Incremental dynamic mode decomposition: A reduced-model learner operating at the low-data limit

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Agathe REILLE, Nicolas HASCOËT, Chady GHNATIOS, Amine AMMAR, Elías G. CUETO, Jean

Louis DUVAL, Francisco CHINESTA, Roland KEUNINGS - Incremental dynamic mode

decomposition: A reduced-model learner operating at the low-data limit - Comptes Rendus

Mécanique - Vol. 347, n°11, p.780-792 - 2019

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Incremental dynamic mode decomposition: A reduced-model

learner operating at the low-data limit

Agathe

Reille

a

,

Nicolas

Hascoet

a

,

Chady

Ghnatios

b

,

Amine

Ammar

c

,

Elias

Cueto

d

,

Jean

Louis

Duval

e

,

Francisco

Chinesta

a

,∗

,

Roland

Keunings

f

a _{ESIGroupChair@PIMM,ArtsetMétiersInstituteofTechnology,CNRS,CNAM,HESAMUniversity,151,boulevarddel’Hôpital,75013Paris,} France

b_{NotreDameUniversity–Louaize,P.O.Box72,ZoukMikael,ZoukMosbeh,Lebanon}

c_{ESIGroupChair@LAMPA,ArtsetMétiersParisTech,2,boulevardduRonceray,BP93525,49035Angerscedex01,France} d_{ESIGroupChair@I3A,UniversityofZaragoza,MariadeLuna,s.n.,50018Zaragoza,Spain}

e _{ESIGroup,BâtimentSeville,3bis,rueSaarinen,50468Rungis,France}

f _{ICTEAM,UniversitécatholiquedeLouvain,av.GeorgesLemaître,4,B-1348Louvain-la-Neuve,Belgium}

a

b

s

t

r

a

c

t

Keywords: Machinelearning Advancedregression Tensorformats PGD Modedecomposition Nonlinearreducedmodeling

Thepresentworkaimsatproposinganewmethodologyforlearningreducedmodelsfrom asmall amountof data. It is based onthe fact that discretemodels, ortheir transfer functioncounterparts,have alow rankand then theycan beexpressedvery efficiently usingfewtermsofatensordecomposition.Anefficientprocedureisproposedaswellasa wayforextendingittononlinearsettingswhilekeepinglimitedtheimpactofdatanoise. Theproposed methodologyisthenvalidatedbyconsideringanonlinearelasticproblem andconstructingthemodelrelatingtractionsanddisplacementsattheobservationpoints.

1. Introduction

In physics in general andin engineering in particular, when addressing a generic problem, the first step consists in selectingthebestsuitedmodelforthedescriptionoftheevolutionofthesystemunderconsideration.Nowadays,avariety of models exists that are well established and validated, covering most of the domains of physics, e.g., solid andfluid mechanics,electromagnetism,heattransfer...

These models are based on the combination of two types of equations of very different nature. From one side, the so-calledbalance equations,whosevalidity intheframework ofclassicalphysics isout ofdiscussion.The second type of equations are the so-called constitutive models. These relate primal variables and dual variables, e.g., stress and strain, temperatureandheat flux.Theirexpression andvalidity dependon thenature oftheconsideredsystemandthe applied loading—inthemostgeneralsense.

*

Correspondingauthor.

E-mailaddresses:Agathe.REILLE@ensam.eu(A. Reille),Nicolas.Hascoet@ensam.eu(N. Hascoet),cghnatios@ndu.edu.lb(C. Ghnatios), Amine.AMMAR@ensam.eu (A. Ammar),ecueto@unizar.es (E. Cueto),Jean-Louis.Duval@esi-group.com (J.L. Duval),Francisco.CHINESTA@ensam.eu (F. Chinesta),roland.keunings@uclouvain.be (R. Keunings).

Constitutiveequationsareoftenphenomenological,andingeneralinvolvesomeparameters assumedtobeintrinsicto thesystemunderconsideration.

Forthisreason,andeveninthecontextofclassicalengineering,beforesolvingagivenproblem,onemustdecideonthe mostappropriatemodel.Thereobviouslyexistsariskonthepertinenceofthechosenmodelforaddressingthephenomena understudy.Byperforming some engineeredexperimentsinorderto calibratethe model,thatis,to extractthe valueof theparametersthatitinvolves,theseriskscanbealleviated.

Thecalibratedmodelisexpressedgenerallyasasystemofcouplednonlinearpartialdifferentialequationsandmustbe solvedforgivenloadingandboundaryconditions.

Veryoften,thesolutionoftheseproblemsis nottractable by ananalytical procedure.Numericalmethods were intro-ducedinthetwentiethcentury forthat purpose,andare nowadayswidelyemployed, inindustryaswellasinacademia. Thus, approximateprocedureswere proposed,mostofthem computationallymanipulable.Thefirst optionconsistsin as-suming the solution expressible as a combination of a reduced number of functions with a physical or mathematical foundation. The Ritz methodfollows thisrationale. Thus,if thesefunctionsare notedby

G

i

(

x

)

,i

=

1

,

. . . ,

G

, thesolution approximationreads u

(

x

,

t

)

≈

G



i=1

α

i

(

t

)

G

i

(

x

)

(1)

wherethetime-dependentcoefficients

α

i

(t)

arecalculatedbyinvokingaprojectionmethod,liketheGalerkinmethod,for instance.

However, theknowledgeofthesefunctionsbecomes atrickyissueinmanycases.Whenthe choiceofan appropriate, physically soundbasis is unavailable, thebest option is considering a generalpurpose approximation basis, forexample consistingofpolynomials(Lagrange,Chebyshev,Legendre,Fourier...):

P

i

(

x

)

,i

=

1

,

. . . ,

P

,fromwhichtheapproximationnow reads u

(

x

,

t

)

≈

P



i=1

β

i

(

t

)

P

i

(

x

)

(2)

andwhere,forensuringaccuracy,asufficientnumberoftermsmustbeconsidered,implyingaquitelargenumberofterms inthefinitesum(2).

Inordertobetteradapttocomplexdomains,globalfunctions

P

i

(

x

)

areoftenreplacedbycompactlysupportedfunctions

N

i

(

x

)

,leadingtotheso-calledFiniteElementMethod,FEM.Ifweconsider

N

nodes,theapproximationreads

u

(

x

,

t

)

≈

N



i=1

Ui

(

t

)

N

i

(

x

)

(3)

wherenowUi

(t)

representsthesearchedsolutionatnodexi andattimet.

This formulationbecomes generalenough atthe price ofcomputing many—in some casestoo many—unknowns, the nodalvaluesUi

(t)

,

∀

i.

Inthesequel,forthesakeofnotationalsimplicity,thedependenceontimeisnotmadeexplicit,anditisassumedthat the discrete systemthat results fromintroducing approximation (3) into theweak form ofthe problem(for thesake of simplicityassumedtobelinear)leadstothelinearsystem

K

U

=

F (4)

Here,matrix

K

representsthediscreteformofthemodel,whereasvectorsU andF containthenodalunknownsandloads, respectively.Theirrespectivesizesare

N

× N

,

N

×

1 and

N

×

1.

1.1. Reducedmodelling

Inmanyapplicationsofpracticalinterest,despitetherichnessofEq. (4) thatisabletoconsideranychoiceoftheloading inavectorspaceofdimension

N

,usualloadingscontaininpracticemanycorrelations.ThisresultsinthesolutionU being

definedinasubspaceofdimension

n

,with

n

 N

.

ProperOrthogonalDecompositionproceedsbyembeddingthesolutionontothatsubspaceofdimension

n

[1] [2] [3] [4]. Nonlinearmanifoldlearningconstitutesitsnonlinearcounterpart.Forthatpurpose,fromasetofsnapshotsofthesolution

U1

,

. . . ,

US,areducedbasis

R

i

(

x

)

,i

=

1

,

...,

n

,with

n

 N

isextracted.Thesolutionisfinallyexpressedby u

(

x

,

t

)

≈

n



i=1

whosematrixformreads

U

= B

γ

(6)

The columns ofmatrix

B

contain the reducedfunctions at node locations. Thus, the component

B

i j

=

R

j

(

xi

)

with i

=

1

,

. . . ,

N

and j

=

1

,

...,

n

.

Now,byinjecting (6) into(4) andpremultiplying bythetransposeof

B

(somethingequivalent toaGalerkinprojection inthereducedbasis),itresults

(

B



KB)

γ

= B

F (7)

whoserespectivesizesare

n

× n

,

n

×

1 and

n

×

1.

Inthenonlinearcase,differentprocedureshavebeenproposedforalleviatingtheconstructionof

K

.Among them,one canfindtheDiscreteEmpiricalInterpolationMethod[5] ortheHyper-Reduction[6],tocitebutafew.

Inordertoavoidthenecessityofperformingan“apriori”learning,asimultaneoussearchingofspaceandtimefunctions was proposed in the context of the so-called ProperGeneralized Decomposition, PGD[7] [8] [9] [10] [11]. It was then extendedforaddressingspaceseparation[12] orparametricsolutions[13] [14] [15] [16] insomeofournumerousprevious works.Thus,thespace-timeseparatedrepresentationnowreads

u

(

x

,

t

)

≈

D



i=1

T

i

(

t

)

X

i

(

x

)

(8)

wherebothgroupsoffunctions,

T

iand

X

i,areobtainedbyinjectingtheapproximation(8) intotheproblemweakformand then usingarank-one enrichmentforincrementallyconstructingtheseparatedrepresentation.Ateachenrichmentstepof thatgreedyalgorithm,analternateddirectionstrategyisconsideredforaddressingthenonlinearityarisingfromtheproduct ofbothunknownfunctions

T

i and

X

i.

Despiteitsgenerality,thecomputedsolution(8) dependsontheloadingterm.InthecaseofPOD-basedtechniques,the reducedalgebraicsystemcaneasilybeinverted,leadingtothereducedtransferfunction

(B



KB)

−1_{.Notethatthereduced} modelmatrix(

B



KB

)doesnotdependexplicitlyontheloading.Thus,assoonasanewloadingF comesintoplay, itis tobeprojectedonthereducedbasis

B

T_F

_≡

_{f.Then,thereducedsolutionisobtainedfrom}

_γ

_{= (B}



_KB)

−1_{f.Torecoverthe} solutionintheoriginalfiniteelementbasis,onesimplyhastoperformtheprojectionU

= B

γ

.

PGD, on the contrary, does not compute such a reduced model (or its transfer function counterpart), but instead it proceedsbyeither

(i) calculatingthesolutionforeachloading,or

(ii) expressingtheloadinginanreducedbasis(e.g.,POD)accordingto

f

(

x

,

t

)

=

F



i=1

μ

i

F

i

(

x

,

t

)

(9)

thatallowstransformingtheinitialproblemintoaparametricone.WithinthePGDrationale,thesolutionissearched intheparametricformu(x,t,

μ

)

,with

μ

= (

μ

1

,

...,

μ

F

)

.

AcriticismthatisusuallyattributedtothePGDtechnique,ifcomparedtoPOD-basedprocedures,ispreciselythe neces-sityoftheloadtoberepresentableintermsofthereducedbasis

F

1

,

. . . ,

F

F.However,itisimportanttonotethatevenif

such aconstraintislessexplicitwhenusingthePODrationale,italsoappliesimplicitly[17].Thus,itisimportanttonote thatthePODrationaleisbasedontheexistenceofareducedbasis

B

,andthatthisbasiswasconstructedfromaparticular choiceofsnapshotsU1

,

. . . ,

US,associatedwithaparticularloadingF1

,

. . . ,

FS.Thus,ifaquitedifferentloadingcomesinto

play—one whosesolution cannot be accurately approximated by the reducedbasis

R

i involved in

B

—the transfer func-tioncanbeapplied,butnothingguaranteestheaccuracyoftheresultingsolution.Finally,theapplicabilityofModelOrder Reductiontechniquesremainssubordinatetothefulfillmentoftheconditionsassumedduringtheirconstruction.

All thejustdiscussedtechniquesbelong tothevast familyofModelOrderReductiontechniques.In fact,modelswere notreallyreduced,onlythesolutionprocedure.

1.2. Learningmodels

Wearguedinsomeofourformerworksthat,insomecircumstances,modelsaretoouncertaintorepresentthephysical system[18].Inthatcase,adata-drivenprocedurebecomesanappealingchoice[19] [20] [21] [22].Forthesakeofsimplicity, we assumetheproblemtobelinearandthat

N

couples

(

Ui

,

Fi

)

—assumedindependent,thatis,spanning thewholevector spaceofdimension

N

—areavailable.Wealsoassumethatnothingisknownonthemodelwhosediscreteexpressionconsists ofthematrix

K

.

Thus,withallthecouplesfulfillingthealgebraicrelationship

K

Ui

=

Fi

,

∀

i (10)

onecould constructthevectorK (vector

˜

expressionofthematrix

K

),the extendedloadingvectorF that

˜

includesallthe loads,i.e.F

˜



= (

F1

,

. . . ,

FN

)

andfinally matrix

U

˜

fromthedifferent Ui organized inan adequatemanner soasto enable Eqs. (10) tobeexpressedas

˜U ˜

K

= ˜

F (11)

TheycanbesolvedtoobtainthemodelK,

˜

oritsmatrixcounterpart

K

.

There aremanywaysofperforming such an identification. Inthelinear orthenonlinearcases as,forexample, deep-learning(basedonneuralnetworks,NN),dynamicmodedecomposition,DMD...tocitebutafew.

It isimportantto note thatthe complexity ofstandard solutions involvingknownmodels

K

,scales withthe number ofunknowns

N

,however,whentrying toidentifythemodelthecomplexityscales with

N

2 (numberofcomponentof

K

), justifyingthat alotofdataismandatory forextractingthehiddenmodel,situationthat becomes evenmorestringentin thenonlinearcasethatrequiresidentifyingamodelK

|

UaroundanypossiblevalueofU.

Thisapparentcomplexityjustifiesthefactofassociatingthetermbig-datatomachinelearning.However,inengineering, the available data is insome circumstances very scarce: collecting datacould be technologically challenging, sometimes simplyimpossible,andinallcases,generallyexpensive.

Thus,thebig-dataisbeing,orshouldbe,transformedinasmartercounterpart,withsmart-dataparadigm rationalizing theamountofneededdata,whiledrivingitsacquisition(collection):whichdata,whereandwhen?

The presentwork tries to exploit the fact previously discussed, that despite the richness of loadings and responses (beingthemodeltheapplicationthatrelatesboth),ingeneralbotharelivinginlow-dimensionalmanifolds,andtherefore the model relating both is expectedto reflect this fact. This consideration was exploited when proposing the so-called hyper-reductiontechniques,howevertheywereappliedonpre-assumedmodels,whereasinwhatfollowsweapplyitwhile assumingthemodelunknown.

After thisintroduction,the next section proposesa procedureforextractingthe reducedmodelfrom asmallamount ofdata,then theprocedure willbe illustrated fromtheanalysisofa quitesimplenonlinear problem, beforefinishing by addressingsomegeneralconclusions.

2. Methods

Forthesakeofsimplicity,westartbyassumingthediscretelinearproblem

K

U

=

F (12)

whereF andU representtheinputandoutputvectorsrespectively.Inageneralcase,theycouldhavedifferentnatureand representthevaluesoftheirrespectivefieldsattheobservationpoints.Therespectivesizesare

N

×

1 and

N

×

1.

Asinthecaseofreduced-ordermodeling,weassumethatinputsandoutputsare(toacertaindegreeofapproximation) livinginasub-spaceofdimension

n

,muchsmallerthan

N

.Thus,therankof

K

isexpectedtobealso

n

,evenifapriori,it wasreadytooperateinalargerspaceofdimension

N

.

Thequestion isthereforehowto extractthereducedmodel? Amongthenumerouspossibilities weexplored,we com-menthereontwoofthem.

2.1. Rank-

n

constructor

Here,we considera setof

S

input-output couples

(

Fi

,

Ui

)

, andassume themodel tobe expressiblefromitslow-rank form

K

LR

K

≈ K

LR

=

n



j=1 Cj

⊗

Rj

=

n



j=1 CjR_j (13)

where

⊗

denotesthetensorproduct,andCj andRjaretheso-calledcolumn androw vectors.Thisexpressionissomehow similartotheseparatedrepresentationusedinthePGD,theSVD(SingularValueDecomposition)ortheCURdecomposition [23] [24].

Remark1.Ifthemodelisexpectedtobesymmetric,onecouldenforcethatsymmetryinthesolutionrepresentation,i.e.

K

≈ K

LR

=

n



j=1

Letusdefinethefunctional

E(K

LR

₎

_accordingto

E

(

K

LR

)

=

S



i=1



Fi

− K

LRUi



_p (15)

Thechoiceofmanydifferentnormscouldbeenvisagedhere.Forexploitingsparsity,forinstance,thechoiceshouldbethe

L1-norm.Inwhatfollows,weconsiderthestandard L2-norm,i.e.wetake p

=

2.

By consideringmatrices

F

and

U

containing intheir columnsvectors Fi andUi respectively,the previous expression canberewrittenusingtheFrobeniusnorm(relatedtop

=

2)as

E

(

K

LR

)

=

F

− K

LR

U



_F (16)

whoseminimizationresultsin

K

LR

(

U U

T

)

= FU

 (17)

or,equivalently,

K

LR

= (FU



)(

U U

T

)

−1 (18)

Thisprovesthat

K

LR_{and,moreparticularly,itscolumnandrowvectors,correspondtotheSVD(orrank-}

_n

_{truncatedSVD)}

decompositionof

(F U



)(U U

T

₎

−1_.

Remark2.WithinthestandardPGDrationale,onecouldcomputetheseparatedrepresentationof

K

LR _bycomputing

pro-gressivelytheseparatedrepresentationusingthestandardgreedyalgorithmappliedonEq. (17) from

K

∗

: K

LR

₍

_{U U}

T

₎

_{= K}

∗

_{: (FU}



₎

₍₁₉₎

with“

:

”thetwice-contractedtensorproduct,and

K

LR_{approximatedby(13).Thetestmatrix}

_K

∗ _{isobtainedfrom:}

K

∗

=

n



j=1

C∗R_j (20)

oritssymmetrizedcounterpart.

Thisprocedureisspeciallysuitablewhen

N

becomeslarge,thusmakingdifficultthecalculationoftheSVDdecomposition of

(F U



)(U U

T

₎

−1_.

Remark3.Thisprocedureensuresthatwhenaddressingalinearproblem,

N

linearlyindependentloadingsFj

,

j

=

1

,

...,

N

, ensuretheconstructionofarank-

N

model

K

.Ifthemodeliscomputedbyincorporatingprogressivelytheavailabledata— from one loadingleading to model

K

1 _{(asingle data model),to}

_N

_{leading to}

_K

N_{—we can define themodel enrichment}



j

_K

_{= K}

j

_{− K}

j−1_{, j}

₌

₂

_,

_...,

_N

_.

Note that addingmore data,Fj

,

j

> N

, tothe modeldoes notmake it evolve anymore, asexpected. Inother words:

K

j

_{= K}

N_{, j}

_{> N}

_{.Inthenonlinearcase,whereamodelisaprioriexpectedtoexistaroundeachU,themodelcontinuesto}

evolvewhen j

> N

.

Withthelow-rankmodelthuscalculated,assoonasanewdatumarrives,F,thesolutionisevaluatedfrom

U

=



K

LR



−1F (21)

Remark4.Insection2.3weproposeanalternativemethodologybasedonthecalculationoftransferfunctionsthat avoids modelinversion.

2.2. Progressivegreedyconstruction

Inthiscaseweproceedprogressively.Weconsiderthefirstavailabledatum,thepair(F1

,

U1).Thus,thefirst,one-rank, reducedmodelreads

K

1

= (

C1R₁

)

(22)

ensuring

andonepossiblesolution,ensuringsymmetry,consistsofR1

=

F1 andC1

=

_ϕF1₁,with

ϕ

i

=

F1U1.Manyothersolutionsexist, sincethereare

N

availabledataforcomputing2

N

unknowns(the componentsoftherowandcolumnvectors).Ofcourse, theproblemcanbewrittenasaminimizationproblemusinganadequatenorm.

Supposenowthataseconddatumarrives,(F2

,

U2),fromwhichwecanalsocomputeitsassociatedrank-one approxima-tion,andsoon,foranynewdatum(Fi

,

Ui).Thisleadsto

K

i

= (

CiRi

)

(24)

ensuringatitsturn

(

CiRi

)

Ui

=

Fi (25)

whereCi

=

_ϕFi_i (with

ϕ

i

=

F_iUi)andRi

=

Fi.

ForanyotherU,themodelcouldbeinterpolatedfromthejustdefinedrank-onemodels,

K

i,i

=

1

,

...,

S

,accordingto

K|

U

≈

S



i=1

K

i

I

i

(

U

)

(26)

with

I

i

(

U

)

theinterpolationfunctionsoperatinginthespaceofthedataU.

Thisconstructorisnotappropriatewhenaddressinglinearbehaviors.Ifweassumeknowntwolocalrank-onebehaviors

K

1 and

K

2ensuringthefulfillmentofrelations

K

1U1

=

F1 and

K

2U2

=

F2,itfollowsthatifforexampleF

=

0

.

5

(

F1

+

F2

)

, byconsidering

K

=

0

.

5

(K

1

+ K

2

)

,theresultingoutputdoesnotsatisfylinearity,i.e.U

=

0

.

5

(

U1

+

U2

)

.

However,inthenonlinearcase,bydefiningthesecantbehavioratthemiddlepointassociatedwithF

=

0

.

5

(

F1

+

F2

)

as

˜K =

0

.

5

(K

2

− K

1

)

,wewillhave

K

= K

1

+ ˜K =

0

.

5

(K

1

+ K

2

)

,factthatallowsviewingtheprogressivegreedyconstruction anditsassociatedinterpolationasalinearizationprocedure.

2.3. Generalremarks

– Allthe previousanalysiswas basedon thecalculationof thereducedmodel

K

LR_{.However, thisprocedureneeds its}

inversion

K

LR−1 _{forevaluatingthesolutionU}

_{= K}

LR−1_{F.Thisissuecouldbeeasilycircumventedasfollows.}

The discretemodel

K

U

=

F canbe rewritten asU

= K

−1F andbyintroducing thediscrete transferfunction

T

(that representstheinverseof

K

),onecouldlearndirectly,byusingallthepreviousrationale,thereducedexpressionofthe transferfunction,i.e.

T

LR_.

Theadvantageoflearningthereduceddiscretetransferfunction isthat, assoonasadatumF isavailable—andunder theassumptionthat itis livinginthesamesubspacethat servedforconstructing thereducedmodel—theevaluation becomesstraightforward:asimplematrix–vectorproduct.Itcanevenbemoresimplified,sincematrix

T

LR_isexpressed

inaseparateformat,U

= T

LR_F,with

T

≈ T

LR

=

n



j=1

ˆ

CjR

ˆ

_j (27)

– Nothingreallychangeswhenconsideringnonlinearmodels,exceptthefactthatthemodels,

K

or

T

and,inturn,the vectors they involve, must be assigned to the neighborhood of the data U or F. In practice, data can be classified, creatinganumberclusterswherethemodelsareconstructed[25].

Thus,eachtimeadatumF arrives,theclustertowhichitbelongs—say,

κ

—isfirstidentified.Then,thelow-rankdiscrete transferfunctionsofthatcluster,

T

LR

κ ,ischosenandthesolutionevaluatedaccordingto

U

= T

_κLRF (28)

– Again withinthePGDrationale, allthediscussion abovecan beextended toparametric settings.Thedescription and resultsconstituteaworkinprogressthatwillbereportedinongoingpublications.Anotherworkinprogressconcerns theobtentionoferrorbounds,neededforcertifyingtheconstructedmodelsandtheirpredictions.Thereisavastcorps ofliteratureonverificationandvalidation,buttheywhereassociatedwithmodel-basedsimulations.Here,forthebest identified model,theexisting errorestimationprocedures can be applied.However, theremaining question concerns theestimationoftheerrorassociatedwiththemodelitself,whosequalitydependsonthecollecteddata,i.e.validation, morethanverification.Thisissuealsoappliesinmodel-basedsimulations:howtobesurethattheconsideredmodels aretheappropriateones?

Fig. 1. Domain equipped with a finite-element mesh.

Despitetheapparent difficulty,sincethe modelisdefinedwiththeonlysupportofa discretebasis, someestimators couldbeeasilydefined,andwillbereportedinfutureworks.

– All thepreviousdevelopmentscouldbe accomplishedbyfirstextractingthereducedbasesassociatedwithinputand output vectors(actionandreaction)andthenlearningtheassociatedreducedmodelswithintheframeworkofa pro-gressive POD. However, the procedure hereconsidered seems more physicallysound. A fully reducedcounterpart of theprocedureshereproposedanddescribedconstitutesaworkinprogress,whoseresultswillbereportedinongoing publications.

– The present procedure becomes quite close to theso-called dynamic model decompositionas soon asthe model is assumedtobeexpressedinatensorformandextendedtononlinearsettingsfromanappropriateclustering.

– The main issuewhen considering data-driven modelsis the noise impacton the predictions.There are many possi-bilities, mostofthembased onthe useoffilters.Here,we considered, asprovedlater, a simpleprocedure.The data (Fi

,

Ui) is obtained with a resolution

N

, on which noise applies. A quite efficient filterconsists in simply consider-ing themodel,i.e.itsrowandcolumnvectors, described witha coarserresolution

M

< N

.In fact,inputsandoutputs couldexhibitfastfluctuations,butinmanyapplications,inwhichmodelorderreductionapplies,thesefastfluctuations are almost due to noise, andfor that reason, the model could be expected being described with a coarser resolu-tion.

3. Results

3.1. Anonlinearexample

Inordertoprovethepotentialinterestofthemethodologyheredescribed,we considera nonlinearelasticmodelwith Poissoncoefficient andYoung’s modulusgivenby respectivelyby

ν

=

0

.

3 and E

=

105

₍

₁

₊

₁₀₀₀

_ε

I I

)

,with

ε

I I thesecond invariantofthestraintensor

ε

.UnitsareMPaforstresses,N/mmforappliedtractionsandmmforlengths.

The mechanical problemis definedinthe two-dimensional domaindepicted inFig. 1, that alsoshowsthe finite ele-ment meshemployed forgenerating thesyntheticpseudo-experimental data.Displacementsare preventedon its bottom boundary. Left andright boundariesare traction-free anda distributedtension applies on its upper boundary, given by

t

= (

0

,

a0

+

a1x).

Fig.2depicts,fora0

=

100 anda1

=

0,thedisplacementfield andtheYoungmodulusdistribution,whileFig.3depicts thedisplacementalongtheupperboundary.

From thereference elastic modelwe explore theparameter space

(a

0

,

a1

)

by computing realizationsgenerated from

Fig. 2. Displacement field (left) and Young modulus distribution (right).

Fig. 3. Displacement on the upper boundary.

0

,

a1

=

0

)

,

(a

0

=

100

,

a1

=

0

)

,

(a

0

=

100

,

a1

=

10

)

and

(a

0

=

0

,

a1

=

10

)

, defining four loading clusters around those points.

Thus,thenonlinearelasticmodelwasemployedforgeneratingthesyntheticdata.Inthepresentcase,itconsistsof trac-tionanddisplacementontheupperboundaryofthedomain.Forexample,fortheloadingclusteraround

(a

0

=

100

,

a1

=

0

)

, theinput(loading)andoutput(verticaldisplacement)dataareshowninFig.4.

From all the available data, the discrete, low-rank transfer function

T

LR _{is calculated at each loading cluster. Fig. 5}

depictsthe columnandrow vectorsinvolvedin the rank-two modelassociated withthe cluster

(a

0

=

100

,

a1

=

0

)

.The obtainedrank,two,correspondstotheexpectedone,sincetheloadingisdefinedinamanifold ofdimension2,consisting ofparametersa0anda1.

Whenevaluatingthemodelatanypointintheparametricdomain

(a

0

,

a1

)

,themodel

˜T

LRisinterpolatedfromthefour computedmodels

˜T

LR

₌

4



κ=1

T

_κLR

Iκ

(

a0

,

a1

)

(29)

Fig. 4. Top: input (applied tension). Bottom: output (vertical displacement).

Fig.6comparesthereferencesolutionandtheoneobtainedbyinterpolatingmodelsatpoint

(a

0

=

50

,

a1

=

5

)

,

˜T

LR,and thencomputingtheresponsefromU

= ˜T

LR_F.

3.2. Influenceofnoisydata

Inthepresentcasetheloadingwasrandomlyperturbed,aswellthecomputeddisplacementfield(usingtheunperturbed loading). AsmentionedinSection2,asimplefilteringprocedureconsistsintruncatingtheapproximation toanumberof

M

< N

terms. Our experience indicates that this simple procedure calculates first those modes with a lower frequency content,thusfilteringnoiseinpractice.

Fig. 7 compares column and row vectors involved in the discrete low-rank transfer function (whose dimensional-ity increased due to the noise but was truncated to rank 3) in absence of any noise filtering (i.e.

M

= N

) and when using

M

 N

. A performing filter capability is appreciated, with a beneficial effect on the accuracy of the predic-tions.

Fig. 5. Column and row vectors defining the discrete low-rank transfer functionTLR .

Fig. 7. Comparingcolumnsandrowvectorsinvolvedinthediscretelow-rankdiscretetransferfunctionwithM= N(top)and M Nenablingfiltering (bottom).

3.3. Equivalentneuralnetwork

It iseasy to transformthe proposed procedure into an equivalent artificial neural-network (ANN). Fig. 8compares a standard ANN with oneemulating the proposed reducedmodellearning, combined withthe modelinterpolation for in-ferring responses fromapplied inputs. The proposed NN tries to emulate a linear combinationof locally linear learned behaviors. AsdiscussedinSection2.2thelinearcombinationofbehaviorscanbeviewedasalinearizationofanonlinear behavior.

IntheproposedNNfourclusterswereconsidered(evenifforthesakeofclaritythefigureonlydepictstwo)representing the four locallylinear behaviors(two parameters –a0 anda1 –with toclasses each),while the output layerensures a bilinearinterpolationofthem.

ThestandardANNconsistsofthreeneuronlayers,theinputandoutputinvolving66andtheintermediatewith100.The intermediatelayerforthephysics-informedneuralnetworkcontained100 neuronspercluster.

BothNNweretrainedby300input/outputvectorsgeneratedt

= (

0

,

a0

+

a1x),witha0 anda1twouniformlydistributed randomnumbernumbersintheintervals

[

0

,

100

]

and

[−

10

,

10

]

respectively.

Fig. 9 compares the displacement predictions obtained by the trained neural networks, the standard one (ANN) and thephysics-informed onefora0

=

100 anda1

=

0,withthereferencesolution,fromwhichthesuperiorityofthe physics-informedNNcanbestressed.

Fig. 8. Standard one-layer NN (top) and the physics-informed one related to the low-rank reduced model (bottom).

4. Conclusions

The present work succeeded at proposing a newmethodology for learning reducedmodels froma smallamount of data by assuming a tensor decomposition ofthe discrete reduced model orits transfer function counterpart. It was ex-tendedtononlinearsettingsandprovedthattheassociatedphysics-informedNNexhibitsahigheraccuracythanastandard one.

Ourpresentworksinprogressconcernthevalidationandverificationaswellasitsextensiontoparametricsettings.

Acknowledgements

TheworkofE.C.hasbeenpartiallysupported bytheSpanishMinistryofEconomyandCompetitivenessthroughGrant numberDPI2017-85139-C2-1-RandbytheRegional GovernmentofAragon andtheEuropean SocialFund,research group T88. The other authors acknowledge the ANR (Agence nationale de la recherche, France) through its grant AAPG2018 DataBEST.

Fig. 9. Predictionsobtainedfromthestandardone-layerANNandthephysics-informedone(Low-RankReducedModel)relatedtothelow-rankreduced modelfora0=100 anda1=0.Twoclustersare depicted intheso-calledhiddenlayer,eveniftheNNcontainsfour.

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