application of the wnn-based scg optimization algorithm

Research ArticleApplication of the WNN-Based SCG Optimization Algorithm forPredicting Soft Soil Foundation Engineering Settlement

Guihua Li Chenyu Han Hong Mei and Shuai Chen

School of Earth Sciences and Engineering Hohai University Nanjing 210098 China

Correspondence should be addressed to Chenyu Han 191309020011hhueducn

Received 10 August 2021 Revised 8 September 2021 Accepted 15 September 2021 Published 14 October 2021

Academic Editor Mian Ahmad Jan

Copyright copy 2021 Guihua Li et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Settlement prediction in soft soil foundation engineering is a newer technique Predicting soft soil settling has long been one of themost challenging techniques due to difficulties in soft soil engineering To overcome these challenges the wavelet neural network(WNN) is mostly used So after assessing its estimate performance two elements early parameter selection and system trainingtechniques are chosen to optimize the traditional WNN difficulties of readily convergence to the local infinitesimal point lowspeed and poor approximation performance e number of hidden layer nodes is determined using a self-adaptive adjustmenttechnique e wavelet neural network (WNN) is coupled with the scaled conjugate gradient (SCG) to increase the feasibility andaccuracy of the soft fundamental engineering settlement prediction model and a better wavelet network for the soft groundengineering settlement prediction is suggested in this paper Furthermore we have proposed the technique of locating the earlyparameters based on autocorrelation e settlement of three types of traditional soft foundation engineering including metrotunnels highways and high-rise building foundations has been predicted using our proposed model e findings revealed thatthe model is superior to the backpropagation neural network and the standard WNN for solving problems of approximationperformance As a result the model is acceptable for soft foundation engineering settlement prediction and has substantial projectreferential value

1 Introduction

Prediction of soft soil settlement has always been one of thetechnical problems in soft soil engineering As there aremanyfactors affecting the settlement of soft foundations how topredict the settlement of soft foundations correctly becomes acommon problem for researchers in design and constructionTo improve the accuracy calculation and prediction based onmeasured data are a general method in engineering at present[1] Commonly used deformation prediction methods basedon measured data include statistical analysis time seriesanalysis grey system theory Kalman filter and neural net-works but these have their limitations [2 3] Optimizing themodel and improving the prediction accuracy are an im-portant content of the deformation prediction modelAccording to the actual application research a single theoryor model is difficult to accurately predict the magnitude ofdeformation e close combination and comprehensive

comparison of multiple theoretical models is an effective wayto study the prediction of deformation [3 4] By integratingthe advantages of wavelet analysis with artificial neuralnetworks the wavelet neural network (WNN) has beenswiftly established and has played a key role in deformationmonitoring On the other hand as the economy has growncommunities have seen an increase in the number of high-rise structures e settling of the building foundation mustbe checked and anticipated to assure its safety during theconstruction and operation phases However due to thedifficulty of physical circumstances and the ambiguity of thecauses driving settlement using traditional certainty con-cepts to forecast settlement is extremely challenging Al-though the inclusion of WNN simplifies the problem it hasseveral drawbacks in its current form

e wavelet neural network (WNN) is a new explorationunder this guiding ideology which combines waveletanalysis and neural network successfully and thus provides a

HindawiScientific ProgrammingVolume 2021 Article ID 9936285 13 pageshttpsdoiorg10115520219936285

scientific theoretical basis and analysis tool for modernforecasting [5 6] ere are many forms of WNN Compactwavelet neural networks that use wavelet functions to re-place the hidden layer functions of conventional neuralnetworks seem common but they display convergence tolocal minima low speed and poor approximation perfor-mance frequently and some other shortcomings as well[6ndash9] Accordingly the present study proposes an optimizedWNN built on the scaled conjugate gradient algorithm(SCG) Within the study the settlement data of metrotunnels highway soft foundations and roads and high-risebuildings are taken as soft foundation engineering examplesree neural networks BP neural network the traditionalWNN based on backpropagation (BP algorithm) andWNNbased on SCG algorithm are compared and analyzedcomprehensively is article designed an optimization ofWNN built on the SCG method to predict soft soil foun-dation engineering settlement under the complex geologysettings to overcome issues mentioned above e resultsshow that the optimization model achieves a better per-formance and is more suitable than the other two networksfor soft soil foundation engineering settlement prediction

e following are the offerings of this research work

(1) We investigate the soft soil foundation engineeringsettlement prediction and wavelet neural networkWe explain the methodology that we have adoptedduring our proposed work and perform experimentfor our three techniques ie BP Neural NetworkWNN-based BP Algorithm and Improved WaveletNeural Network

(2) We compare the prediction accuracy obtainedduring the three techniques mentioned above

(3) Our proposed work improves the stability andconvergence accuracy of WNN Hence the initialparameter setting method can be linked with wavelettype wavelet time-frequency parameters andlearning samples

(4) From our proposed scheme we concluded that SCGalgorithm combined with the autocorrelation cor-rection can determine the number of hidden layernodes us an improved WNN can be derivedsuccessfully

e rest of the research work consists of the followingSection 2 explains the related work Section 3 illustratesmaterial and methodology used during our proposed workSection 4 deliberates our experimental work that we haveperformed during our work and finally the paper is con-cluded in Section 5

2 Related Work

Soft soil is found all around the world Its unique charac-teristics ie high void ratio high water content highcompressibility low shear strength low permeability andunique structural characteristics necessitate particularconsideration in the study construction and maintenanceof geotechnical structures built on them Large-scale

construction of high-speed transit systems high-risestructures and subterranean works for numerous urbancenters built on such soils is a huge problem Many re-searchers have attempted to overcome the issues that arise inthe foundation of soft soils e authors in [10] used a three-dimensional (3D) analytic approach to investigate the in-fluence of pavement smoothness on the dynamic behavior ofsubsurface movement loads e impact of unconventionaland low-carbon additions on the long-term treatment ofsoils for building and paving materials is investigated Datafrom three centrifuge tests during traffic loads are used tostudy the cyclic behavior of mud with a ground comprised ofovercompacted soil at the top

e researchers of [6] used a 3D dynamic finite-elementanalysis to estimate the permanent settling of a segment ofthe cross-river tunnel allowing for the influence of primarystress rotation and verifications for test data e disruptedstate concept (DSC) model for normally consolidated clayswas described in [11] which included the impacts ofcracking particle breaking heating softness and hardeninge overconsolidation ratio on strength militancy anddistortion is used to develop a novel disruption functionConsolidation theory numerical computation and curvefitting are three types of approaches for forecasting groundsettlement ere have been several recent works on con-solidation theory [12ndash14]

In a novel approach the authors of [15] described thesettlement of an inspired embankment on a soft basis basedon a classic hyperbolic approach and used the system de-formations features reflected in preloaded embankments toforecast the settlement during the later stage In [16] theauthors proposed a staged observational technique forpredicting embankment settlement on soft ground withstaged construction they discovered that immediate set-tlement adds to the shift distance of the parallel lines duringstaged construction e researchers in [17] used a geneticalgorithm to optimize a BP neural network to forecast thesummer electrical short-term load Because of its highconvergence rate and low memory use the scaled conjugategradient (SCG) method is used in NFC training As a resultthe authors of [18] trained a type-1 fuzzy system using acustomized form of SCG According to them the improvedSCG accelerates convergence in the steepest descent ap-proach of fuzzy system training As a consequence the SCGappears to be a good candidate for NFC training for large-scale issues Training NFC with SCG for large-scale issueson the other hand might take days or weeks on any personalcomputer Training NFC with SCG for large-scale issues onthe other hand might take days or weeks on any personalcomputer An alternate method for reducing training time isto calculate the Hessian matrix using first-order gradients asin conjugate gradient (CG) algorithms [19] A transfor-mation wavelet is a useful tool for data processing and time-frequency representation development e wavelet theoriesare described thoroughly in [20 21] In the context of neuralnetworks the application of wavelet transform is not newPrevious research [22 23] proposed a theoretical frameworkfor neural feed-forward networks based on wavelets estudy of [24] has investigated the capacity to employ

2 Scientific Programming

wavelet-based cross-pollination for an unknown real-timefunction Because wavelets have a high compression capacityand have fewer coefficients the results were achieved in thiscircumstance

In [25] the authors offer a statistical model identificationframework for using wavelet networks which is studied overa wide range of topics including architecture initializationvariable selection and model selection Because of theircapacity to extract varied information wavelet-based tech-niques have been employed in numerous computer visionapplications using Convolutional Neural Networks (CNN)Wavelet CNN texture classification [26] multiscale facesuperresolution [27] picture superresolution [28] and edgefeature boosting [29] are only a few examples In [30] amultilevel wavelet CNN model for picture restoration waspresented e researchers in [31] suggested a new layer thatconducts wavelet-based convolution filtering and activationbefore returning to pixel space

Similarly [32] created a hybrid wavelet deep learningnetwork based on the wavelets scattering transform [33]is presented a basic classification that was subsequentlyenhanced [34] In [35] the authors also suggest a wavelet forthe segmentation of the brain tumors which is strengthenedby an evolutionary network design and for this applicationDWT is coupled with the neural network classification [36]To lower the spatial resolution and expand the receptive fieldto dense pixel-specific prediction encoder-decoder CNNarchitecture with encoder DWT and inverse decoder DWTwas presented in [37] In addition a neural wavelet networkfor speech and noise separation was proposed in [38] eresearchers in [39] built WNN optimization based on theSCG algorithm to forecast the settling of the foundation ofthe structure under complex geological circumstances efindings indicated that WNN optimization was optimal andthat this had a positive effect compared to the BP neuralnetwork and the BP WNN Inspired from the work of abovescholars this research work combines wavelet neural net-work (WNN) with optimized scaled conjugate gradientalgorithm to successfully predict the soft soil foundationengineering settlement by performing numerousexperiments

3 Materials and Methodology

31 Materials Used during Our Research Work

311 Soft Soil Foundations Soft soil foundation consists ofsoft soil with finer particles and organic soil with wider gapsbecause its texture comprises mucky soil with silt and otherhighly compressible soils with clay components and siltcomponents [40] in fine soil particles e soft foundationsof soil are mostly based on the changes in soil foundationsproduced by geographical circumstances geological struc-tures and the soil conditionsrsquo features and qualities

(1) Characteristics of Soft Soil Foundation e majorcharacteristics of soft soil include low water permeabilityand high water content according to the basic attributes ofsoil However the water permeability is poor and the shear

strength is exceedingly low [41] e shear foot brake andcompression system with high compressibility or mucky soilgenerally exhibit considerable settlement after the externalload is transmitted to the foundation section As a resultconstructions erected on soft soil foundations such asbuildings roads and bridges have a significant inclinationor settlement It is straightforward to cause the injury andcrack of the building and therefore the increase of themacrospore which can cause the collapse of the buildingonce it is serious erefore the municipal constructionunits should actively analyze and study the soft soil foun-dation treatment and scientifically discover and take a lookat the shear resistance and cargo resistance level of the softsoil foundation [42]

(2)e Impact of Soft Soil Foundatione impact of soft soilfoundation can be seen in Figure 1 and its major compo-nents are as follows

(1) Poor bearing capacity due to the high water contentand tiny seepage of the soil conditions in soft soilfoundations the bearing strength of the foundationis fairly low making settlement foundations ex-tremely simple to produce posing a severe danger toresidentsrsquo travel quality [43]

(2) Large settlement the significant settlement is one ofthe features of a soft soil foundation e settlingfeatures of soft soil foundations will create additionaldifficulties for engineering building projects andpose a severe danger to the progress and quality ofengineering projects

(3) Strong compressibility the soft soil ductility isstrong due to the macrospore structure dependingon water content and soil quality [12] Duringconstruction the macrosporersquos soft soil layer is en-dangered which prevents appropriate controlmeasures from being taken and further impacts theefficiency and progress of the building and the sta-bility of the basee consequence is easy to dislocatethe path and the subgrade of the cityrsquos engineeringstructure collapses

312 Wavelet Neural Network Wavelet Networks are anovel network type that brings together traditional SigmoidNetworks (NNs) with Wavelet Analysis (WA) [43] estretch factor denoted by aj and the panning elementdenoted by bj are two new parameters introduced by thistechnique ese new parameters replace the respectiveweights and thresholds of the neural network by using thewave element to replace neurons and establish a connectionbetween the transforming and the neural networks throughan approximation of wavelet decomposition [25] ewavelet neural network contains three layers according toFigure 2 input layer hidden layer and layer output Duringthe forward propagation learning phase the data from theinput layer is processed and sent to the hidden layer edata is subsequently processed in the output layer by ahidden layer After that at the backpropagation stage the

Scientific Programming 3

output layer determines the output value of each unit bycomputing the difference between the output value and theintended output values Finally the weight to modify eachinput layer and the hidden layer is the product of eachreceiving unit error value and transmission unit activationvalue [44 45]

32 Methodology

321 Convergence Analysis After considering the WNNrsquoslimitations the main issue is poor convergence When thewavelet-based neural network uses the BP neural networkrsquosinitialization and training technique there will be poorconvergence difficulties and recommended improvements[7] is is due to the differing activation functions of thehidden layer nodes is article optimizes the initial pa-rameter selection technique as well as the network trainingmethod based on this

322 Optimization of the WNN

(1) Selection of Initial Values of Network Parameters einitialization of network parameters has an impact onwhether or not the networkrsquos subsequent learning con-verges and how quickly it converges Currently randomvalues are used to create the initial parameters of WNNswhich significantly increases the number of learning timesand even causes the network to fail to converge e authorsuggested an autocorrelation correction initial parametersetting technique in [46] which links the initial parametersettings of WNN with wavelet types wavelet time-fre-quency parameters and learning samples Excellentstarting parameters may be obtained with a high degree ofcertainty using this approach and the wavelet networkrsquosfollow-up learning speed will be substantially increased Asa result this article uses this approach to determine theinitial values of network parameters which will be dis-cussed further in this article

Prediction of Softsoil foundation

engineeringsettlement

Ψ 2

Ψn

Ψ11

2

n

1

2

m

Training data+

Input data

Input layer Output layerHidden layer

X1

X2

X3

y1

y2

y3

Figure 2 Wavelet neural network

Impact of soft soilfoundation

Poor bearing capacity

Large settlement

Strong compressibility

Figure 1 Impact of soft soil foundation


(2) Determination of the Network Structure In this sectionwe discuss the structure of our proposed Wavelet NeuralNetwork based on optimized scaled conjugate gradient al-gorithm for the prediction of soft soil foundation engi-neering settlement

(1) Number of hidden layers the researcher of [47]demonstrated that a three-layer neural networkmodel can handle general function fitting and ap-proximation issues Because settlement prediction isa function fitting issue a three-layer wavelet neuralnetwork will suffice

(2) Number of hidden layer nodes the number ofhidden layer nodes directly influences the networkrsquosgeneralization ability and training time therefore itis crucial in the development of the neural networkmodel ere is however no theoretical direction inthis area e primary approaches used in practicalapplications are testing or the use of empiricalequations [48]

is study presents an adaptive technique based onempirical formulas for obtaining a higher number of hiddenlayer nodes To begin use equation (1) to find the number ofhidden layer nodes and the maximum number of learningperiods for the network When the network reaches itsmaximum number of nodes the number of hidden layernodes will increase since it is still unable to fulfill the errorcriteria Similarly when the network does not meet thelearning number specified and the error criterion is fulfilledconcurrently the number of hidden layer nodes willdecrease

O m + n

radic+ 1 (1)

Here n is the number of input nodes m is the number ofoutput nodes is the number of hidden layer nodes and l is aconstant between one and twenty

323 Optimization of Learning Algorithm Because thetraditional BP network utilizes the steepest descent methodthe primary drawback is that this slows down networkconvergence and is readily confined to the best local solution[41] too For that reason there were several optimizationmethods among these for its similarity in nature to theSDBP algorithm but with a higher convergence time theconnected gradient algorithm is frequently employed fortackling big optimization issues For this reason the SCGmethod [49] is used in this article for network training in thecombined gradient algorithm e following is the proce-dure of detailed application

We take the error energy function as

E(θ) 12

1113944(f(x θ) minus y)2 (2)

Here x is the input value y is the output value and θ is aparameter

For the objective function E(θ) of the wavelet neuralnetwork with P input samples the gradient of the θ (iewki wjk ak bk) is

E

wki

1113944P

p11113944

m

j1fj minus yj1113872 1113873wjk

ψxprime

xi (3)

E

wjk

1113944P

p1fj minus yj1113872 1113873ψ

1113936ni1 wkixi minus bk

ak

1113888 1113889 (4)

E

ak

1113944

P

p11113944

m


ψak

(5)

E

bk

1113944P

p11113944

m


ψbk

(6)

If the Morlet wavelet function is used namelyψ(t) e(minus t22) cos(5t) then

xprime 1113944n

i1wkixi tprime

xprime minus bk

ak

(7)

us the network parameters in equations (3) to (6) are

ψak

cos 5tprime( 1113857eminus tprime

221113872 1113873tprime

2

ak

+ 5 sin 5tprime( 1113857eminus tprime

221113872 1113873 tprime

ak

ψbk


221113872 1113873 tprime

ak


221113872 1113873 1

ak

ψtprime

minuscos 5tprime( 1113857eminus tprime

221113872 1113873 tprime

ak

minus 5 sin 5tprime( 1113857eminus tprime

221113872 1113873 1

ak

(8)

Substituting these into the SCG algorithm the optimal θcan be solved

In the formulas above fj(j 1 2 m) is the outputof the network ωki is the connection weight between the kthneuron in hidden layer and the ith neuron in input layer ωjk

is the connection weight between the jth neuron in outputlayer and the kth neuron in the hidden layer and ak bk arethe scale parameters and translation parameters of thewavelet basis function

324 Optimized Construction of the WNN Based on theforegoing research the improved WNN model in this paperis constructed in the following steps

Step 1 select an appropriate amount of trainingsamples define the training samples according tocertain rules and determine the number of inputneurons and output neurons of the networkStep 2 the learning algorithm of the network SCGalgorithmStep 3 set the network training period target error andother parameters


Step 4 calculate the number of hidden layer nodes inthe network by empirical formulas make adaptiveoptimization adjustments and rebuild the networkstructureStep 5 transfer function the hidden layer to the outputlayer adopts the sigmoid function and the input layerto the hidden layer adopts the Morlet wavelet functionStep 6 select a set of weights randomly and use theweights optimized by autocorrelation correctionmethod as the initial weights for network trainingReset the training parameters of the network and usethe SCG algorithm to train and establish an improvedWNN

325 Settlement Prediction Model Equation (9) is used forsample normalization to make full use of the sensitivity ofthe Sigmoid function and improve the convergence speed oftraining [50]

xlowast

x minus xmin

2 xmax minus xmin( 1113857 (9)

Here xmax and xmin are the maximum and minimumvalues of each group of input components and x and xlowast arethe values before and after normalization of each group ofinput components

Apply the improved WNN described aboveSample training first the measured samples

xi i 1 2 n1113864 1113865 are divided into k(k isin N kle n) groupsand each group has m + 1(m n minus k) valuee first value isused as the input node value of the network and the latter isused as the expected value of the output node Next thenetwork connection weight is trained en using theconverged connection weight xk xk+1 xk+mminus1 are usedas the network input to calculate the predicted value xk+m Atlast after removing xk and adding xk+m xk+1 xk+2 xk+m

are set as the new input of the network to calculate thepredicted value xk+m+1 and so on to make furtherpredictions

326 Training Plan e three neural networks used havethe same structure and training samples and the differencesare shown in Table 1 e training sample is the cumulativesettlement value e average of 10 times prediction will bethe result to reduce the randomness of the predicted valuee model is evaluated by the relative error of the predictionresult and themodel accuracyemodel accuracy is given by

Model accuracy

1113936|predicted value minus measured value|2

n minus 1

1113971

(10)

Here n is the number of predicted valueTable 1 represents a comparison of three models such as

BP Neural Network which uses the SDBP Algorithm as alearning method where Sigmoid is the function of hiddenlayer by randomly generating the initial parameter Whilethe SDBP Algorithm is used as the learning technique in the

BPWavelet Neural Network the Morlet Wavelet function ofthe hidden layer is generated randomly Similarly ourImprovedWavelet Neural uses SCG Algorithm as a learningmodel instead of the SDBP Algorithm e initial parameterin the case of our Improved Wavelet Neural Network isgenerating by the autocorrelation correction method

4 Experimental Work and Results

Many soft foundation projects have emerged in recent yearsas a result of the steady building of national fundamentalprojects Soft foundation engineering settlement predictionhas always been a challenging topic in engineering due to theintricacy of soft foundation deformation As a result threeneural networks are used to forecast the settlement of threecommon soft foundation projects metro tunnels roads andhigh-rise structures to compare and assess the convergenceof the optimization model in this study

41 Settlement Prediction of the Metro Tunnel e westextension of one cityrsquos metro tunnel is located in a softflowing murky salty clay layer with high moisture contenthigh compressibility high sensitivity low strength anddeformability It is a floodplain of the Yangtze River with athick covering layer deep bedrock and poor geologicalcondition As to the tunnel its surrounding area is at thepeak of the development period there are many construc-tion sites and the settlement of its structure is obvious eexperimental data are 20 periods of the measurement pointswhich settled significantly e first 15 periods of data areused as training samples to predict the settlement of the next5 periods e first 15 periods of data are divided into 8training samples Each group has 8 values the first 7 valuesare used as the input of the network node and the latter isused as the expected value of the output nodee predictionresults of the three models are shown in Tables 2ndash5

Table 2 illustrates the relative error and accuracy ofsettlement prediction of Metro Tunnel using BP NeuralNetwork with training times 3503 During BP NeuralNetwork we obtain an accuracy of 245 for the number ofmeasurements 16 17 18 19 and 20

Table 3 explains the relative error and accuracy of set-tlement prediction of Metro Tunnel using WNN-based BPAlgorithm with training times 931 During this algorithmwe obtain an accuracy of 132 for the number of mea-surements 16 17 18 19 and 20

Table 4 describes the relative error and accuracy of set-tlement prediction of Metro Tunnel using Improved WaveletNeural Network with training times 267 During ImprovedWavelet Neural Network the accuracy of 089 for the numberof measurements 16 17 18 19 and 20 can be obtained

Table 5 shows the relative error and accuracy of set-tlement prediction of Metro Tunnel using all the threetechniques with training times 3503 931 and 267 re-spectively is reflects that the model accuracy obtainedduring BP Neural Network which is 245 is greater than theWNN-based BP Algorithm and Improved WNN for thenumber of measurements 16 17 18 19 and 20


42 Settlement Prediction of the Highway Soft Soil RoadbedYangtze River Bridge opened to traffic in 2001 is one of thenational key construction projects during the ninth five-yearplan period Its lead is a soft soil foundation and settlementmonitoring points are laid out according to its sections e13 periods of monitoring data from a monitoring point on acertain section are selected for prediction experimentamong which the first 9 periodrsquos data are used as trainingsamples to predict the settlement of the last 4 periods esettlement date of the first 9 periods is divided into 4 training

samples Each group has 6 values the first 5 values are usedas the input of the network node and the latter is used as theexpected value of the output nodee three networkmodelsare used to predict respectively the results are given inTables 6ndash9

Table 6 illustrates the relative error and accuracy ofsettlement prediction of soft soil roadbed of highway usingBP Neural Network with training times 2589 During BPNeural Network we obtain an accuracy of 610 for thenumber of measurements 10 11 12 and 13

Table 1 Comparison of three models

S no Model Learning method Hidden layer function Initial parameter1 BP neural network SDBP algorithm Sigmoid function Randomly generated2 BP wavelet neural network SDBP algorithm Morlet wavelet function Randomly generated3 Improved wavelet neural network SCG algorithm Morlet wavelet function Autocorrelation correction method

Table 2 Settlement prediction of the metro tunnel using the BP neural network

S no Measurement number Training times Relative error ()1 16

3503

01522 17 03433 18 01644 19 01095 20 0278

Model accuracy 245

Table 3 Settlement prediction of the metro tunnel using WNN-based BP algorithm


931

00792 17 00733 18 01184 19 00605 20 0200

Model accuracy 132

Table 4 Settlement prediction of the metro tunnel using the improved wavelet neural network


267

00582 17 00953 18 01034 19 00295 20 0095

Model accuracy 089

Table 5 Settlement prediction of the metro tunnel using all three techniques

Measurement numberBP neural network WNN-based BP algorithm Improved wavelet neural network

Training times Relative error () Training times Relative error () Training times Relative error ()16

3503

0152

931

0079

267

005817 0343 0073 009518 0164 0118 010319 0109 0060 002920 0278 0200 0095Model accuracy 245 132 089


Table 7 describes the relative error and accuracy ofsettlement prediction of soft soil roadbed of highway usingWNN-based BP Algorithm with training times 616 Duringthis technique we obtain an accuracy of 365 for the numberof measurements 10 11 12 and 13

Table 8 explains the relative error and accuracy of set-tlement prediction of soft soil roadbed of highway usingImprovedWavelet Neural Network with training times 132During Improved Wavelet Neural Network the accuracy of182 for the number of measurements 10 11 12 and 13 canbe obtained

Table 9 shows the relative error and accuracy of set-tlement prediction of soft soil roadbed of highway using allthe three techniques with training times 2589 616 and 132respectively is reflects that the model accuracy obtainedduring BP Neural Network is 610 for the number ofmeasurements 10 11 12 and 13 is reflects that thisaccuracy is greater than the WNN-based BP Algorithm andImproved WNN

43 Settlement Prediction of the Building Foundation e 21periods monitoring data of a high-rise building soft foun-dation are taken for analysis e first 13 periods of data areused as training samples to predict the settlement of the last 8periods of observation Use the three network models tomake predictions and the results are listed in Tables 10ndash13

Table 10 illustrates the relative error and accuracy ofsettlement prediction of building foundation using BPNeural Network with training times 3201 During BP NeuralNetwork we obtain accuracy of 043 for the number ofmeasurements 14 15 16 17 18 19 21 and 21

Table 11 describes the relative error and accuracy ofsettlement prediction of building foundation using WNN-based BP Algorithm with training times 1145 During thistechnique we obtain an accuracy of 047 for the number ofmeasurements 14 15 16 17 18 19 20 and 21

Table 12 shows the relative error and accuracy of set-tlement prediction of building foundation using ImprovedWavelet Neural Network with training times 254 DuringImproved Wavelet Neural Network the accuracy of 035 forthe number of measurements 14 15 16 17 18 19 20 and 21can be obtained

Table 13 shows the relative error and accuracy of set-tlement prediction of building foundation using all the threetechniques with training times 3201 1145 and 254 re-spectively is reflects that the model accuracy obtainedduring WNN-based BP Algorithm is 047 for the number ofmeasurements 14 15 16 17 18 19 20 and 21 is reflectsthat this accuracy is greater than the rest of the twotechniques

Figures 3ndash5 show a comparison of the three techniqueswhere the mean relative error and maximum absolute errorof the prediction results obtained using the BP neuralnetwork model are larger than those obtained using theWNN-based BP Algorithm It demonstrates that the WNN-based BP Algorithmrsquos generalization (prediction) capacityoutperforms the BP neural network model e forecastfindings from the Improved WNN approach are larger thanthe measured settlement values which is consistent with theactual engineering experience When these two approachesare compared the prediction power of the WNN-based BPAlgorithm is superior to that of the Improved WNNmethod

Table 6 Settlement prediction of the soft soil roadbed of the highway using the BP neural network


2589

02162 11 01233 12 02064 13 0172

Model accuracy 610

Table 8 Settlement prediction of the soft soil roadbed of the highway using the improved wavelet neural network


132

00152 11 00963 12 00554 13 0054

Model accuracy 182

Table 7 Settlement prediction of the soft soil roadbed of the highway using WNN-based BP algorithm


616

01282 11 00963 12 01124 13 0103

Model accuracy 365


e WNN outperforms the BP neural network in termsof prediction accuracy and convergence speed and has ex-cellent adaptive prediction capabilities when compared tothe WNN based on the SDBP algorithm according to

settlement prediction findings for three types of softfoundation engineering As a result the improved WNNbased on the SCG algorithm greatly increases predictionaccuracy and convergence speed

Table 9 Settlement prediction of the soft soil roadbed of the highway using all three techniques

Measurement numberBP neural network WNN based on BP algorithm Improved wavelet neural network


2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182

Table 11 Settlement prediction of the building foundation using WNN-based BP algorithm


1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047

Table 10 Settlement prediction of the building foundation using the BP neural network


3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043

Table 12 Settlement prediction of the building foundation using the improved wavelet neural network


254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035


Table 13 Settlement prediction of the building foundation using all three techniques



3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror

BP Neural NetworkWNN based BP AlgorithmImproved WNN

Figure 3 Settlement prediction of the metro tunnel

10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror


Figure 4 Settlement prediction of the soft soil roadbed of the highway


5 Conclusion

Aiming at the convergence defects in the application oftraditional settlement prediction models this paper opti-mizes the WNN based on the traditional BP algorithm andapplies it to soft foundation engineering Here we haveintroduced BP Neural Network and Improved WaveletNeural Network in the soft ground foundation engineeringforecast based on the basic concept of the wavelet neuralnetwork e neural wavelet network method has beenenhanced and the scale conjugate gradient technique hasbeen updated to maximize the excellent value neural waveletnetwork approach e enhanced wavelet neural networksettlement model is constructed and settlements are pre-dicted based on the conventional BP technique e im-proved wavelet neural network model based on the classicBP method has greater prediction accuracy and the soft soilsettling has a good prediction effect with a specified refer-ence value according to the comparison of calculation ac-curacy evaluation indexes

Data Availability

e datasets used andor analyzed during the current studyare available from the corresponding author upon reason-able request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Guihua Li and Chenyu Han designed the study analyzed thedata and wrote the manuscript Hong Mei and Shuai Chenanalyzed the data and contributed to writing the manuscript

Acknowledgments

is study was supported by the Fundamental ResearchFunds for the Central Universities (B210205013)

References

[1] P Tao Y An-ying L Xing and Y Qin ldquoPrediction of softground settlement based on BP neural network-grey systemunited modelrdquo Rock and Soil Mechanics vol 26 no 11pp 1810ndash1814 2005

[2] T-sheng Wang Li-ping Zhang and Xi-sheng Hua ldquoReviewon research of deformation prediction of metro tunnelconstructionrdquo Advances in Science and Technology of WaterResources vol 5 pp 62ndash65 2003

[3] H Shen-xiang H Yin and Z Jiang Deformation MonitoringData Processing and Methods Wuhan University PressWuhan China 2003

[4] Z Guo-qiang H Wang Li Li W Cheng-tang and X I E Bi-ting ldquoPrediction of maximum settlement of foundation pitbased on SFLA-GRNN modelrdquo Rock and Soil Mechanicsvol 40 no 2 pp 792ndash798 2019

[5] X Xia X Liu and J Lou ldquoA network traffic prediction modelof smart substation based on IGSA-WNNrdquo ETRI Journalvol 42 no 3 pp 366ndash375 2020

[6] Na Yang Q Fu W Shu-li and Li Rong-dong ldquoImprovementof wavelet neural networks model and applicationrdquo Journal ofSystems Scienceamp Information vol 29 no1 pp 168ndash173 2009

[7] Bi Yan-qiu W-lian Zhou and Da-hai Zhang ldquoInvestigationon the convergence performance of continuous waveletneural networkrdquo Proceedings of the CSU-EPSA vol 12 no 6pp 51ndash53 2005

[8] Y Zhang H Yang H Cui and Q Chen ldquoComparison of theability of ARIMA WNN and SVM models for droughtforecasting in the Sanjiang Plain Chinardquo Natural ResourcesResearch vol 29 no 2 pp 1447ndash1464 2020

[9] Q Qinghua Zhang ldquoUsing wavelet network in nonparametricestimationrdquo IEEE Transactions on Neural Networks vol 8no 2 pp 227ndash236 1997

14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror


Figure 5 Settlement prediction of the building foundation


[10] J Qian R Zhou S Chen X Gu and M Huang ldquoInfluence ofpavement roughness on dynamic stresses in saturated subsoilsubjected to moving traffic loadingrdquo International Journal ofGeomechanics vol 18 no 4 Article ID 04018012 2018b

[11] X Yu F Jiang J Du and D Gong ldquoA cross-domain col-laborative filtering algorithm with expanding user and itemfeatures via the latent factor space of auxiliary domainsrdquoPattern Recognition vol 94 pp 96ndash109 2019

[12] J Yu and R Yang ldquoStudy on the predictive algorithm of plantrestoration under heavy metalsrdquo Scientific Programmingvol 2021 Article ID 6193182 2021

[13] R A Barron ldquoClosure of ldquoconsolidation of fine-grained soilsby drain wellsrdquordquo Transactions of the American Society of CivilEngineers vol 113 no 6 pp 751ndash754 1948

[14] L Kok M Jamiolkowski and S Hansbo ldquoConsolidation byvertical drainsrdquo Gacuteeotechnique vol 31 no 31 pp 45ndash661981

[15] Q Zheng A C F Chiu G H Lei and C W W Ng ldquoAnanalytical solution for consolidation with vertical drainsunder multiramp loadingrdquo Gacuteeotechnique vol 65 no 7pp 531ndash547 2015

[16] T Yang G W Li and W Q Yang ldquoSettlement prediction ofstage constructed embankment on soft ground based on thehyperbolic methodrdquo Rock and Soil Mechanics vol 25 no 10pp 1551ndash1554 2004 in Chinese

[17] S Y Liu and F Jing ldquoSettlement prediction of embankmentswith stage construction on soft groundrdquo Chinese Journal ofGeotechnical Engineering vol 25 no 2 pp 228ndash232 2003

[18] M V Ribeiro C A Duque and J M T Romano ldquoAninterconnected type-1 fuzzy algorithm for impulsive noisecancellation in multicarrier-based power line communicationsystemsrdquo IEEE Journal on Selected Areas in Communicationsvol 24 no 7 pp 1364ndash1376 2006

[19] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993

[20] I Daubechies Ten Lectures on Wavelets Vol 61 Society forIndustrial and Applied Mathematics Philadelphia PA USA1992

[21] M Yu T Quan Q Peng X Yu and L Liu ldquoA model-basedcollaborate filtering algorithm based on stacked autoencoderrdquoNeural Computing amp Applications vol 6 no 12 2021

[22] Y C Pati and P S Krishnaprasad ldquoAnalysis and synthesis offeedforward neural networks using discrete affine wavelettransformationsrdquo IEEE Transactions on Neural Networksvol 4 no 1 pp 73ndash85 1993

[23] Q Zhang and A Benveniste ldquoWavelet networksrdquo IEEETransactions on Neural Networks vol 3 no 6 pp 889ndash8981992

[24] C Bernard S Mallat and J jacques Slotine ldquoWaveletinterpola- tion networksrdquo in Preprint Centre de Mathema-tiques AppliqueesEcole Polytechnique 1999

[25] A K Alexandridis and A D Zapranis ldquoWavelet neuralnetworks a practical guiderdquo Neural Networks vol 42pp 1ndash27 2013

[26] F Shin K Takayama and T Hachisuka ldquoWavelet con-volutional neural networks for texture classificationrdquo arXivpreprint arXiv170707394 2017

[27] H Huang R He Z Sun and T Tan ldquoWavelet-srnet awavelet- based cnn for multi-scale face super resolutionrdquo inProceedings of the IEEE Conference on Computer Vision andPattern Recognition pp 1689ndash1697 Silver Spring MD USAJuly 2017

[28] T Guo H S Mousavi T H Vu and V Monga ldquoDeepwavelet prediction for image superresolutionrdquo in Proceedingsof the IEEE Conference on Computer Vision and PatternRecognition (CVPR) Workshops Seattle WA USA July 2017

[29] S F Ddn De Silva I T S Piyatilake andA V S Karunarathne ldquoWavelet based edge feature en-hancement for convolutional neural networksrdquo InternationalConference in Machine Vision vol 11041 2018

[30] P Liu H Zhang K Zhang L Lin and W Zuo ldquoMulti-levelwavelet-cnn for image restorationrdquo 2018 arXiv preprintarXiv180507071

[31] B Zhang and L Meng ldquoEnergy efficiency analysis of wirelesssensor networks in precision agriculture economyrdquo ScientificProgramming vol 2021 Article ID 8346708 2021

[32] E Oyallon E Belilovsky and S Zagoruyko ldquoScaling thescattering transform deep hybrid networksrdquo in Proceedings ofthe International Conference on Computer Vision (ICCV)Venice Italy October 2017

[33] S Mallat ldquoGroup invariant scatteringrdquo Communications onPure and Applied Mathematics vol 65 no 10 pp 1331ndash13982012

[34] X Yu Q Peng L Xu F Jiang J Du and D Gong ldquoA se-lective ensemble learning based two-sided cross-domaincollaborative filtering algorithmrdquo Information Processing ampManagement vol 58 no 6 Article ID 102691 2021

[35] B Alizadeh Savareh E Hassan M Hajiabadi S Majid Azimiand M Ghafoori ldquoWavelet-enhanced convolutional neuralnetwork a new idea in a deep learning paradigmrdquo BiomedicalEngineeringBiomedizinische Technik vol 64 no 2 pp 195ndash205 2018

[36] L Rui X Xiaoyu and D Xueyan ldquoFatigue load spectrum ofhighway bridge vehicles in plateau mountainous area basedon wireless sensingrdquo Mobile Information Systems vol 2021Article ID 9955988 2021

[37] L Ma J umlorg Stuumlckler T Wu and D Cremers ldquoDetaileddense inference with convolutional neural networks via dis-crete wavelet transformrdquo arXiv preprint arXiv1808018342018

[38] Z Zhang Y Shi H Toda and T Akiduki ldquoA study of a newwavelet neural network for deep learningrdquo in Proceedings ofthe Wavelet Analysis and Pattern Recognition (ICWAPR)2017 International Conference on pp 127ndash131 IEEE XianChina July 2017

[39] S Bo and B Kexin ldquoOpen innovation mode of green in-novation system for manufacturing industryrdquo Mobile Infor-mation Systems vol 2021 Article ID 9948683 2021

[40] G Li T Huang M Jiang and R Yue ldquoOptimization andapplication research of wavelet neural networkrdquo in Pro-ceedings of the 2009 International Workshop on IntelligentSystems and Applications Wuhan China May 2009

[41] L Guo-dong and D Jin ldquoDiscussion on problems of BPneural networks applied to hydrological predictionrdquo Journalof Hydraulic Engineering vol 1 pp 65ndash70 1999

[42] X Ni ldquoDiscussion on soft soil foundation treatment tech-nology in municipal road and bridge engineeringrdquo E3S Webof Conferences vol 165 Article ID 04002 2020

[43] J Lin ldquoDiscussion on the technical characteristics of softground foundation treatment in municipal road and bridgeengineering constructionrdquo Henan Building Materials vol 3pp 224-225 2019

[44] F Lu J H Xu and Z Y Wang ldquoStudy on genetic waveletneural network model of water demand prediction in HefeiCityrdquo Science of Surveying and Mapping Papers vol 38 no 5pp 28ndash31 2013


[45] J Y Hu H Y Wen and L Zhou ldquoApplication of geneticwavelet neural network in prediction of dam deformationrdquoYellow River Papers vol 36 no 10 pp 126ndash128 2014

[46] Z Xue-zhi Z Chun-hua and C Tong-jian ldquoA research on theinitialization of parameters of wavelet neural networksrdquoJournal of South China University of Technology vol 2 no 2pp 77ndash79 2003

[47] K-l Hsu H V Gupta and S Sorooshian ldquoArtificial neuralnetwork modeling of the rainfall-runoff processrdquo WaterResources Research vol 31 no 10 pp 2517ndash2530 1995

[48] B Ye and L Yan ldquoAnalysis of choosing the number of thehiddenlayers and its nodes number in back propagationnetworkrdquo Journal of Shangqiu Vocational and TechnicalCollege vol 12 no 6 pp 52-53 2004

[49] S O Sada ldquoImproving the predictive accuracy of artificialneural network (ANN) approach in a mild steel turningoperationrdquo International Journal of Advanced ManufacturingTechnology vol 112 no 9-10 pp 2389ndash2398 2021

[50] W Bi and G Wang ldquoLocal cultural IP development andcultural creative design based on big data and internet ofthingsrdquo Mobile Information Systems vol 2021 Article ID5521144 2021


scientific theoretical basis and analysis tool for modernforecasting [5 6] ere are many forms of WNN Compactwavelet neural networks that use wavelet functions to re-place the hidden layer functions of conventional neuralnetworks seem common but they display convergence tolocal minima low speed and poor approximation perfor-mance frequently and some other shortcomings as well[6ndash9] Accordingly the present study proposes an optimizedWNN built on the scaled conjugate gradient algorithm(SCG) Within the study the settlement data of metrotunnels highway soft foundations and roads and high-risebuildings are taken as soft foundation engineering examplesree neural networks BP neural network the traditionalWNN based on backpropagation (BP algorithm) andWNNbased on SCG algorithm are compared and analyzedcomprehensively is article designed an optimization ofWNN built on the SCG method to predict soft soil foun-dation engineering settlement under the complex geologysettings to overcome issues mentioned above e resultsshow that the optimization model achieves a better per-formance and is more suitable than the other two networksfor soft soil foundation engineering settlement prediction

e following are the offerings of this research work

(1) We investigate the soft soil foundation engineeringsettlement prediction and wavelet neural networkWe explain the methodology that we have adoptedduring our proposed work and perform experimentfor our three techniques ie BP Neural NetworkWNN-based BP Algorithm and Improved WaveletNeural Network

(2) We compare the prediction accuracy obtainedduring the three techniques mentioned above

(3) Our proposed work improves the stability andconvergence accuracy of WNN Hence the initialparameter setting method can be linked with wavelettype wavelet time-frequency parameters andlearning samples

(4) From our proposed scheme we concluded that SCGalgorithm combined with the autocorrelation cor-rection can determine the number of hidden layernodes us an improved WNN can be derivedsuccessfully

e rest of the research work consists of the followingSection 2 explains the related work Section 3 illustratesmaterial and methodology used during our proposed workSection 4 deliberates our experimental work that we haveperformed during our work and finally the paper is con-cluded in Section 5

2 Related Work

Soft soil is found all around the world Its unique charac-teristics ie high void ratio high water content highcompressibility low shear strength low permeability andunique structural characteristics necessitate particularconsideration in the study construction and maintenanceof geotechnical structures built on them Large-scale

construction of high-speed transit systems high-risestructures and subterranean works for numerous urbancenters built on such soils is a huge problem Many re-searchers have attempted to overcome the issues that arise inthe foundation of soft soils e authors in [10] used a three-dimensional (3D) analytic approach to investigate the in-fluence of pavement smoothness on the dynamic behavior ofsubsurface movement loads e impact of unconventionaland low-carbon additions on the long-term treatment ofsoils for building and paving materials is investigated Datafrom three centrifuge tests during traffic loads are used tostudy the cyclic behavior of mud with a ground comprised ofovercompacted soil at the top

e researchers of [6] used a 3D dynamic finite-elementanalysis to estimate the permanent settling of a segment ofthe cross-river tunnel allowing for the influence of primarystress rotation and verifications for test data e disruptedstate concept (DSC) model for normally consolidated clayswas described in [11] which included the impacts ofcracking particle breaking heating softness and hardeninge overconsolidation ratio on strength militancy anddistortion is used to develop a novel disruption functionConsolidation theory numerical computation and curvefitting are three types of approaches for forecasting groundsettlement ere have been several recent works on con-solidation theory [12ndash14]

In a novel approach the authors of [15] described thesettlement of an inspired embankment on a soft basis basedon a classic hyperbolic approach and used the system de-formations features reflected in preloaded embankments toforecast the settlement during the later stage In [16] theauthors proposed a staged observational technique forpredicting embankment settlement on soft ground withstaged construction they discovered that immediate set-tlement adds to the shift distance of the parallel lines duringstaged construction e researchers in [17] used a geneticalgorithm to optimize a BP neural network to forecast thesummer electrical short-term load Because of its highconvergence rate and low memory use the scaled conjugategradient (SCG) method is used in NFC training As a resultthe authors of [18] trained a type-1 fuzzy system using acustomized form of SCG According to them the improvedSCG accelerates convergence in the steepest descent ap-proach of fuzzy system training As a consequence the SCGappears to be a good candidate for NFC training for large-scale issues Training NFC with SCG for large-scale issueson the other hand might take days or weeks on any personalcomputer Training NFC with SCG for large-scale issues onthe other hand might take days or weeks on any personalcomputer An alternate method for reducing training time isto calculate the Hessian matrix using first-order gradients asin conjugate gradient (CG) algorithms [19] A transfor-mation wavelet is a useful tool for data processing and time-frequency representation development e wavelet theoriesare described thoroughly in [20 21] In the context of neuralnetworks the application of wavelet transform is not newPrevious research [22 23] proposed a theoretical frameworkfor neural feed-forward networks based on wavelets estudy of [24] has investigated the capacity to employ

















32 Methodology






Ψ 2

Ψn

Ψ11

2

n

1

2

m

Training data+

Input data


X1

X2

X3

y1

y2

y3




Large settlement








O m + n

radic+ 1 (1)




E(θ) 12

1113944(f(x θ) minus y)2 (2)



E

wki

1113944P

p11113944

m


ψxprime

xi (3)

E

wjk

1113944P

p1fj minus yj1113872 1113873ψ


ak

1113888 1113889 (4)

E

ak

1113944

P

p11113944

m


ψak

(5)

E

bk

1113944P

p11113944

m


ψbk

(6)


xprime 1113944n

i1wkixi tprime

xprime minus bk

ak

(7)


ψak


221113872 1113873tprime

2

ak


221113872 1113873 tprime

ak

ψbk


221113872 1113873 tprime

ak


221113872 1113873 1

ak

ψtprime


221113872 1113873 tprime

ak


221113872 1113873 1

ak

(8)









xlowast

x minus xmin







Model accuracy


n minus 1

1113971

(10)



















3503

01522 17 03433 18 01644 19 01095 20 0278

Model accuracy 245



931

00792 17 00733 18 01184 19 00605 20 0200

Model accuracy 132



267

00582 17 00953 18 01034 19 00295 20 0095

Model accuracy 089




3503

0152

931

0079

267

005817 0343 0073 009518 0164 0118 010319 0109 0060 002920 0278 0200 0095Model accuracy 245 132 089













2589

02162 11 01233 12 02064 13 0172

Model accuracy 610



132

00152 11 00963 12 00554 13 0054

Model accuracy 182



616

01282 11 00963 12 01124 13 0103

Model accuracy 365







2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror






























































32 Methodology






Ψ 2

Ψn

Ψ11

2

n

1

2

m

Training data+

Input data


X1

X2

X3

y1

y2

y3




Large settlement








O m + n

radic+ 1 (1)




E(θ) 12

1113944(f(x θ) minus y)2 (2)



E

wki

1113944P

p11113944

m


ψxprime

xi (3)

E

wjk

1113944P

p1fj minus yj1113872 1113873ψ


ak

1113888 1113889 (4)

E

ak

1113944

P

p11113944

m


ψak

(5)

E

bk

1113944P

p11113944

m


ψbk

(6)


xprime 1113944n

i1wkixi tprime

xprime minus bk

ak

(7)


ψak


221113872 1113873tprime

2

ak


221113872 1113873 tprime

ak

ψbk


221113872 1113873 tprime

ak


221113872 1113873 1

ak

ψtprime


221113872 1113873 tprime

ak


221113872 1113873 1

ak

(8)









xlowast

x minus xmin







Model accuracy


n minus 1

1113971

(10)



















3503

01522 17 03433 18 01644 19 01095 20 0278

Model accuracy 245



931

00792 17 00733 18 01184 19 00605 20 0200

Model accuracy 132



267

00582 17 00953 18 01034 19 00295 20 0095

Model accuracy 089




3503

0152

931

0079

267

005817 0343 0073 009518 0164 0118 010319 0109 0060 002920 0278 0200 0095Model accuracy 245 132 089













2589

02162 11 01233 12 02064 13 0172

Model accuracy 610



132

00152 11 00963 12 00554 13 0054

Model accuracy 182



616

01282 11 00963 12 01124 13 0103

Model accuracy 365







2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror



















































O m + n

radic+ 1 (1)




E(θ) 12

1113944(f(x θ) minus y)2 (2)



E

wki

1113944P

p11113944

m


ψxprime

xi (3)

E

wjk

1113944P

p1fj minus yj1113872 1113873ψ


ak

1113888 1113889 (4)

E

ak

1113944

P

p11113944

m


ψak

(5)

E

bk

1113944P

p11113944

m


ψbk

(6)


xprime 1113944n

i1wkixi tprime

xprime minus bk

ak

(7)


ψak


221113872 1113873tprime

2

ak


221113872 1113873 tprime

ak

ψbk


221113872 1113873 tprime

ak


221113872 1113873 1

ak

ψtprime


221113872 1113873 tprime

ak


221113872 1113873 1

ak

(8)









xlowast

x minus xmin







Model accuracy


n minus 1

1113971

(10)



















3503

01522 17 03433 18 01644 19 01095 20 0278

Model accuracy 245



931

00792 17 00733 18 01184 19 00605 20 0200

Model accuracy 132



267

00582 17 00953 18 01034 19 00295 20 0095

Model accuracy 089




3503

0152

931

0079

267

005817 0343 0073 009518 0164 0118 010319 0109 0060 002920 0278 0200 0095Model accuracy 245 132 089













2589

02162 11 01233 12 02064 13 0172

Model accuracy 610



132

00152 11 00963 12 00554 13 0054

Model accuracy 182



616

01282 11 00963 12 01124 13 0103

Model accuracy 365







2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror

















































xlowast

x minus xmin







Model accuracy


n minus 1

1113971

(10)



















3503

01522 17 03433 18 01644 19 01095 20 0278

Model accuracy 245



931

00792 17 00733 18 01184 19 00605 20 0200

Model accuracy 132



267

00582 17 00953 18 01034 19 00295 20 0095

Model accuracy 089




3503

0152

931

0079

267

005817 0343 0073 009518 0164 0118 010319 0109 0060 002920 0278 0200 0095Model accuracy 245 132 089













2589

02162 11 01233 12 02064 13 0172

Model accuracy 610



132

00152 11 00963 12 00554 13 0054

Model accuracy 182



616

01282 11 00963 12 01124 13 0103

Model accuracy 365







2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror






















































3503

01522 17 03433 18 01644 19 01095 20 0278

Model accuracy 245



931

00792 17 00733 18 01184 19 00605 20 0200

Model accuracy 132



267

00582 17 00953 18 01034 19 00295 20 0095

Model accuracy 089




3503

0152

931

0079

267

005817 0343 0073 009518 0164 0118 010319 0109 0060 002920 0278 0200 0095Model accuracy 245 132 089













2589

02162 11 01233 12 02064 13 0172

Model accuracy 610



132

00152 11 00963 12 00554 13 0054

Model accuracy 182



616

01282 11 00963 12 01124 13 0103

Model accuracy 365







2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror


























































2589

02162 11 01233 12 02064 13 0172

Model accuracy 610



132

00152 11 00963 12 00554 13 0054

Model accuracy 182



616

01282 11 00963 12 01124 13 0103

Model accuracy 365







2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror




















































2589

0216

616

0128

132

001511 0123 0096 009612 0206 0112 005513 0172 0103 0054Model accuracy 610 365 182



1145

00602 15 00563 16 00704 17 00375 18 00096 19 00017 20 00318 21 0088

Model accuracy 047



3201

00962 15 00183 16 00484 17 00115 18 00126 19 00147 20 00618 21 0035

Model accuracy 043



254

00252 15 00433 16 00534 17 00245 18 00296 19 00147 20 00108 21 0071

Model accuracy 035





3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror


















































3201

0096

1145

0060

254

002515 0018 0056 004316 0048 0070 005317 0011 0037 002418 0012 0009 002919 0014 0001 001420 0061 0031 001021 0035 0088 0071Model accuracy 043 047 035

16 17 18 19 20

Measurement Number

0

005

01

015

02

025

03

035

Rela

tive E

rror



10 11 12 13

Measurement Number

0

005

01

015

02

025

Rela

tive E

rror




5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror















































5 Conclusion


Data Availability






Acknowledgments


References










14 15 16 17 18 19 20 21

Measurement Number

0

002

004

006

008

01

012

Rela

tive E

rror















































application of the wnn-based scg optimization algorithm

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