Modeling electrical properties for various geometries of antidots on a superconducting film

13 Pages·2017·2.38 MB·English

by  Sajjad Ali Haider

ApplNanosci(2017)7:933–945
https://doi.org/10.1007/s13204-017-0633-4
ORIGINAL ARTICLE
Modeling electrical properties for various geometries of antidots
on a superconducting film
Sajjad Ali Haider1 • Syed Rameez Naqvi1 • Tallha Akram1 • Muhammad Kamran1 •
Nadia Nawaz Qadri1
Received:28September2017/Accepted:9November2017/Publishedonline:17November2017
(cid:2)TheAuthor(s)2017.Thisarticleisanopenaccesspublication
Abstract Electrical properties, specifically critical current Introduction
density, of a superconducting film carry a substantial
importanceinsuperconductivity.Inthiswork,wemeasure Various geometries of artificial pinning centers with vortex
andstudythecurrent–voltagecurvesforasuperconducting latticehavebeeninvestigatedtoobservethebehaviorofvortex
Nb film with various geometries of antidots to tune the motion and pinning interaction (Baert 1995; Cuppens et al.
critical current. We carry out the measurements on a 2011;Heet al.2012;Jaccardet al.1998;Kamranet al.2015;
commercially available physical property measurement de Lara et al. 2010; Latimer et al. 2012; Martin et al.
system to obtain these so-called transport measurements. 1999,1997).Allthesestudieshaveresultedinimprovingthe
Weshowthateachoftheusedgeometriesexhibitsavastly currentcarryingpropertiesofsuperconductingmaterialsforuse
different critical current, due to which repeatedly per- applications. The critical current density is one of the most
forming the measurements independently for each geom- importantpropertiesofsuperconductors.Thesuperconducting
etry becomes indispensable. To circumvent this filmwithanarrayofartificialpinningcentersbynanolithog-
monotonous measurement procedure, we also propose a raphytechniquescanenhancethecurrentdensity.Ifeachdefect
framework based on artificial neural networks to predict accommodatesoneflux,thenthecurrentdensityincreasesand
the curves for different geometries using a small subset of resistancedecreases,butasthevortexlatticebecomesdisorder,
measurements, and facilitate extrapolation of these curves thecurrentdensitydecreasesandresistanceincreases.Inthis
overawiderangeofparametersincludingtemperatureand way, the current voltage (IV) characteristics or curves of the
magnetic field. The predicted curves are then cross- pinningcentersonasuperconductingfilmshowabruptchan-
checked using the physical measurements; our results ges.SuchabruptchangesgivetheIVcurvesmeasurement,also
suggest a negligible mean-squared error—in the order of calledthetransportmeasurement,asubstantialprominencein
10(cid:2)9. thefieldofsuperconductivity.
During our experimental work, it has been recognized
Keywords Critical fields (cid:3) Transport properties (cid:3) Critical thattransportmeasurementsarenotoriousandcumbersome
currents (cid:3) Vortex pinning (cid:3) Artificial neural networks to measure, especially when they are repeatedly needed
(Kamran et al. 2016). We believe that an approximation
model, such as one based on artificial neural networks
(ANN) proposed in this work, can help in evading the
repeated measurements, and rather extrapolate the curves
using their smaller subset for unforeseen parameters. The
proposed methodology not just relieves the researchers
from this tedious procedure—saving them time, cost, and
& SyedRameezNaqvi energy, but is applicable to various geometries of anti-
[email protected]
dots—giving this study a significant importance.
1 DepartmentofElectricalEngineering,COMSATSInstitute ANNaresaid tobemathematicalequivalentsofa human
ofInformationTechnology,G.T.Road,WahCantt,Pakistan brainthattendtolearninonesituation,andrepeatinanother
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934 ApplNanosci(2017)7:933–945
(Guojinet al.2007).Theyaresupposedtorecognizepatterns, thicknessisdepositedonanS0 substrateforobtainingthe
i 2
andestablishrelationshipsbetweenindependentvariablesofa desired arrays of antidots. The micro bridges and nanos-
smallersubsetofvalues.Theserelationshipsarethenusedin tructured array of antidots are, respectively, fabricated by
solving problems thatneed approximation and prediction for photo- and e-beam lithography techniques on a resistive
a larger sample space having unforeseen values, where the layer of polymethyl metacrystalline (Mostako and Alika
ANN are typically found extremely useful (Cybenko 1989). 2012; Shahid et al. 2016). Fabrication of the microbridges
ANN may simply be described as a network of intercon- and array is followed by scanning of the samples to the
nected nodes, each having some weight and bias—called desired antidots using scanning electron microscopy
network coefficients. The purpose of such a network is to (SEM). We subsequently mount our sample in the PPMS
providemappingbetweeninputsandoutputs,wherethelatter for transportmeasurements, which are carriedoutby four-
areobtainedafterasuccessfullearningphase.Theprocessof probemethodinPPMSwithtemperaturefluctuationwithin
learning comprises training and validation. In the training ±3 mK, and the external magnetic field applied perpen-
phase, several pairs of inputs and outputs are iteratively dicular to the plane of the film. We have also swept the
provided to the network to enable the ANN to establish a fieldfrom-8to8 mAatconstanttemperature.Duringthe
mathematical relationship. The coefficients are updated in entiremeasurementprocess,whichisalwayscarriedoutin
each iteration until the network converges to an optimal high vacuum, a small amount of liquid helium is placed
solution.Themanner,inwhichthesecoefficientsareupdated, inside the chamber to prevent overheating.
distinguishes one training algorithm from the rest (Hornik Figure 1 presents SEM of the four geometries that we
1991, 1993; Hornik et al. 1989). investigate in this work, and Fig. 2 corresponds to their
While the ANN have been widely ratified in engineering respective IV curves measured at different values of tem-
applications(Elminiret al.2007;Ghanbariet al.2009;Reyen perature. The top left, top right, bottom left, and bottom
et al.2008),theiruseinthefieldofmaterialscience,especially right in both figures, respectively, correspond to the rect-
forthepurposeofmodelingelectricalproperties,isinordinately angular, square, honeycomb, and kagome arrays of anti-
constrained (Haider et al. 2017; Kamran et al. 2016; Quan dots. These curves may be divided into three regions
et al. 2016; Zhao et al. 2016). In this work, we explore the according to their slopes: in the first region, the voltage is
predictionofIVcurvesofasuperconductingfilmwithvarious almostzeroforgraduallyincreasingcurrent,followedbya
geometries of antidots using three commonly used ANN sudden jump in the voltage in the second region, and
architectures,trainedbyvariouslearningalgorithms.Thepre- finally,inthethirdregion,thereexistsalinearrelationship
dictedresultsarethencomparedwiththeactualmeasurements; between the two variables. Two important observations
thisstepistermedvalidation.Althoughslightlydifferentfrom that may be conveniently made from these figures are:
eachother,ourapproximatedresultsfromeachtrainingalgo-
1. The IV curves show a sudden jump at critical current
rithm achieve high accuracy—smaller mean-squared error
(I ) in the second region, which resembles Shapiro
(MSE). Besides describing the approximation methodology, c
steps. These steps usually appear when the interstitial
this work is intended to present a comparison between three
vortex lattice is formed, and due to high vortex
ANN architectures, and three training algorithms in terms of
velocities, instability may occur; as a result of which
predictionaccuracy(giveninMSE),trainingtime,andnumber
the system shows a step.
ofiterationstakentoconvergetoanoptimalsolution.
2. Thesharpnessinthecurvessignificantlyvariesforeach
The rest of the article is organized as follows: Sec-
geometry: those having larger interstitial area may
tion‘‘Physicalmeasurementsystemandreadings’’presents
accommodate a larger number of vortices, leading to
detail of the experimental setup to obtain the transport
increased energy conservation in those geometries.
measurements, and a brief commentary on their charac-
Therefore, the honeycomb and kagome arrays will
teristics. In Section ‘‘Artificial neural network model’’, we
exhibit flatter or smoother curves in comparison with
present the approximation methodology based on ANN,
sharpstepsforrectangularandsquarearraysofantidots.
followed by the results and discussions in Section ‘‘Re-
search methodology and simulation results’’. We conclude After successively performing the transport measure-
the paper in Section ‘‘Conclusion’’. ments, we had obtained a three-dimensional (H, T, I) data
setcomprising½4(cid:4)4(cid:4)1600(cid:5)valuesforeachfilmhavinga
different geometry.Note that we have takenoutone curve
Physical measurement system and readings from this data set for each geometry, and kept it isolated
from the entire ANN modeling process. These four curves
Our experimental setup for transport measurements pri- should be used for cross-checking our approach on, so to
marily comprises a physical properties measurement sys- say, unforeseen data values, once the system had been
tem (PPMS) from quantum design. An Nb film of 16 nm completely designed using the MATLAB’s ANN toolbox
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Fig.1 SEMofrectangular(top
left),square(topright),
honeycomb(bottomleft),and
kagome(bottomright)arrays
Fig.2 Transportmeasurementsatvarioustemperaturesandzeromagneticflux
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936 ApplNanosci(2017)7:933–945
on the modified data set (the one excluding the curves
extracted for cross-checking). The toolbox, by default,
divides the provided data set into three: the first subset is
usedforthetrainingpurpose,50%oftheremainingvalues
isusedforvalidation,whiletherestisstrictlykeptisolated,
which is then used for the testing purposes. The modified
data set values were still copious enough to give us con-
fidence in allocating a large data set exclusively for the
training purpose. However, while performing the simula-
tions, we realized that increasing the size of training set
beyond70%wouldnotgiveusaconsiderableadvantagein
terms of prediction accuracy. MATLAB’s ANN toolbox
alsousesseventypercentofthedatavaluesforthetraining
purpose by default, which further justifies our selection of
training and testing data sets. In the next section, we
elaborate on the ANN’s operation principle, training Fig.3 Structureoffullyconnectedfeed-forwardneuralnetworkwith
algorithms, and architectures used in this work. asinglehiddenlayer
Selectionofthenumberofhiddenlayersandthenumber
Artificial neural network model of neurons per layer are a critical process, having a high
impactonthesystemsstability.MostoftheavailableANN
training algorithms utilize MSE (difference between the
The structure and operation of a neuron—the basic building
expected and observed responses) as their objective
block of ANN—have been described on several occasions
function:
(Guclu et al. 2015). Briefly, it is a cascaded design, from an
input layer to an output layer, where functional blocks are 1XM 1XM
sandwiched either between input and hidden, or hidden and /¼2 ðyk(cid:2)dkÞ¼2 e2k ð2Þ
output layers, where each layer may comprise a specific k¼1 k¼1
number of neurons. It is believed that the number of hidden where y is the kth output value calculated by d , and it
k k
layers in a network is directly proportional to the prediction represents the expected value. To the best of our knowl-
accuracy, i.e., the greater the number of hidden layers, the edge, all the ANN architectures follow backpropagation
more accurate the results will be at the expense of com- technique to minimize their objective function—from the
plexity. However, problems similar to the one addressed in basicfeed-forwardneuralnetworktothosewidelyadopted
this work merely require up to two hidden layers for an architectures,suchasconvolutionalneuralnetwork(CNN),
acceptable accuracy level in most cases (Setti et al. 2014). allthetrainingalgorithmsbackpropagatetheirerrorinthe
The mapping between input and output layers, achieved form of sensitivities from the output to input layer.
throughafewhiddenlayers,mayfollowoneofthethreemost
widely adopted architectures: feedforward, cascaded, and ANN architectures
layer-recurrentneural nets. While thedifferencebetween the
three architectures will be highlighted later, in what follows Feedforward
wemakeuseofthesimplestone,feedforward,justtodescribe
the operation of ANN. Figure 3 depicts a fully connected The simplest architecture that an ANN model may follow
feed-forward neural net with a single hidden layer (dH). R is termed afeed-forward neuralnet, inwhich each layer is
number of inputs are connected to input layer (dI) with S only connected to its immediate neighbors. The mapping
numberofneurons,whereasoutputsgeneratedbyinputlayer from input through the hidden to the output layers, there-
acts as a source for the hidden layer, having T number of fore, is achieved in a serial and linear manner.
neurons. Here, d is termed activation or threshold function,
whichinawayquantizetheoutputofthenetwork.Themost Cascaded
commonlyusedthresholdfunctionsarestep,linear,andtan-
sigmoid (hyperbolic tangent sigmoid): Unlikethefeed-forwardnets,theoutputlayerinacascaded
!! networkisnotonlyconnectedtoitsimmediatepredecessor
M M
Y ¼d Xwyd XwHP : ð1Þ (hidden)layer,butalsohasadirectconnectiontotheinput
k kl H ij R
j¼0 j¼0 layer.Thisallowsthenetworktoexploittheinitialweights
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ApplNanosci(2017)7:933–945 937
in addition to those provided by the predecessor to facili- o/ðkÞ
rwðkÞ¼(cid:2)a
tate optimal convergence at the cost of added complexity. owðkÞ
where a is the learning parameter.
Layer-recurrent
Because of its broad range of contributions, several
variantsofbackpropagationalgorithmhavebeenproposed.
Unlike the feed-forward nets, the layer-recurrent nets are
Although each variant has its own pros and cons, in what
nonlinear, i.e., they have a feedback loop in the hidden
follows, we discuss only the ones that have proven their
layers with additional tap delays. The latter especially
efficiency for the problems as such addressed in the pro-
prove helpful in analyzing time-series data, where the
posed work (Haider et al. 2017).
network is supposed to have a dynamic response.
Parts (a), (b), and (c) in Fig. 4 depict feed-forward,
Levenberg Marquardt framework
cascaded, and layer-recurrent neural nets, respectively,
where circles in (c) depict the additional tap delays.
The Levenberg Marquardt (LM) algorithm is a pseudo-
second-order training algorithm, which works in conjunc-
Benchmark backpropagation algorithms
tionwiththesteepest descentmethod. Ithas beenreported
that this algorithm promises better stability and conver-
The backpropagation technique is an iterative method,
gence speed (Levenberg et al. 1944).
which works in conjunction with gradient descent algo-
Let us consider the output response of feedforward
rithm (Reed et al. 1993). While in each iteration, the net-
neuralnetwork,calculatedusingEq. 1,whereinitialoutput
work coefficients are updated, the method continues to
response is given as y ¼r . The network error is calcu-
compute gradient of the cost function accordingly. The 0 k
lated using Eq. 2.
objective of this iterative method is to minimize the cost
The network sensitivities are backpropagated through
function in terms of MSE:
the network to update the learning rules (Demuth et al.
o/ðwÞ
r/ðwÞ¼ ¼0 8 j ð3Þ 2014). Derived from the Newton algorithm and steepest
ow
j descent method, the update rule for LM algorithm is
defined as
update rule:
wðkþ1Þ¼wðkÞþrwðkÞ DW ¼(cid:2)JwTeJweþdrI(cid:3)(cid:2)1JwTee ð4Þ
where or the above equation can be written as
Fig.4 ANNarchitectures:
Hidden layer 1 Output
afeedforward,bcascaded,and
clayerrecurrent
w w
Output
Input
b b
(a)
w
w
Input w Output
b
b
(b)
1 w
w
1:k w Output
Input
b
b
(c)
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938 ApplNanosci(2017)7:933–945
Dx ¼(cid:2)(cid:4)JT ðx ÞJ ðx Þþd I(cid:5)(cid:2)1JT ðx Þvðx Þ ð5Þ (CGF) (Naqvi et al. 2016; Johansson et al. 1991; Powell
k we k we k r we k k
1977).
where J has dimensions (P(cid:4)Q(cid:4)R) and error vector is Letusconsidersetofinputvectorsr ,whichismutually
we k
thematrixofdimensions(P(cid:4)Q(cid:4)1).TheJacobianmatrix conjugate with respect to positive definite Hessian matrix
is defined using relation: H , according to condition:
wb
2oe11 oe11 ... oe11 oe11 3 rkTHwbrk ¼0: ð7Þ
6 ow1 ow2 owR ob1 7
6 7 Thequadraticfunctionisminimizedbysearchingalongthe
66oe12 oe12 ... oe12 oe12 77 eigenvectors of the Hessian matrix H . For the give
6 ow ow ow ob 7 wb
6 ...1 ...2 ... ...R ..1. 7 iterative function
6 7
6oe oe oe oe 7
66 1Q 1Q ... 1Q 1Q77 5FðwÞ(cid:2)Hwbrkþ- ð8Þ
6 ow1 ow2 owR ob1 7
Jwe ¼66 ... ... ... ... ... 77 ð6Þ 52FðwÞ¼Hwb: ð9Þ
6oe oe oe oe 7
6 P2 P2 ... P2 12 7
6 ow ow ow ob 7 For the iteration kþ1, the change in gradient can be
6 1 2 R 1 7
6oe oe oe oe 7 calculated from equation:
6 P2 P2 ... P2 P27
6 7
66 o.w..1 o.w..2 ... o.w..R o.b.1. 77 MUk ¼Ukþ1(cid:2)Uk ¼ðHwbrkþ1þ-Þ(cid:2)ðHwbrkþ-Þ¼HwbMrk
6 7
4oePQ oePQ ... oePQ oePO5 ð10Þ
ow ow ow ob
1 2 R 1 where
where P is the number of training patterns with Qoutputs, Mr ¼ðr (cid:2)r Þ¼dkr ð11Þ
k kþ1 k R k
R is the number of weights and elements in error vector,
and e is calculated using Eq. 2. Conventionally, Jacobian where d is a learning rate and is selected to minimize
R
matrix J is initially calculated and later computations are function F(w) in the direction of r . The first search
k
performed on stored values for weights and biases upda- direction is arbitrary:
tion.Withfewerpatterns,thismethodworkssmoothlyand
r ¼(cid:2)U ð12Þ
0 0
efficiently while with large sized patterns, calculation of
Jacobian matrix faces memory limitations. This concludes where
LM algorithm performance degraded with larger training
U (cid:6)5FðwÞj : ð13Þ
patterns. k w¼w0
Conjugate gradient TheGram–SchmidtOrthogonalization(Messaoudi1996)is
used to construct r for each iteration, orthogonal to
k
The Conjugate Gradient (CG) algorithm is known for its fMU ;MU ;...;MU g as
0 1 k(cid:2)1
fast convergence rate, and has been employed for solving
r ¼(cid:2)U þb r ð14Þ
k k k k(cid:2)1
spare linear equations on numerous occasions. Few of its
variantsincludeScaledCG(SCG)andFletcher–PowellCG
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ApplNanosci(2017)7:933–945 939
where b is scalar and given as large set of IV curves. We divide the set into two: (1)
k
MUT U training and testing samples and (2) sample for cross-
b ¼ k(cid:2)1 k ð15Þ checking, and use the first subset to train our 90 ANN
k UT U
k(cid:2)1 k(cid:2)1 models (3 architectures (cid:4) 10 configurations (cid:4) 3 training
and d can be calculated using relation: algorithms), where each configuration refers to a different
R
number of neurons in the hidden layers. Once trained, the
(cid:2)UTU
dk ¼ k k: ð16Þ system provides us the training time and epochs taken by
R rTH r
k wb k
Bayesian regularization each ANN model to converge; we record these values.
Following the successful training, validation, and testing
TheBayesianRegularization (BR)algorithmmakes useof phases (combined called ANN Learning), we predict the
LM algorithm (hence not so different) to search for the response for the second sample, and record the MSE
minima from Hessian matrix of a given function. Refer between the predicted and measured responses. The same
to MacKay (1992) for the detailed description. process is repeated for ten configurations.
Objective statement
Research methodology and simulation results
Let X(cid:7)Rl, where l! { R(cid:4)D }, such that R[D, be a
Research methodology bounded experimental data. Let /¼/ðiÞji2X be selec-
ted features set, / ðiÞ;...;/ ðiÞ(cid:7)/ are n features asso-
1 n
Figure 5conciselypresentsourdatacollectionandanalysis ciated with training and testing of ANN’s network to
methodology. We have used the PPMS for obtaining a predictoutputvector/~ .Formally,/ismappedto/~ :
pred pred
Fig.5 Adoptedresearch
Physical
methodology Measurements
Configuration Cross−Validation
Training Sample
Sample
New
Input Response
Training
MSE
Feedforward Cascaded Recurrent
Time
Record
Parameters
Epoch
LM CG BR LM CCGG BR LM CG BR
Prediction
Validation
Terminate
Testing
ANN Learning 10?
Config.
<10? Count
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940 ApplNanosci(2017)7:933–945
Table1 Curvesusedforcomparison MSE, number of epochs, and training time. The cost
function is selected to be MSE and calculated using Eq. 2
Geometry Temperature(K) Magneticflux(Oe)
Rectangle 8.8 0 Simulation results
Square 8.65 41
Honeycomb 8.6 200 Theentirepurposeofthisresearchworkistohighlightthe
Kagome 8.4 2000 ability of the proposed approach to predict the IV curves
for different values of temperature and magnetic flux,
which were not available while training the ANN model.
/!/~pred.Theoutputvector/~pred isthepredictedversion As already stated in ‘‘Physical measurement system and
of / in terms of readings’’ section, we kept four IV curves (one for each
n/~ ¼D(cid:2)dk ;d2 ;d ;d (cid:3)2f(cid:2)1:1g;(cid:2)dL ;dC ;dB (cid:3)2Rlo geometry of antidots) isolated from the modeling data,
pred WB NN R S WB WB WB which we should use for cross-checking of the proposed
approach on unforeseen data. Table 1 presents those IV
where the input parameters, dk is a vector of randomly
WB curves. Figure 6 presents a plot of the predicted values
initialized weights and biases, d2 represents two hidden
NN against the physically measured ones. The thicker curves,
layers with different combinations of neurons, and d and
R for example, for the square array (shown in the top right
d are the learning rate and step size, respectively. The
s corner),representslargernumberofdatapointsavailablein
output parameters comprised of optimized weights and
the transport measurements. The reason for choosing a
biases vector from three different training algorithms of
different number of data points for each geometry was
dL (LM algorithm), dL (CG algorithm) and dB (BR twofold. First, we deliberately wanted to evaluate perfor-
WB WB WB
algorithm). Performance parameters are selected to be mance of each training algorithm in the absence of
Fig.6 PredictedIVcurves:rectangular(topleft),square(topright),honeycomb(bottomleft),andkagome(bottomright)
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Fig.7 MSE:rectangular(topleft),square(topright),honeycomb(bottomleft),andkagome(bottomright)
sufficiently large data sets, and thereby estimate the mini- all the geometries and architectures. Instead, let us com-
mum data points needed for an acceptable MSE. Second, mentontheobtainedresultsinacase-by-casemanner.BR
we wanted to showcase the applicability of our approach algorithm outperforms the other two in terms of MSE for
onvariousgeometrieswithavaryingnumberofdatapoints the square and honeycomb arrays, which had more data
in the transport measurements. It may be clearly observed points than the remaining geometries: rectangular and
that prediction for each geometry results in a negligible kagome. However, this happens only at the cost of
error. increased training time and number of iterations to con-
verge. The increased MSE in case of the latter two
Figures 7, 8, 9, respectively, present MSE, number of geometries by BR reflects its impotence in approximating
iterations to converge, and training time by each of the curves with a dearth of data points. For these two geome-
training algorithms. Note that the horizontal axis in each tries, LM appears to be more promising, except for a few
figurecorrespondstodifferentANNmodels;eachhavinga random peaks, see Fig. 7.
different number of neurons in the hidden layer; we call LM and CGF prove to be a better option if fast con-
this network configuration. In essence, each plot corre- vergence,bothintermsofnumberofiterationsandtraining
spondstoadifferentgeometryofantidots,trainedbythree time, is decisive. This is evident in Figs. 8 and 9. It is
benchmark algorithms for thirty different configurations. interestingtonotethatCGF,incontrasttoBR,takesalarge
Considering the fact that training an ANN model is a number of iterations to converge for square and honey-
stochastic process—immensely relying upon random comb arrays; those having large data sets, and its training
numbers—it is difficult to ascertain the reason behind time is minimal with geometries having smaller data sets.
diversity,especiallysharppeaks,ineach result.Therefore, This advocates its usage in systems requiring real-time
it is not possible to advocate the use of one algorithm for approximation, where accuracy could be slightly
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Fig.8 Epochs:rectangular(topleft),square(topright),honeycomb(bottomleft),andkagome(bottomright)
compromised. However, for the application presented in outperform the rest of the algorithms in terms of MSE,
this work, CGF is not the best available option. On the whereas LM and CGF can be good options for minimum
other hand, LM stands between the other two alternative trainingtimeandepochs,evenintheabsenceoflargedata
algorithms, both in terms of prediction accuracy and con- sets. For the purpose of predicting IV curves in supercon-
vergence rate. While it has better approximation accuracy ductingfilms,thisworkwillbeusedasbenchmark,sinceit
for smaller data sets, it tends to converge faster for the points out the best pairs of architecture and algorithm for
geometries having large number of data points. the most widely adopted assessment parameters, namely:
Table 2 presents the best results, in terms of minimum MSE, epochs, and training time.
MSE, epochs, and training time, obtained from the pre-
dictionprocessfor eachgeometry.Notethatnumber(No.)
of neurons, expressed as [x, y], represents x and y neurons Conclusion
in the first and second layers of each neuron, respectively.
The table should be interpreted as follows: for the rectan- Motivated by the experience that transport measurements
gular array, the layer-recurrent architecture having eleven insuperconducting filmsare notoriousand cumbersometo
andfiveneuronsinthehiddenlayers,oncetrainedwithLM obtain, a predictive model, based on artificial neural net-
algorithm,achievestheMSEof3:3(cid:4)10(cid:2)7,whichisbetter works,hasbeenproposed.Themodeltakesafiniteamount
thananyotherpairorarchitectureandalgorithm.Similarly, of data points, for each of the four geometries of antidots
for the same geometry, BR converges in the least number including rectangular, square, honeycomb, and kagome,
of iterations of 11 with the layer-recurrent architecture, and extrapolates the curves for awide range ofunforeseen
having 18 and 10 neurons in the hidden layer, while CGF valuesoftemperatureandmagneticflux.Wehaveassessed
trains the cascaded network with [5, 2] neurons in just three different architectures of artificial neural networks,
0.093 s, which is faster than all other options. It is evident trained by three renowned training algorithms for the
that BR, if provided with a sufficiently large data set, can purpose of predicting these current–voltage curves. Our
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