Static and Dynamic Modelling of Materials Forging

CorynA.L. Bailer-Jones,David J.C.MacKayCavendishLaboratory, Universityof Cambridge

email: calj@mrao.cam.ac.uk;mackay@mrao.cam.ac.uk

TanyaJ.Sabin and Philip J.WithersDepartmentof MaterialsScienceandMetallurgy, Universityof Cambridge

email: tjs1005@cus.cam.ac.uk;pjw12@cus.cam.ac.uk

ABSTRACT

The ability to model the thermomechanicalprocessingof materialsis an increasinglyimportantrequirementin many areasof engineering. This is particularly true in the aerospaceindustry wherehigh materialandprocesscostsdemandmodelsthat canreliably predictthe microstructuresof forgedmaterials.We analysetwo typesof forging,cold forging in which themicrostructuredevelopsstaticallyuponannealing,andhot forgingfor which it developsdynamically, andpresenttwo differentmodelsforpredictingtheresultantmaterialmicrostructure.For thecold forging problemwe employ theGaussianprocessmodel.This probabilisticmodelcanbeseenasa generalisationof feedforwardneuralnetworkswith equally powerful interpolationcapabilities. However, as it lacks weightsand hiddenlayers, itavoids ad hoc decisionsregardinghow complex a ‘network’ needsto be. Resultsarepresentedwhichdemonstratetheexcellentgeneralisationcapabilitiesof thismodel.For thehot forgingproblemwehavedevelopedatypeof recurrentneuralnetworkarchitecturewhichmakespredictionsof thetimederivativesof statevariables.This approachallows us to simultaneouslymodelmultiple time seriesoperatingondifferenttimescalesandsampledatnon-constantrates.Thisarchitectureis verygeneralandlikely to becapableof modellingawide classof dynamicsystemsandprocesses.

Australian Journalon IntelligentInformationProcessingSystems, 5(1),10–16,1998

1. Introduction

Theproblemin the modellingof materialsforging canbe broadlystatedas follows: Given a certainmaterialwhich undergoesa specifiedforging process,what arethefinal propertiesof this material?Typical final prop-ertiesin which weareinterestedarethemicrostructuralproperties,suchas the meangrain sizeandshapeandthe extent of grain recrystallisation. Relevant forgingprocesscontrol variablesarethe strain,strainrateandtemperature,all of whichmaybefunctionsof time.

A trial-and-errorapproachto solving this problemhas often been taken in the materialsindustry, withmany differentforging conditionsattemptedto achieveagivenfinal product.Theobviousdrawbacksof thisap-proacharelargetimeandfinancialcostsandthelackofany reliablepredictivecapability. Anothermethodis todevelop a parameterised,physically-motivatedmodel,andto solvefor theparametersusingempiricaldata[2].However, the limitation with this approachis that intermsof the physicaltheorythe microstructuralevolu-tion dependsuponseveral “intermediate”microscopicvariableswhich have to be measuredin orderto applythemodel.Someof thesevariables,suchasdislocationdensity, are difficult and time-consumingto measure,

making it impracticableto apply suchan approachtolarge-scaleindustrialprocesses.

Our approachto thepredictionof forgedmicrostruc-tures is therefore to develop an empirical model inwhich we definea parameterised,non-linearrelation-ship betweenthe microstructuralvariablesof interestand thoseeasily measuredprocessvariables. Suchamodelcouldbe implemented,for example,asa neuralnetwork with thehiddennodesessentiallyplayingaroleanalogousto the“intermediate”microscopicvariables.

2. Materials Forging

When a material is deformed,potentialenergy is putinto the systemby virtue of work having beendonetomove crystalplanesrelative to oneanother. Themate-rial is thereforenot in equilibrium andhasa tendencyto lower its potentialenergy by atomicrearrangement,through the competingprocessesof recovery, recrys-tallisation and grain growth. Theseprocessesare en-couragedby raisingthetemperatureof thematerial(an-nealing). Forge deformationprocessescanbe dividedinto two classes.In cold working the recrystallisationrateis so low that recrystallisationessentiallydoesnotoccurduring forging. Recrystallisationis subsequently

10

Fig. 1: Deformationgeometries. (a) (left) Plane-straindiametricalcompression. The workpieceis subsequentlysectionedinto manynominally identicalspecimenswhich areannealedat differentcom-binationsof temperatureandtime. The compressiongives rise to anon-lineardistribution of strainsacrossthespecimen(seeFigure2b).Theseallow usto obtainmany input trainingvectors,� , for ourmodelusinga singlecompressiontest. (b) (right) Axisymmetricaxial com-pression.

achievedstaticallyby annealing.In contrast,hot work-ing refersto thehightemperatureforgingof materialsinwhich recrystallisationoccursdynamicallyduringforg-ing. This processis considerablymore complex thancoldworkingasnow thefinal microstructureof thema-terial is generallya path-dependentfunctionof thehis-tory of the processvariables. This is particularly trueof the Aluminium–Magnesiumalloy consideredhere,which have a relatively long ‘memory’ of the process,thusnecessitatingamodelwhichkeepstrackof thehis-tory of thematerial.We shallconsidera modelfor thisdynamicprocessin Section4.

The ultimategoal of forge modelling is the inverseproblem: Given a set of desiredfinal propertiesfor acomponent,what is the optimal material and forgingprocesswhich will realisetheseproperties?This is aconsiderablyharderproblemsincetheremaybea one-to-many mappingbetweenthe desiredpropertiesandthe necessaryforging process. This problemwill notbeaddressedin thispaper.

3. Static Modelling

Cold forgingcanin generalbemodelledwith theequa-tion ��������� (1)where� is amicrostructuralvariable,� is thesetof pro-cessvariablesand � is somenon-linearfunction. In ourparticularimplementationwe areinterestedin predict-ing asinglemicrostructuralvariable,namelygrainsize,in agivenmaterial(anAl-1%Mg alloy) asa functionofthetotalstrain,� , annealingtemperature, , andanneal-ing time, � . Theexperimentalset-upfor obtainingthesedatais asfollows. A workpieceof thematerialis com-pressedin plane-straincompressionat room tempera-ture,asshown in Figure1a.After thespecimenhasbeenannealed,it is etchedandthegrainsizesmeasuredwith

Fig. 2: (a) The left half of this diagramshows the microstructureofhalf of asectionedspecimenwhichhasbeendeformedunderaplane-strain compression.The materialhasbeenannealedat ��������� for30 mins producingmany recrystallisedgrains. (b) The right half ofthis diagramis the correspondingstrain contourmap producedbythe Finite Elementmodel. Note that the areasof high strain in (b)correspondto smallgrainsin (a).

an optical microscope.The local strainexperiencedateachpoint in thematerialis evaluatedusingaFiniteEl-ement(FE) model,the parametersof this modelbeingdeterminedby the known materialproperties,forginggeometries,friction factorsandsoon. Figure2b showsan exampleof an FE map. Many grain sizeswithina singlesmall areaareaveragedto give a meangrainsize. Thus we now have a set of model inputs, � , and � , associatedwith a singlemeangrain sizewhichcanbeusedto developastaticmicrostructuralmodelofforging. Furtherdetailsof the experimentalprocedurecanbefoundin Sabinetal. [9].

3.1. The Gaussian Process Model

The Gaussianprocessmodel[4] [10] assumesthat theprior joint probabilitydistribution of a setof any � ob-servationsis givenby an � -dimensionalGaussian,i.e.� �������� ���"!$#&%'#)(*�+ (2), -�.0/�1�2435 ����627%8:9;(=� ����@2A%8CBD# (3)where ��� = �E� > �E� > �#F�HGI���GJ�#�KLK�KL#F�H������"F is the set� of observationscorrespondingto the set of � in-put vectors, �L� � !NMO�L� > #F� G #�K�KLK�#F� � ! . % and ( � ,respectively themeanandcovariancematrix for thedis-tribution, parameterisethis model. Theelementsof thecovariancematrix arespecifiedby thecovariancefunc-tion, which is a function of the input vectors, � � � ! ,and a set of hyperparameters. A typical form of thecovariancefunctionisPRQ�S �@T > -�.0/VU&2435"WYX�Z[ WYX >

�E\�] WY^Q 27\�] W_^S G` GWa=b TJG b Tdc�e Q�S K

(4)Thisequationgivesthecovariancebetweenany two val-ues � Q and � S with correspondingf -dimensionalinputvectors� Q an � S respectively, andis capableof imple-mentinga wide classof functions,� , thatcouldappearin equation1. (TheGaussianprocessmodelhasascalar‘output’, � ; to modelseveral microstructuralvariableswe would useseveral independentmodels.) The first

11

termin equation4 expressesourbelief thatthefunctionwe aremodellingis smoothlyvarying, where ` W is thelengthscaleoverwhich thefunctionvariesin the gEhEi in-putdimension.Thesecondtermallows thefunctionstohave a constantoffsetandthethird is a noiseterm: thisparticularform is a modelfor input independentGaus-siannoise.Thehyperparameters,` W ( g = 3 K�KLK f ), T > , T G ,T c , specifythefunction,andaregenerallyinferredfromasetof trainingdatain a fashionanalogousto traininganeuralnetwork. They arecalledhyperparametersratherthanparameters becausethey explicitly parameteriseaprobability distribution rather than the function itself.This distinguishesthemfrom weightsin a neuralnet-work, which are rather“arbitrary”, in that addingan-otherhiddennodecould changethe weightsyet leavetheinput–outputmappingessentiallyunaltered.

Oncethe hyperparametersareknown, the probabil-ity distribution of a new predictedvalue, �H�Rj > , corre-spondingto anew ‘input’ variable,���kj > , is� �E� �Rj > � � � #��L� � !I#&� �Rj > #)( �kj > (5), -�.0/mln2 �E�H�kj > 2po�H�kj > G5Hq G rsut?vxw y # (6)i.e. a one-dimensionalGaussian,where o�H�kj > andq rs t?vxw areevaluatedin termsof thecovariancefunctionand the training data. We would typically report ourpredictionas o�H�kj >kz q rs t?vxw . Theseerrorsreflectboththe noisein the data(third term in equation4) andthemodeluncertaintyin interpolatingthetrainingdata.ThefactthattheGaussianprocessmodelnaturallyproducesconfidenceintervals on its predictionsis important inthe materialsindustry wherematerialpropertiesmustoftenbespecifiedwithin certaintolerances.

Our modelassumesthat the measurementnoiseandthe prior probability of the unknown function can bedescribedby a Gaussiandistribution. In our applicationit is moresensibleto assumethat it is the logarithmofgrain sizeswhich aredistributedasa Gaussian,ratherthan the grain sizesthemselves. This is becauseun-certaintiesin measuringgrainsizescalewith themeangrain size, and are thereforemore appropriatelyex-pressedasafractionof themeangrainsizeratherthanafixedabsolutegrainsize.Moreover, empiricalevidencesuggeststhatgrainsizedistributionsarewell describedby a log normaldistribution [7].

3.2. Model Predictions

A Gaussianprocessmodelwastrainedusingasetof 46datapairsobtainedfrom theplane-straingeometry, with{ K {}|~ � ~�{ Kd , 5I} C ~ ~ } C, 1 mins ~ � ~60minsastheinputs.Oncetrained,themodelwasusedto producepredictionsof grainsizesfor a rangeof theinput variables.Thesepredictions,shown in Figure3,agreewell with metallurgicalexpectations.

Oneof theassumptionsimplicit in ourmodelof coldforging is thatgiventhe local strainconditions,themi-crostructureis independentof the materialshapeand

Fig. 3: Grain size predictionsobtainedwith the Gaussianprocessmodel trainedon datafrom the plane-straincompressiongeometry.In eachof thethreeplots,two of theinput variablesareheldconstantand the other varied. When not being varied, the inputs wereheldconstantat: @@����� � C; R@��� mins; @� � . The crossesinthestrainplot aredatafrom thetrainingset.As theGaussianprocessis an interpolationmodel,predictionsat any valuesof the inputsareconstrainedby theentiretrainingset.

forgegeometry. In otherwords,we assumethatpredic-tionscanbeobtainedgivenonly the local accumulatedstrain (and annealingconditions). This is an impor-tant requirementasit meansthat a singlemodelcouldbe appliedto a rangeof industrial forging geometries,provided that the local strainscould be obtained(e.g.with an FE model). We testedthe validity of this as-sumptionby usingthe Gaussianprocessmodeltrainedon plane-straindata to predict grain sizesin a mate-rial compressedusinga differentgeometry, namelyanaxial compression(Figure1b). As before,after com-pressionthematerialwasannealed,sectionedandgrainsizesmeasured.A new FE modelgavetheconcomitantlocal strains. Theseprocessinputs were then usedtoobtainpredictionsof thegrainsizesusingthe previousGaussianprocessmodel. Figure 4 plots thesepredic-tions againstthe measurements.We seeremarkableagreement—wellwithin thepredictederrors—thusval-idatingourmodellingapproach.A practicalapplicationof ourmodelis to producediagramssuchasthatshownin Figure5, a mapof thegrainsizes.Sucha mapis im-portantfor engineerswho needto know thegrainsizesatdifferentpointsin thematerial,andcanthusassessitsresistanceto phenomenasuchascreepandfatigue.

It should be noted that this methodcontainsotherimplicit assumptions.Thefinal materialmicrostructureis very stronglydependentuponthe materialcomposi-tion. It is well known that even small changesin thefractionsof thealloying constituents(andby extension,impurities)canhave a strongeffect on the thermome-

12

Fig. 4: Gaussianprocessmodelpredictionscomparedwith measuredvalues. The Gaussianprocessmodelwas trainedon datafrom onecompressionalgeometry(plane-strain)andits performanceevaluatedusingdatafrom anothergeometry(axial-compression)whichwasnotseenduring training. The line is to guidethe eye. Note thatnot even a perfectmodelwould producepredictionson this line dueto finite noisein thedata.

Fig. 5: The left half is an imageof the microstructurein the axiallycompressedspecimen.Theright half is thecorrespondinggrainsizepredictionsfrom theGaussianprocessmodelshown asacontourmap.

chanicalprocessingof the material. Oneway forwardis to include further input variablescorrespondingtocomposition[3]. A secondimplicit assumptionhasbeenthe constancy of the initial microstructure.Dependinguponthe materialandthe degreeof thermomechanicalprocessing,the final microstructuremay retain some‘memory’ of its initial microstructure,thusnecessitat-ing amodelwhichhas“initial conditions”asadditionalinputvariables.

4. A Recurrent Neural Network forDynamic Process Modelling

4.1. Model Description

For thehot working problem,we assumethat therearetwo setsof variableswhich are relevant in describingthe behaviour of the dynamicalsystem. The first, � ,areexternalvariableswhich influencethebehaviour ofthe system,suchasthe strain,strainrateandtempera-ture. It is assumedthat all of thesecanbe measured.The secondset of variables, � , are the statevariableswhich describethe systemitself. Theseare split into

1.0

1.0

 ¡ ¢ £ ¢ ¤ Fig. 6: A recurrentneuralnetwork architecture(‘dynet’) for mod-elling dynamicalsystems.The outputs, , from the network arethetimederivativesof thestatevariablesof thedynamicalsystem.There-currentinputs, ¥ , arethesestatevariables.Thevaluesof ¥ at thenexttime stepareevaluated(usingequation9) from the outputs(via therecurrentconnections)andtheprevious valuesof ¥ . All connectionsandthetwo biasnodesareshown.

two categories. The first are measured,suchas grainsize,andthe secondareunmeasured,suchasdisloca-tion density. Notethattheunmeasuredvariablesarenotintrinsicallyunmeasurable:this is simplyacategory forall of thestatevariableswhichwebelieveto berelevantbut which, for whatever reason,we donot measure.

Both � and� arefunctionsof time. A generaldynam-ical equationwhich describesthetemporalevolutionofthe statevariablesin responseto the externalvariablesis ¦ �§� � ¦ � ��¨"�E�§� � �#F�©� � FV# (7)where ¨ is somenon-linearfunction. To a first-orderapproximation,wecanwrite�§� � b e � ©�@�§� � b ¦ �§� � ¦ � e � K (8)This dynamicalsystemcanbemodelledwith therecur-rentneuralnetworkarchitectureshown in Figure6. Thisis a discretetime network in which the input dataareprovidedasadiscretelist of valuesseparatedby knowntime intervals. The input–outputmappingof this net-work implementsequation7 directly: Ratherthanpro-ducingthestatevariablesattheoutputof thenetwork,asis often the casewith recurrentnetworks (e.g.[8]), weproducethe time derivativesof the statevariables,forreasonsthatareexpoundedonbelow. Thehiddennodescomputea non-linearfunctionof both theexternalandtherecurrentinputswith a sigmoidfunction(e.g.tanh),asconventionallyusedin feedforwardnetworks. A lin-earhidden–outputfunction is usedto allow for an ar-bitrary scaleof the outputs. The recurrentpart of thedynamicalsystem,viz. equation8, is implementedwiththe recurrentloops shown in Figure 6, by settingtheweightsof theserecurrentloopsto the sizeof the time

13

step, e � , betweensuccessive epochs.Explicity, the ªxhEirecurrentinput at timestep« is givenby�}¬0� « §���}¬� « 2 3 b® ¬x� « 2 3 Fe � � « (9)where

® ¬ � « 2 3 ©� ¦ �§� « 2 3 &¯ ¦ � and e � � « is thetimebetweenepoch � « 2 3 andepoch � « .

Theprincipalreasonfor developinga network whichpredictsthetimederivativesof thestatevariablesis thatit canbe trainedon time-seriesdatain which thesepa-rationsbetweentheepochs,e � � « , neednotbeconstant:at eachepoch« we simply settheweightsof therecur-rentfeedbackloopsto e � � « . Furthermore,thenetworkcanbetrainedonmultiple timeseriesin which thetimescalesfor eachtime seriesmaybevery different. Thisis important in forging applicationsas the forging oflargecomponentswould occurovera longertime scalethanfor smallcomponents,whereasthemicroscopicbe-haviour of the materialswould essentiallybe the same(for a given material). In sucha casewe would wantto incorporatedata from both forgings into the samemodel, but without having to obtain measurementsatthesameratein bothcases.

While our network is similar to that of Jordan[5],our architecturehastheimportantattributesthat: 1) theoutputsare time derivativesof the statevariables,and2) in training the network the error derivativescanbepropagatedvia therecurrentconnectionsto thearbitrar-ily distantpast. Our training algorithmcanbe seenasageneralisationof themethoddescribedby Williams &Zipser[11] extendedto multiple time series.Althoughnecessarilyonly the feedforwardweightsaretrainable,theinput–hiddenweights,for example,arenonethelessdependentuponthevaluesof thehiddennodesby virtueof the recurrentconnections,and this dependency istaken into account. Training proceedsby minimizinganerrorfunction,typically thesumof squareserror, bygradientdescentor aconjugategradientalgorithm.Theweightscanbe updatedafter eachepochof eachtime-series(i.e.RealTimeRecurrentLearning[11]), afterallepochsof all patterns,or at any intermediatepoint.

We use a Bayesianimplementationof the trainingprocedure[6]: multiple weight decayconstantsreg-ularize the determinationof the weights,and a noiseterm specifiesthe assumednoise levels in the targets( � ). Someform of regularisationis probablyessentialon accountof the many intermediateoutputvaluesforwhich their areno targets.Thenoiseandweightdecayhyperparametersarecurrentlysetby hand,but couldinprinciplebeinferredfrom thedata.

To train thenetwork weneedat leastonetargetvalueatat leastoneepoch.Notethatthetrainingalgorithmisnotrestrictedto usetargetsonly for the‘outputs’: errorscanbepropagatedfrom any node.Generallywe wouldhave valuesof the statevariables(recurrentinputs)forthefinal epoch.However, in metallurgical applicationswe may sometimesbe able to obtain additionalmea-surementsatsomeintermediateepochs,thusimprovingthe accuracy of the derived input–outputfunction. We

will of coursenot have any target valuesfor the ‘un-measured’statevariables. Hencethesevariablesmaynot even correspondto any physicalvariables,insteadacting as ‘hidden’ variableswhich convey somestateinformationnot containedin the ‘measured’statevari-ables. Nonethelesswe may be able to provide someloosephysicalinterpretationfor unmeasuredvariables.

Oncetrained,thenetwork producesa completetimesequencefor all statevariablesgivena sequenceof theexternalinputs,i.e. thedetailsof theforgingprocess.

4.2. Model Demonstration

We now demonstratethe performanceof the modelononeparticularsyntheticproblemin which therearetwoexternalinput variables,\ > � � and \°GI� � , andtwo statevariables,� > � � and �HG}� � , definedby¦ � >¦ � � \ > 2 5 � > b | � G 27\ > � > (10)¦ �HG¦ � � \ G 2 � > b � G 2m\ G � G K (11)To mimic realprocessesin whichtheexternalinputsareoften constrainedto be positive (e.g. temperature),the\ > and \ G sequencesweregeneratedfrom constrainedrandomwalks: at eachepoch, \ > ( \ G ) changeswith aprobabilityof 0.1 (0.5) by a randomamountuniformlydistributedbetween2 { K and b { K ( 2 3 and b 3 ). Themodulusof \ is takento ensureapositivesequence.Theinitial � valueswererandomlyselectedfrom a uniformdistributionbetween2 3 and b 3 .

Equations10 and11 arenot analyticallysolvable,sowe usedthe first-orderTaylor expansionin equation8to evaluatethe � > � � and �HGI� � sequences.A very smallvalueof e � wasusedto yield anaccurateapproximationto thetruecontinuoussequence.

Onehundredsynthetictime seriesweregeneratedinthis fashion,with differentsequencesfor \ > and \±G anddifferent initial valuesof � > and �HG . The sequencesweregeneratedfrom � = 0 to � = 8 inclusive. We choseto samplethesesequencesat a constantepochsepara-tion of e � = 0.1,giving a sequencelengthof 81 epochs.This time extent andvalueof e � samplesthe dynami-cal sequencewell: If all \ dependenceis removed inequations10and11,weareleft with dampedsinusoidaloscillationsin � > and � G , with oscillationperiod1 and² dampingtimescale2. The \ termsmake theseriessomewhatmorecomplex, but theoveralloscillatorybe-haviour is retained.

Thenetwork wastrainedon 50 of thetime series.Indoing so, the network was only given the initial andfinal � valuesand the complete \ sequencesfor eachtime series: it did not seethe intermediate� values.Oncetrained,the network wasappliedto the other50sequences,i.e. given the initial � valuesand the \ se-quences,thenetwork producessequencesfor � . We seefrom Figure7 that thenetwork makesexcellentpredic-tionsof � > and �HG at thefinal epoch.

14

Fig. 7: Network predictionsof the final valuesof the ¥ ³ and ¥�´ se-quencesfor the 50 time seriesin the testdatasetplottedagainstthetarget values. The rms errorsare0.0067and0.0084for ¥ ³ and ¥�´respectively.

It is now interestingto askwhetherthe network hasmanagedto correctly learn the entire � sequencesforthe test data,or whetherit has learnedsomesimplerinput–outputfunctionwhich just givescorrect � valuesatthefinal epoch.Figure8 showsthe \ inputsequences,and the predicted� sequencesin comparisonwith thetruesequences:Theagreementis generallyvery good.Someof thepredictedsequences(suchastheloweroneshown) deviatefrom thetruesequenceat earlyepochs.This is probablydueto the differencebetweenthe sta-tionarydistributionsof thestatevariablesat � =8 andatearly epochs:The network is explicity trainedto givepredictionsfor the target valuesof � . Theselie in therangesshown in Figure 8, which is narrower than therangeof � valuesat � = 0. Therefore,we would notexpectthenetwork to give suchgoodpredictionsin thenon-overlappingpartof this latterrange.As thedynam-ical systemis damped,the poorerpredictionstend tooccurearlieron. Nonetheless,we seethat thenetworkhaslearnedagoodapproximationof thetrueunderlyingdynamicalsequencefor therangeof � valuespresentinthetargets.

The network hasbeenappliedto a numberof othersyntheticproblems(includingoneswith non-linear, e.g.� > �HG , terms),anda goodpredicitive capablility is ob-

Fig. 8: Externalinput andstatevariablesequencesfor two differenttime seriesin the testdataset. In eachplot the solid lines from topto bottomare ¥ ³ , ¥�´ , x³ and }´ . For ¥ ³ and ¥�´ the solid lines arethe nework predictionsandthe dashedlines the true sequences.Forclarity, the sequencesfor ¥�´ , x³ and }´ have beenoffset from theirtruepositionsby -1, -3 and-6 respectively.

served. The network hasalsobeenable to learnfromtraining data sets in which e � varies within eachse-quence,andthelengthsof thesequencesdiffer [1].

5. Summary

We have introducedtwo techniquesfor modellingma-terialsforging,onein thestaticdomainrelevantto coldforging,andanotherin thedynamicdomainrelevanttohot forging. The Gaussianprocessmodel usedin theformer casehasbeendemonstratedto provide a goodpredictive capability, even when using a small, noisydataset.Furthermore,thesufficiency of our input vari-ables(for agivenmaterial)hasbeendemonstratedfromthe successfulapplicationof the trainedGaussianpro-cessmodelto adifferentforginggeometry.

For the hot forging problem we have introducedageneralneuralnetwork architecturefor modelling dy-namical processes. The power of this network wasdemonstratedon a syntheticproblem. Furtherdetailsof this model, as well as its applicationto a rangeofsyntheticproblems,will be presentedin a forthcomingpaper[1]. Futurework with thismodelwill focuson itsapplicationto realdatasetsobtainedfrom hot forging.

15

Acknoµ

wledgements

The authorsaregrateful to the EPSRC(grantnumberGR/L10239),DERA andINCO Alloys Ltd. for finan-cial supportandto Mark Gibbsfor useof his Gaussianprocesssoftware.

References

[1] C.A.L. Bailer-Jones,D.J.C.MacKay, in prepara-tion, 1998.

[2] T. Furu, H.R. Shercliff, C.M. Sellars, M.F.Ashby, “Physically-basedmodelling of strength,microstructureand recrystallisationduring ther-momechanicalprocessingof Al–Mg alloys”, Ma-terials Sci.Forum, 217–222,pp.453–458,1996.

[3] L. Gavard,H.K.D.H. Bhadeshia,D.J.C.MacKay,S. Suzuki, “Bayesianneural network model forausteniteformationin steels”,MaterialsSci.Tech-nol., vol. 12,pp.453–463,1996.

[4] M.N. Gibbs,BayesianGaussianprocessesfor re-gressionandclassification. PhDthesis,Universityof Cambridge,1997.

[5] M.I. Jordan,“Attractor dynamicsandparallelismin a connectionistsequentialmachine”,Proc. ofthe Eight Ann. Conf. of the Cognitive Sci. Soc.,Hillsdale,NJ:Erlbaum,1986.

[6] D.J.C.MacKay, “Probablenetworksandplausiblepredictions:a review of practicalBayesianmeth-ods for supervisedneural networks”, Network:Computationin Neural Systems, vol. 6, pp. 469–505,1995.

[7] F.N. Rhines,B.R. Patterson,“Effect of thedegreeof prior coldwork onthegrainvolumedistributionandthe rateof graingrowth of recrystallisedalu-minium”, Metallurgical TransationsA, vol. 13A,pp.985–993,1982.

[8] A.J. Robinson, F. Fallside, “A recurrent errorpropagationnetwork speechrecognitionsystem”,ComputerSpeech andLanguage, vol. 5, pp. 259–274,1991.

[9] T.J. Sabin, C.A.L. Bailer-Jones,S.M. Roberts,D.J.C.MacKay, P.J.Withers,“Modelling theevo-lution of microstructuresin cold-worked andan-nealedaluminiumalloy”, in T. Chandra,T. Sakai(eds),Proc. of the Int. Conf. on Thermomechani-cal Processing, Pennsylvania,PA: The Minerals,MetalsandMaterialsSociety, 1997.

[10] C.K.I. Williams, C.E.Rasmussen,“Gaussianpro-cessesfor regression”, in D.S. Touretzky, M.C.Mozer, M.E. Hasselmo(eds),Neural InformationProcessingSystems8, Boston,MA: MIT Press,1996.

[11] R.J. Williams, D. Zipser, “A learning algorithmfor continuallyrunningfully recurrentneuralnet-works”, Neural Computation, vol. 1, pp.270–280,1989.

16