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Neuralnetworkpotentials

JinzheZenga,b,LiqunCaoa,b,andTongZhua,b,

aShanghaiEngineeringResearchCenterofMolecularTherapeutics&NewDrugDevelopment, SchoolofChemistryandMolecularEngineering,EastChinaNormalUniversity,Shanghai,China bNYU-ECNUCenterforComputationalChemistryatNYUShanghai,Shanghai,China

∗Correspondingauthor:E-mailaddress:tzhu@lps.ecnu.edu.cn

Abstract

Recently,artificialneuralnetwork-basedmethodsfortheconstructionofpotentialenergysurfacesandmoleculardynamics(MD)simulationsbasedonthemhavebeenincreasinglyusedinthefieldoftheoreticalchemistry.Theneuralnetworkpotentials(NNP)strikeagoodbalancebetweenaccuracyandcomputational efficiencyrelativetoquantumchemicalcalculationsandMDsimulationsbasedonclassicalforcefields.Thus, NNPisbecomingapowerfultoolforstudyingthestructureandfunctionofmolecules.Inthischapter,we introducethebasictheoryofNNP.TheconstructionstepsandtheusageofNNParealsointroducedindetail withtheMDsimulationofmethanecombustionasanexample.Wehopethatthischaptercanhelpthose readerswhoarenewbutinterestedinenteringthisfield.

Keywords: Neuralnetwork,Potentialenergysurface,Moleculardynamicsimulation,Chemicalreaction

Introduction

Moleculardynamics(MD)simulationhasbeenakeytheoreticaltoolforstudyingthestructuralanddynamicalpropertiesofawiderangeofchemicalsystems.OneoftheessentialelementsdeterminingthereliabilityofMDsimulationsistheunderlyingpotentialenergy surface(PES),whichdescribestheintra-andintermolecularinteractions.Therearetwomain waystoconstructthePESforMD.Thefirstoneistoperformquantummechanical(QM)calculationon-the-fly,whichisalsoknownasabinitioMDsimulation(AIMD) [1].Theotherone istheempiricalinteratomicpotentials(forcefields)whicharebasedonclassicalmolecular mechanics(MM) [2,3].Theforcefieldsundoubtedlyhavetheadvantageintermsofcomputationalefficiency.However,thesefunctionalformsofforcefieldscanonlyaccuratelydescribeclassicalphysicalinteractionsandusuallydonothaveagoodaccuracywhentaking

intoaccountthemotionofelectrons.Thus,theiraccuracyisoftenquestionedastheylackimportantquantumeffectssuchaspolarizationandchargetransfer.Thisisalsothereasonthat onlyafewforcefieldscandescribechemicalreactions.TheAIMDismoreaccurateinprinciple.However,itsapplicationsaresignificantlylimitedbyitscomputationalcost.Itseemsto bedifficulttoensurebothefficiencyandaccuracyofPESsatthesametime.

Inthepasttwodecades,thedevelopmentofmachinelearningpotentials(MLP)provides thepossibilitytosolvethiscontradiction.MLPemploysmachinelearningmodelsratherthan empiricalformulasorSchr € odingerequationtoestablishtherelationshipbetweenthepotentialenergyofthesystemandthestructureaswellasthechemicalinformation.Compared withempiricalformulas,MLmodelshaveabetterfitpowerandarethereforemoreaccurate. Theyhavebeenusedsincethe1990s [4–9].FollowingtheworkofBehlerandParrinello [10],a seriesofMLPconstructionapproacheshavebeenproposedforawidevarietyofchemical systems.Forexample,Csa ´ nyiandco-workers [11] proposedtheGaussianApproximationPotentials,Mulleretal.proposedtheGDML,DTNNandtheSchNetmethods [12–14],Hammer etal.proposedthekCONmodel [15],Eandco-workersdevelopedtheDeepPotential (DeePMD)model [16,17],Jiangetal.developedtheEANNmodel [18],andDralandcoworkersproposedtheKREGmethod [19,20].Thereareseveralmachinelearningmethods thatcanbeusedtobuildthePES.Amongthem,theartificialNNiswidelyusedduetoits bettercomputationalscalingforperformanceandmemoryrequirements [21,22].Recently, universalNNPshavebeenproposedformanyelements [23,24]

ComparedwithAIMD,theefficiencyoftheM LPissubstantiallyimproved(typicallyby morethan3ordersofmagnitude).Withcodeoptimizationorondedicatedhardware,MD simulationwithMLPhasbeenabletosimulatesystemscontaining100millionatoms [17]. However,thereisstillanorderofmagnitude differenceintheefficiencyofMLPcompared toclassicalforcefields.Therefore,itwasma inlyusedtostudyscientificproblemsthat traditionalforcefieldsarenotcapableoforrequirehighaccuracy.Forexample,thedesign/discoveryofnewmaterials,homogeneouscatalysis,etc [25– 39].Thelatestdevelopmentsandapplicationsinthisfieldcanbefoundinseveralrecentcomprehensivereviews [40– 51].

Recently,wehaveusedtheneuralnetwork-basedpotentials(NNP)tosimulateanumber oftypicalchemicalsystems,suchasthehydrationofzincions [52],metalloproteins [53],and combustionreactions [54,55] (Fig.1).Metals,especiallysometransitionmetals,areimportant cofactorsintheregulationofproteinstructureandfunction.However,importantquantum effectsassociatedwithmetalcoordinationareoftennotaccuratelydescribedbyclassicalforce fields.Wefoundthatbenefittingfromitspowerfulfittingability,theNNPcaneasilyachieve higheraccuracythanclassicalforcefieldsinthesesystems.Combustionisanevenmorecomplexchemicalsystemthatinvolvesthousandsofchemicalreactionsandgenerateshundreds ofmolecularspeciesduringtheprocess.Traditionally,MDsimulationsbasedonreactive forcefieldssuchasReaxFF [56] wereusedtoinvestigatethecombustionreactionprocesses. However,theaccuracyoftheReaxFFhasoftenbeencriticized [57,58].Wehavedeveloped NNPsasaccurateasDFTmethodtoachieveefficientsimulationsofmethanecombustion andlong-chainalkanepyrolysis.FromtheMDtrajectories,onecannotonlyobtaindetailed reactionnetworksbutalsodiscovernewreactions.Withthischapter,wehopetopassonour experiencetohelpmorebeginnersbetterunderstandanduseNNPandtheMDsimulations basedonit.

FIG.1 (A)RadialdistributionfunctionofZn Odistance(solidlines)andcorrespondingintegrationofradialdistributionfunction(RDF, dashedlines)calculatedfromthetrajectorysimulatedbyNNP.The gray areaistheexperimentalmeasurementofRDF [52].TheP1 andP2 denote,respectively,thepeakexperimentalradialdistributionof thefirstandsecondsolvationshells,whileN1 andN2 denote,respectively,theexperimentalcoordinationnumber ofthefirstandsecondsolvationshells.(B)TheworkflowtoconstructNNPESforzincproteins [53]. (C)SnapshotsofthepartialcombustionsystemextractedfromthereactiveMDsimulationofmethanecombustion [54] Panel(A)ThisfigureisreprintedwithpermissionfromM.Xu,T.Zhu,J.Z.H.Zhang,Moleculardynamicssimulationof zincioninwaterwithanabinitiobasedneuralnetworkpotential,J.Phys.Chem.A123(2019)6587–6595.Copyright2019 AmericanChemicalSociety.Panel(B)ThisfigureistakenfromM.Xu,T.Zhu,J.Z.H.Zhang,Automaticallyconstructedneural networkpotentialsformoleculardynamicssimulationofzincproteins,Ftont.Chem.9(2021)692200.Panel(C)Thisfigureis takenfromJ.Zeng,L.Cao,M.Xu,T.Zhu,J.Z.H.Zhang,Complexreactionprocessesincombustionunraveledbyneural network-basedmoleculardynamicssimulation,Nat.Commun.11(2020)5713.

Methods

AgoodNNPmodelshouldmeetthefollowingrequirements:(1)Itsaccuracymustbevery closetothequantumchemistrymethodthatusedtolabelthedata.(2)Theonlyinputneeded bythemodelshouldbetheelementinformation,atomiccoordinate,chargesandspinstate. (3)IfusedforMDsimulation,themodelshouldbesmoothanddifferentiable.(4)Inmost cases,thePESmodelshouldpreservethetranslational,rotational,andpermutationalsymmetriesofthesystem,whichisessentialtoguaranteethetransferabilityofthemodel.

Inmost,butnotall,NNPmodels,thepotentialenergyofthesystemisexpressedasthesum ofatomiccontributionstotheenergy(Fig.2)

FIG.2 TheneuralnetworkmodelthatgeneratesthepotentialenergysurfaceforMDsimulation. Thisfigureistaken fromJ.Zeng,L.Cao,M.Xu,T.Zhu,J.Z.H.Zhang,Complexreactionprocessesincombustionunraveledbyneuralnetworkbasedmoleculardynamicssimulation,Nat.Commun.11(2020)5713.

Whiletheenergyofatom i isdeterminedbyitschemicalenvironment.Therelationship betweenthemisdescribedbyaneuralnetwork N :

Thechemicalenvironmentreferstotheelementinformationandrelativepositions e Ri ofthe atom i andallatomswithinapre-defineddistance/angularcut-offwhichcenteredatatom i. Tosatisfytherequirement4mentionedabove,weusemathematicsormachinelearning methodstotransformtheCartesiancoordinatesofatomsinthechemicalenvironment. Themathematicalrepresentationafterthetransformationiscalledamoleculardescriptor D. Adeepneuralnetworkfunction N isthecompositionofmultiplelayers Li :

Alayer L isusuallycomposedby y ¼L x; w, b ðÞ¼ ϕ xT w + b ,(4) where x istheinputvectorand y istheoutputvector. w and b areweightsandbiases,respectively,bothofwhicharetrainable. ϕ istheactivationfunction,suchasReLU,softplus,sigmoid,tanh,GeLU,andetc.

OneofthesimplestdescriptorsistheCoulombmatrix [59]:

where Z isthenuclearcharge.Sinceallelementsinthematrixaredeterminedonlybythe relativepositionsofatoms,thetranslationalandrotationalsymmetriescanbewellpreserved, whilethepermutationinvariancecanbeachievedbysortingtherowsandcolumnsbytheir respectivenorm.However,thesortedCoulombmatrixcannotensurethesmoothnessand differentiabilityofthePESmodel,anditcanbeonlyusedwhenthetotalenergyislearned directly,nodecompositionintoatomiccontributions.

Currently,therearealotofmoleculardescriptorsusedintheconstructionofNNP.Some ofthem [18] arementionedorintroducedinotherchaptersofthisbook(suchas Chapters11, 12,and 19).Herewewillintroducethreeofthem,theatom-centeredsymmetryfunctions (ACSFs) [60],theSchNetmodel [14],andtheDeepPotential-smoothedition(DeepPot-SE) [61] method.

Atomic-centeredsymmetryfunctions

Theatomic-centeredsymmetryfunctions(ACSFs)wereproposedbyBehlerandParrinllo in2007 [10],whichcontainsaseriesofGaussian-typefunctions:

1 and G2 areradicalandangularterm,respectively.

where Rs, λ, η,and ζ arehyperparameters,

,andthecut-offfunction fc isdefinedby

RC istheuser-definedcut-off.Thecut-offfunctionmakesatomsoutofthecut-offradius havenocontributiontoeither G1 or G2 . Onthisbasis,Gasteggeretal.proposedtheweightedatomic-centeredsymmetryfunctions (wACSFs)descriptor [62],whichintroducedweightingfunctionsofthechemicalelementsto symmetryfunctions(Eqs. 7–8):

where g(Zj) ¼ Zj and h(Zj, Zk) ¼ ZjZk.

wACSFsovercometheundesirablescalingofACSFswithanincreasingnumberofdifferentelements.Meanwhile,toobtainacomparablespatialresolutionofthemolecularstructures,asubstantiallysmallernumberofwACSFsthanACSFsisneeded.

Intheirpreviouswork [63],Smithetal.alsooptimizedtheangulartermofACSFsandproposedtheANAKIN-ME(ANI)model.TheyaddtwonewhyperparameterstoEq.(8):

whichgivesthedescriptoranobviouslymoreexpressivepower.

SchNet

ItisworthtomentionthattheACSFs(anditsvariants)arefixedbeforetraininganNNP.In fact,thereareothermoleculardescriptorssuchasSchNet [14] andDeepPot-SE [61] whichcan belearnedinthetrainingprocess.SchNetcontainsatrainableconvolutionalNN.Itsdescriptor D isaNNfunction N ofthecoordinates Rij andatomictype α ofthegivenchemical system.

TheNNfunctioncontainsboththedenselayers Ld andtheconvolutionallayers Lc :

where x istheinputtensorsthatcouldbeeither α ortheoutputfromthelastlayer, n isa hyperparametertodecidehowmanytimes I isappliedto x, ϕ isthesoftplus,andthe convolutionallayers Lc aregivenby

where W istrainable. Ld canbefoundinEq. 4.DetailsofthismodelcanbefoundinRef. [14]

DeepPot-SE

InDeepPot-SE,theexpressionfortheatomiccontribution Ei isaneuralnetworkconsisting ofthreehiddenlayers.TheinputlayeristhemoleculardescriptorD Ri ,whichdetermined bythe“environmentmatrix” e Ri ,the“embedding”matrix Gi ,andareduceddimensionembeddingmatrix G< i [64].

s(Rij)isaswitchedreciprocaldistancefunctionthatcontrolstherangeofthechemicalenvironmenttobeconsidered.Ifanatomisseparatedfromatom i byadistancegreaterthan Roff, thentheatomisnotincludedinthechemicalenvironmentofatom i.Ifaneighboringatomis withinadistanceof Ron,thenitisgivenfullweightinthedescriptor.Theweightcansmoothly changefrom Ron to Roff.Theenvironmentmatrixisa Nneigh 4array,where Nneigh isthenumberofatomswithin Roff

The“embedding”matrix Gi isactuallyanotherneuralnetwork N whichiscalled “embeddingnetwork”: Gi ¼N sRij (20)

Theequationoftheneuralnetwork N isgiveninEq. 3.Usually,weput3hiddenlayersin theembeddingnetwork.Eachrowoftheembeddingmatrixcorrespondstoaneighbor.If thereare M3 nodesinthethirdlayerof Gi ,thesizeoftheembeddingmatrixis Nneigh M3. Thereduceddimensionalembeddingmatrixhasthesamevaluesas Gi ,butonlythefirst M0 columnsarestored,where M0 issmallerthan M3.Themainpurposeoftheusageof G< i istoreducethecomputationalcost.

Intheirrecentwork [22],PinheiroJr.etal.systematicallybenchmarkedaseriesofmoleculardescriptorsindifferentapplicationcontextsandgiveselectionrecommendationsbased onthetargetandthesizeofthetrainingset.Westronglyrecommendthatreadersofthisbook readthisworkbeforeconstructtheirownNNPs.

Exampleofsimulationwithneuralnetworkpotentials

Afterselectingthemoleculardescriptor,oneonlyneedtofocusontwoissues,thepreparationofthereferencedataset(includingthetrainingandtestset)andthetrainingoftheneuralnetwork.Amongthem,thetrainingofneuralnetworksisrelativelystraightforward, currentlymainlybasedonopen-sourceframeworkssuchasTensorFloworPyTorch.On thecontrary,theconstructionofareferencedatasetthatcoversthetargetchemicalspace

FIG.3 Theworkflowofreferencedatasetconstruction. ThisfigureistakenfromJ.Zeng,L.Cao,M.Xu,T.Zhu,J.Z.H. Zhang,Complexreactionprocessesincombustionunraveledbyneuralnetwork-basedmoleculardynamicssimulation,Nat. Commun.11(2020)5713.

whilebeingassmallaspossibleisthemostcriticalanddifficult.Whenonecannotpredictthe chemicalspacethatMDwillexplore,suchassimulatingcombustionreactions,theconstructionofreferencedatasetsbecomesevenmoredifficult.AfeasibleapproachistorunMDsimulationwhilesampling.PreviousstudiessuggestedusingmultipleNNPmodelstoidentify poorlysampledregionsoftheconfigurationspace [65,66].Inthismethod,severalNNP modelswillbetrainedbasedonthesamereferencedataset.DuringtheMDsimulation,a largenumberoftrialconfigurationsareevaluatedbyallofthesemodels.Ifagivenstructure differsobviouslyfromalltraineddata,thepredictionsofthesemodelsshouldbesignificantly different.Thenoneneedtoaddthisstructureintothedataset.Conversely,ifthetraining setalreadycontainssimilarstructuresofthegivenone,thepredictedresultsofthesemodels shouldbeconsistent.Thisalgorithmwascalled“activelearning”or“learningon-the-fly”and hasbeenusedbymanyworks [66–76],see Chapter14 Fig.3 showsan“activelearning” workflowweusedinthesimulationofmethanecombustion.

Firstly,weneedtoprepareaninitialdatasetbyusingashortMDsimulationwiththe ReaxFFforcefield.Foreachatomineachsnapshotofthetrajectory,webuiltamolecularclusterwhichcontainsthisatomandspeciesthatwithinaspecifiedcut-offcenteredonit.Thenthe MiniBatchKMeans [77] isusedtoremovetheredundancy.Startingfromtheinitialdataset, fourdifferentNNPmodelsweretrainedbasedonthedatasetfromthelaststep.Thenseveral shortMDsimulationswereperformedbasedononeoftheseNNPmodels.Duringthesimulation,theatomicforcesofthecentralatominthemolecularclusterareevaluatedbyallfour NNPmodelssimultaneously.Foraspecificatom,ifthepredictedforcesbythesefourmodels

areconsistentwitheachother,thensimilarmolecularclustersshouldalreadybeincludedin thedataset.Onthecontrary,iftheresultsofthesefourmodelsareinconsistentwitheachother andtheerrorbetweenthemisinaspecificrange,thecorrespondingmolecularclusterwillbe addedintothedataset.Theupdateofthedatasetwillbecontinueduntilthepredictionsofthe fourmodelsarealwaysconsistentorthetargetlengthoftheMDsimulationisreached.More techniquedetailscanbefoundinourpreviousstudy [54].

Casestudies

Simulationoftheoxidationofmethane

Inthissection,wewilltakethesimulationofmethanecombustionasanexampleandintroducetheprocedureofNNP-basedMDsimulation.Allfilesneededinthissectioncanbe downloadedfromthefollowinglink:https://github.com/tongzhugroup/Chapter13tutorial.

Attheverybeginning,oneneedstoconsiderwhichsoftwaretouse.Thereareseveral open-sourceandwidelyusedNNPbuildingsoftware,suchasTensorMol [38],MLatom [19,20],EANN [18],andDeePMD-kit [78].Forbeginners,thereislittledifferencebetween thesesoftwarepackages.TheMLatomandEANNmethodareintroducedinotherchapters ofthisbooksuchas Chapters13 and 19,respectively.HereweemploytheDeePMD-kit.The followingstepsareusedtotraintheNNPandperformMDsimulation.

Step1:Preparingthereferencedataset

Inthereferencedatasetpreparationprocess,onealsohastoconsidertheexpectaccuracyof thefinalmodel,oratwhatQMleveloneshouldlabelthedata.InRef.54,theGaussian16 [79] softwarewasusedtocalculatethepotentialenergyandatomicforcesofthereferencedataat theMN15/6-31G** level.TheMN15functionalwasemployedbecauseithasgoodaccuracy forbothmulti-referenceandsingle-referencesystems [80],whichisessentialforoursystemas wehavetodealwithalotofradicalsandtheirreactions.Here,weassumethatthedatasetis preparedinadvance,whichcanbedownloadedfromtheabovelink.

Step2.TrainingtheNNPES

Beforethetrainingprocess,weneedtoprepareaninputfilecalled methane_param.json whichcontainsthecontrolparameters.Thetrainingcanbedonebythefollowingcommand: $$deepmd_root/bin/dptrainmethane_param.json

Thereareseveralparametersweneedtodefineinthe methane_param.json file.The type_map referstothetypeofelementsincludedinthetraining,andtheoptionof rcut isthecut-offradiuswhichcontrolsthedescriptionoftheenvironmentaroundthecenteratom.Thetypeof descriptoris“se_a”inthisexample,whichrepresentstheDeepPot-SEmodel.Thedescriptor willdecaysmoothlyfrom rcut_smth (Ron)tothecut-offradius rcut (Roff).Here rcut_smth and rcut aresetto1.0A ˚ and6.0A ˚ ,respectively.The sel definesthemaximumpossiblenumberof neighborsforthecorrespondingelementwithinthecut-offradius.Theoptions neuron in descriptor and fitting_net isusedtodeterminetheshapeoftheembeddingneuralnetworkand thefittingnetwork,whicharesetto(25,50,100)and(240,240,240)respectively.Thevalueof

axis_neuron representsthesizeoftheembeddingmatrix,whichwassetto12.Andtheoptions start_lr, decay_rate,and decay_steps inthe learning_rate modulecontrolsthelearningratefor n th batchaccordingtothefollowingformula:

Theinitiallearningratewassetto0.001anditwilldecayevery400steps.Thelossfunction isdefinedas:

The Δε and Δ Fi inthelossfunctionrepresentthedifferencebetweenthepredictionofNNP inthelabeledenergyandforce,respectively. pε and pf areprefactorsthatdecaysexponentially duringtraining.Byassigningdifferentrandomseedsintheneuralnetworkinitializationprocess,onecantrainmultiplemodelsatthesametime.

Afterthetrainingiscompleted,thepredictivepowerofthemodelmustbecheckedfirst.As showninthe Fig.4,themeanabsoluteerrors(MAE)ofpotentialenergyareonly0.04eV/atom

Training Set / Test Set

FIG.4 (A)Energypredictionerrorsforthereferenceset.Themeanabsoluteerrors(MAEs)androotmeansquared errors(RMSEs)areineV/atom.(B).ThecorrelationofatomicforcesbetweenNNpredictionsandDFTcalculations. ThisfigureistakenfromJ.Zeng,L.Cao,M.Xu,T.Zhu,J.Z.H.Zhang,Complexreactionprocessesincombustionunraveledby neuralnetwork-basedmoleculardynamicssimulation,Nat.Commun.11(2020)5713.

and0.14eV/atominthetrainingsetandthetestset,respectively.Asfortheatomicforces,the correlationcoefficientbetweenthepredictedandlabeledvaluesis0.999andtheMAEis 0.12eV/A ˚ .

Step3:Freezethemodel

Thisstepistoextractthetrainedneuralnetworkmodel.Tofreezethemodel,thefollowing commandwillbeexecuted:

$$deepmd_root/bin/dpfreeze-ograph.pb

Afilecalledgraph.pbcanbefoundinthetrainingfolder.Thenthefrozenmodelcanbe compressed [81]:

$$deepmd_root/bin/dpcompress-igraph.pb-ograph_compressed.pb-tmethane_param.json

Step4:RunningMDsimulationbasedontheNNP

ThefrozenmodelcanbeusedtorunreactiveMDsimulationstoexplorethedetailedreactionmechanismofmethanecombustion.TheMDengineisprovidedbytheLAMMPSsoftware [82].Hereweusethesamesystemfromourpreviouswork [54],whichcontains100 methaneand200oxygenmolecules.TheMDwillbeperformedundertheNVTensemble at3000Kfor1ns.TheLAMMPSprogramcanbeinvokedbythefollowingcommand:

$$deepmd_root/bin/lmp-iinput.lammps

The input.lammps istheinputfilethatcontrolstheMDsimulationindetail,techniquedetailscanbefoundinthemanualofLAMMPS(https://docs.lammps.org/).TousetheNNP, the pair_style optioninthisinputshouldbespecifiedasfollows: pair_styledeepmdgraph_compressed.pb pair_coeff **

Step5:Analysisofthetrajectory

Afterthesimulationisdone,wecanusetheReacNetGenerator [83] softwarewhichwas developedinourpreviousstudytoextractthereactionnetworkfromthetrajectory.Allspeciesandreactionsinthetrajectorywillbeputonaninteractivewebpagewherewecananalyzethembymouseclicks.Eventuallyweshouldbeabletoobtainreactionnetworksthat consistentwith Figs.2–4 inRef.[54].

$ reacnetgenerator-imethane.lammpstrj-aCHO–dump

Themostexpensivestepsarestep2(trainingtheNNP)andstep4(RunningMDsimulation basedontheNNP).Performanceofthesetwostepsisbenchmarkedusingdifferenthardware asshownin Table1.

TABLE1 PerformanceoftrainingandMDsimulationsusingdifferenthardware(unit:ms/step)a

NVIDIAGeForceRTX3080Ti16.3910.773.1643.325

NVIDIAGeForceRTX309024.2626.786.7386.965

NVIDIATeslaV10027.0719.175.8635.558

a TrainingisconductedwiththemethanecombustiontrainingsetgivenintheGitHubrepository,usingtheDeePMD-kitsoftware.MD simulationsareconductedwithamethanecombustionsystemof900atoms,usingLAMMPSinterfacetotheDeePMD-kit.

Conclusionsandoutlook

Inthischapter,weintroducethebasicconceptsandideasthatabouttheconstructionand usageofNNPfromabeginner’sperspective.Inthepastyears,machinelearning-basedmoleculardynamicssimulationshavegainedsignificantdevelopmentsandadvancesthathave changedtheresearchparadigmthroughoutthetheoreticalchemistrycommunity.Intheprocessofgrowingfrombeginnertoexpert,webelievethatthefollowingissuesareneededtobe considered [84,85].(1)Dataisattheheartofallmachinelearningmethods.Therefore,the questforbettermethodstobuildreferencedatasetswillneverstop.Somerecentusefuldiscussionscanbefoundintheliterature [86,87].(2)Theselectionofpropermoleculardescriptorswhichcanrepresentthechemicalenvironmentwell,whilewithminimalcomplexity. Althoughwell-establishedmoleculardescriptorssuchasACSFs,EANNandDeepPot-SE arealreadyavailable,onemaystillwanttodesignnewdescriptorsforspecificsystemsofinteresttofurtherenhancethetransferabilityoftheNNPmodelandimprovetheaccuracyand efficiencyofthetrainingprocessandMDsimulation.(3)Theselectionofhyper-parameters forNN.Therearemanyhyper-parameterssuchaslearningrateandnetworkstructureparameterswhichhavehugeimpactontheaccuracyandefficiencyofNNPs.Althoughsome automatedmethodshavebeenproposed [60,87,88],thechoiceoftheseparametersisstill largelyempiricallydominated.Whencomputationalresourcesarelimited,oneneedstocarefullycompareandselecttheappropriatesetofparameters.(4).Thetreatmentoflong-range interactions.Asmentionedabove,distancecut-offs(usuallyaround5A ˚ )areusedindefining thechemicalenvironment.Thepurposeofthisistoavoidhavingtoomanydegreesoffreedominthechemicalenvironment,whichwouldgreatlyincreasethedifficultyofthetraining processaswellasthesizeofthereferencedataset.Forsmallsystemswiththeperiodicboundary,thelong-rangeinteractionscanbeeffectivelyincludedintheshort-rangeinteractions. However,forlargeand/ornon-periodicsystemsincondensedphases,suchasbiomolecules likeproteins,long-rangeinteractionsmustbeexplicitlyconsidered.Onlyveryrecentlysome encouragingprogresshasbeenmade [89,90]

Acknowledgment

ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina(GrantsNo.22173032,21933010).J.Z. wassupportedinpartbytheNationalInstitutesofHealth(GM107485)underthedirectionofDarrinM.York.Wealso thanktheECNUMultifunctionalPlatformforInnovation(No.001)andtheExtremeScienceandEngineeringDiscoveryEnvironment(XSEDE),whichissupportedbyNationalScienceFoundationGrantACI-1548562.56(specifically,theresourcesEXPANSEatSDSCthroughallocationTG-CHE190067),forprovidingsupercomputertime.

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Elfin Land.

But far the greatest miracle Which Elfin land can show, A hostel is, like that which stood In Eastcheap long ago.

Before the entrance, in the blast There swings a tusky sign; And when at night the Elfin moon And constellations shine,

A ruddy glow illumes the panes, And looking through you see, With merry faces seated round, A famous company.

Prince Hal the royal wassailer, And that great fount of fun, Diana’s portly forester, The merry knight Sir John,

With all their losel servitors, Mirth-shaken cheek by cheek; Cambysean Pistol, Peto, Poins, And Bardolph’s fiery beak.

A grove there is in Elfin land, Where closely intertwine The Grecian myrtle’s branches light With Gothic oak sublime.

Beneath its canopy of shade,

Their temples bound with bays, Are grouped the minstrels, that adorn

The mediæval days.

The laurelled Ghibelline, who saw The Stygian abyss, The fiery mosques and walls, that gird The capital of Dis;

The realms of penance, and the rings Of constellated light, Whose luminous pavilions hold The righteous robed in white;

Uranian groves and spheral vales, Saturnian academes, Where sainted theologues abide, Discoursing mystic themes;

The Paradisal stream, that winds Through Heaven’s unfading bowers, And on its banks the beauteous maid, Who culled celestial flowers.

Him next the sweet Vauclusian swan, Love’s Laureate, appears, Who bathed his mistress’ widowed urn With Heliconian tears;

Certaldo’s storied sage,—a bard, Though round his genius rare The golden manacles of verse He did not choose to wear.

Those rosy morns, that usher in Each festal-gladdened day, His prose depicts in hues as bright

As could the poet’s lay

His ultramontane brother, born In Albion’s shady isle; Dan Chaucer, of his tameless race Apollo’s eldest child;

The Medecean banqueter, Whose Fescennines unfold The deeds of heathen Anakim Restored to Peter’s fold;

Ferrara’s Melesigenes, Who o’er a wide domain Of haunted forests, mounts, and seas, Exerts his magic reign;

A glowing Mœnad, with her locks Dishevelled in the wind, His fancy wantons far and near, From Thule unto Ind;

Now from her griffin steed alights Alcina’s palace near, Now in the Patmian prophet’s car Ascends the lunar sphere;

Or with Rinaldo wanders through The Caledonian wood, Amid whose shades and coverts green Heroic trophies glowed;

Or paints the mighty Paladin Transformed to monster gross, Whose mistress drank in Ardennes lone The lymph of Anterōs.

Next hapless Tasso, pale and wan, Released from dungeon grates; The sacred legions of the cross

His genius celebrates;

Armida’s mountain paradise

Amid the western seas; Her dragon-yoke, whose nimble hooves Could run upon the breeze.

The sombre forest, where encamped Dark Eblis’ minions lay, With shapes evoked from Orcus’ gloom To fright his foes away.

Lo, marble pontifices spring, To arch illusive streams, And swans and nightingales rehearse Their moist melodious threnes!

The centuried trees are cloven wide, And forth from every plant A maiden steps, whose tears might melt A heart of adamant.

A sudden darkness veils the sky, And fortresses of fire, With ruddy towers of pillared flame, Above the woods aspire.

Transfigured in the morning beam. On Zion’s holy height

Rinaldo puts the dusky swarms Of Erebus to flight.

Nor absent from the shining throng That dainty bard, I ween, Who hung the maiden empress throne With garlands ever green.

The Elfin Court’s Demodocus,

Chapter 12 - neural network potentials jinzhe zeng & liqun cao & tong zhu - The ebook is ready for d

A

Curse of Queens Amanda Bouchet

Why

Neuralnetworkpotentials

Abstract

Introduction

Methods

Atomic-centeredsymmetryfunctions

SchNet

DeepPot-SE

Exampleofsimulationwithneuralnetworkpotentials

Casestudies

Simulationoftheoxidationofmethane

Step1:Preparingthereferencedataset

Step2.TrainingtheNNPES

Step3:Freezethemodel

Step4:RunningMDsimulationbasedontheNNP

Step5:Analysisofthetrajectory

Conclusionsandoutlook

Acknowledgment

References

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Its realms of mystery

Elfin Land.