[25]等人基于PINNs思想同時集合數(shù)據(jù)驅(qū)動模型和物理模型的優(yōu)勢，提出了AutoODE框架，通過機器學習實現(xiàn)方程參數(shù)的近似，通過自動微分技術(shù)進行結(jié)果的預測，該物理信息神經(jīng)網(wǎng)絡框架對于新冠肺炎疫情的預測的誤差相較于傳統(tǒng)純數(shù)據(jù)驅(qū)動神經(jīng)網(wǎng)絡的誤差降低約57.4%。綜上所述，物理信息神經(jīng)網(wǎng)絡在其被提出后的短短幾年時間已經(jīng)在許多領(lǐng)域內(nèi)得到了成功應用，其自身也在廣泛的應用之中不斷發(fā)展深入。PINNs基于傳統(tǒng)深度學習神經(jīng)網(wǎng)絡的逼近理論及自動微分技術(shù)，同時將物理模型的先驗條件、物理信息規(guī)律等有效嵌入網(wǎng)絡之中，有效降低了網(wǎng)絡的復雜度以及對于訓練數(shù)據(jù)量的需要，同時也保證了優(yōu)異的效率和良好的精度。參考文獻RumelhartDE,HintonGE,WilliamsRJ.Learningrepresentationsbyback-propagatingerrors[J].nature,1986,323(6088):533-536.HintonGE,SalakhutdinovRR.Reducingthedimensionalityofdatawithneuralnetworks[J].science,2006,313(5786):504-507.HintonGE,OsinderoS,TehYW.Afastlearningalgorithmfordeepbeliefnets[J].Neuralcomputation,2006,18(7):1527-1554.HintonGE.Learningmultiplelayersofrepresentation[J].Trendsincognitivesciences,2007,11(10):428-434.HintonGE.ApracticalguidetotrainingrestrictedBoltzmannmachines[M]//Neuralnetworks:Tricksofthetrade.Springer,Berlin,Heidelberg,2012:599-619.RifaiS,VincentP,MullerX,etal.Contractiveauto-encoders:Explicitinvarianceduringfeatureextraction[C]//Icml.2011.LeeH,GrosseR,RanganathR,etal.Convolutionaldeepbeliefnetworksforscalableunsupervisedlearningofhierarchicalrepresentations[C]//Proceedingsofthe26thannualinternationalconferenceonmachinelearning.2009:609-616.FerreiraA,GiraldiG.ConvolutionalNeuralNetworkapproachestogranitetilesclassification[J].ExpertSystemswithApplications,2017,84:1-11.YangJ,YuK,GongY,etal.Linearspatialpyramidmatchingusingsparsecodingforimageclassification[C]//2009IEEEConferenceoncomputervisionandpatternrecognition.IEEE,2009:1794-1801.張軍陽,王慧麗,郭陽,扈嘯.深度學習相關(guān)研究綜述[J].計算機應用研究,2018,35(07):1921-1928+1936.CollobertR,WestonJ,BottouL,etal.Naturallanguageprocessing(almost)fromscratch[J].Journalofmachinelearningresearch,2011,12(ARTICLE):2493?2537.SocherR,HuvalB,ManningCD,etal.Semanticcompositionalitythroughrecursivematrix-vectorspaces[C]//Proceedingsofthe2012jointconferenceonempiricalmethodsinnaturallanguageprocessingandcomputationalnaturallanguagelearning.2012:1201-1211.LeQ,MikolovT.Distributedrepresentationsofsentencesanddocuments[C]//Internationalconferenceonmachinelearning.PMLR,2014:1188-1196.RaissiM,PerdikarisP,KarniadakisGE.Physicsinformeddeeplearning(parti):Data-drivensolutionsofnonlinearpartialdifferentialequations[J].arXivpreprintarXiv:1711.10561,2017.BakerN,AlexanderF,BremerT,etal.Workshopreportonbasicresearchneedsforscientificmachinelearning:Coretechnologiesforartificialintelligence[R].USDOEOfficeofScience(SC),Washington,DC(UnitedStates),2019.RaissiM,YazdaniA,KarniadakisGE.Hiddenfluidmechanics:Learningvelocityandpressurefieldsfromflowvisualizations[J].Science,2020,367(6481):1026-1030.MaoZ,JagtapAD,KarniadakisGE.Physics-informedneuralnetworksforhigh-speedflows[J].ComputerMethodsinAppliedMechanicsandEngineering,2020,360:112789.FangZ,ZhanJ.Deepphysicalinformedneuralnetworksformetamaterialdesign[J].IEEEAccess,2019,8:24506-24513.ZhangE