencywhendealingwithnewtasksofsolvingBoltzmannequations.TovalidatetheperformanceofMeta-PINN,theBoltzmannequationsunderdifferentreducedelectricfieldsandgasmixingratiosaresolvedundertheframeworkofMeta-PINN.Theresultsshowthat,whendealingwithnewtasks,thelossfunctionvaluesandL2errorsofMeta-PINNarereducedfasterthanthatofPINN.Specifically,theminimumandmaximumaccelerationspeedsincreaseby75%and22timesrespectively,whichindicatesthattheMeta-PINNoutperformsthePINNinsolutionefficiency.Additionally,theeffectsofnetworkcapacitiesandinnerstepsonthecomputationalefficiencyareinvestigatedinmulti-tasksofsolvingBoltzmannequation.Theresultrevealsthatthecomputationalefficiencydoesnotimprovewiththeincreaseofnetworkcapacitiesandislessaffectedbytheinnersteps.Thefollowingconclusionscouldbedrawnfromthenumericalexperiments:(1)Meta-PINNcanimprovetheefficiencyofsolvingBoltzmannequationdescribingelectrontransportingasdischargeplasmas.Moreover,insomecases,thelossfunctionvalueandL2errorofMeta-PINNcanbereducedtoalowerorderofmagnitudethanPINN,indicatingthatsolutionaccuracyofMeta-PINNisalsobetterthanthatofPINN.(2)InsolvingtheBoltzmannequation,thecomputationefficiencyisnotimprovedwiththeincreaseofnetworkcapacity.ThemostsuitablenetworkforsolvingtheBoltzmannequationunderdifferentreducedelectricfieldsinargonplasmaistheneuralnetworkwith3hiddenlayersand300neuronsperlayer.(3)Thecomputationefficiencyiseithernotimprovedwiththeincreaseofinnersteps.ThemostsuitableinnerstepsforsolvingtheBoltzmannequationunderdifferentreducedelectricfieldsinargonplasmais5steps.Generally,theMeta-PINNcanbeextendedeasilytoothernumericalsolutionofplasmagoverningequations.
作者：仲林林 吴冰钰 吴奇Author：ZhongLinlin WuBingyu WuQi
作者单位：东南大学电气工程学院南京210096
刊名：电工技术学报 ISTICEIPKU
Journal：TransactionsofChinaElectrotechnicalSociety
年，卷(期)：2024, 39(11)
分类号：TM11
关键词：气体放电等离子体 Boltzmann方程 元学习 物理信息神经网络
Keywords：Gasdischargeplasma Boltzmannequation metalearning physics-informedneuralnetwork
机标分类号：TP3O241.8TN925.93
在线出版日期：2024年6月18日
基金项目：国家自然科学基金，江苏省科协青年科技人才托举工程，东南大学至善青年学者支持计划(中央高校基本科研业务费资助项目基于元学习的气体放电等离子体电子Boltzmann方程数值求解[
期刊论文] 电工技术学报--2024, 39(11)仲林林 吴冰钰 吴奇在气体放电等离子体中,电子的输运行为可由Boltzmann方程精确描述,该方程的解是许多等离子体仿真模型的基础.物理信息神经网络作为一种求解Boltzmann方程的新型方法,虽克服了传统数值方法网格剖分和方程离散的缺陷,但...参考文献和引证文献
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