The Prediction of Pile Foundation Buried Depth Based on BP ...

Research ArticleThe Prediction of Pile Foundation Buried Depth Based on BPNeural Network Optimized by Quantum ParticleSwarm Optimization

Fei Yin Yong Hao Taoli Xiao Yan Shao and Man Yuan

School of Urban Construction Yangtze University Jingzhou Hubei China

Correspondence should be addressed to Yong Hao 518004yangtzeueducn

Received 19 April 2021 Revised 28 May 2021 Accepted 16 June 2021 Published 24 June 2021

Academic Editor Faming Huang

Copyright copy 2021 Fei Yin et al+is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to the fluctuation of the bearing stratum and the distinct properties of the soil layer the buried depth of the pile foundationwill differ from each other as well In practical construction since the designed pile length is not definitely consistent with theactual pile length masses of piles will be required to be cut off or supplemented resulting in huge cost waste and potential safetyhazards Accordingly the prediction of pile foundation buried depth is of great significance in construction engineering In thispaper a nonlinear model based on coordinates and buried depth of piles was established by the BP neural network to predict thesamples to be evaluated the consequence of which indicated that the BP neural network was easily trapped in local extreme valueand the error reached 31 Afterwards the QPSO algorithm was proposed to optimize the weights and thresholds of the BPnetwork which showed that the minimum error of QPSO-BP was merely 94 in predicting the depth of bearing stratum and29 in predicting the buried depth of pile foundation Besides this paper compared QPSO-BP with three other robust modelsreferred to as FWA-BP PSO-BP and BP by three statistical tests (RMSE MAE and MAPE) +e accuracy of the QPSO-BPalgorithm was the highest which demonstrated the superiority of QPSO-BP in practical engineering

1 Introduction

Pile foundation is one of the oldest foundation forms Withthe development of history pile foundation has become themost commonly used foundation form of high-rise build-ings significant structures tunnels bridges offshore plat-forms and other structures on soft ground [1] Pile as amember of foundation structure is vertically or aslant set inthe soil and has certain stiffness and bending shear capacityIt allows itself to pass through the soft compressible soillayer compact the weak soil and transmit part or all of theload from the superstructure to the soil layer or rock belowwith low compressibility and certain bearing capacity thusavoiding the excessive settlement of the foundation andimproving the bearing capacity of the soil layer [2] As one ofthe most important steps in foundation construction theconstruction of pile foundation is a large-scale projectmeanwhile there is also a problem of waste of materials It is

usually found that the designed length of pile is much higheror lower than the required value in actual engineering whichcan be seen from Figure 1

Each pile to be bored would be surveyed in advanceaccording to current construction technique +ere is aconstruction technique of ldquoone pile with one investigationrdquoor ldquoone pile with lots of investigationrdquo [3] namely each pileto be driven will be investigated in advance and the pilelength is designed by the elevation of pile bottom which isestimated by the most unfavorable principle but thistechnique is not adopted in every project [4] Only a fewsurvey boreholes are arranged to predict the soil distributionof the whole site in general engineering

When the irregular fluctuation of the bearing stratumchanges greatly it may result in a large elevation differencebetween the undrilled area and the nearby drilled area+erefore in the area where the bearing stratum is relativelyshallow the pile will reach the bearing stratum too early and

HindawiAdvances in Civil EngineeringVolume 2021 Article ID 2015408 15 pageshttpsdoiorg10115520212015408

cannot continue going deeper At this time the pile head willextend too long from the soil and needs to be cut off whichcan be seen from Figure 2 below On the contrary in the areawhere the bearing stratum is deeply distributed the pilelength will be insufficient and needs to be supplemented+ereason for this situation is that the variety of pile length islimited especially the prefabricated pile If the constructionis carried out according to the designed pile length a largenumber of piles will be cut off or supplemented resulting inunnecessary waste in pile foundation engineering+ereforethis paper predicted the buried depth of pile foundation andbearing stratum then made targeted technical schemepreparation and construction deployment in practical en-gineering according to the prediction results

At present there are few research studies on predictingthe buried depth of pile foundation and the fluctuation ofbearing stratum however in the field of pile foundationengineering many scholars have made some achievementsIn the static load test of pile foundation finished by Qi et al[5] a first-order linear dynamic differential equation wasderived by studying the settlement of pile under variousloads On the basis of gray system theory the GM (1 1)model of load-settlement relationship of single pile wasestablished and employed to predict the ultimate bearingcapacity and the complete load-settlement relationshipDespite the accurate prediction results obtained by thismethod the uniformity of original data should be ensuredAccording to the measured data Gao et al [6] adopted thehyperbolic method to predict the bearing capacity ofsqueezed branch pile Although the error between thepredicted results and the measured curve was not great thevalues predicted by this method were generally too large andhad certain limitations Deng et al [7] used the superlongand large-diameter cast-in-place pile foundation of theSutong Yangtze River Bridge as an example to calculate itssettlement amount by adopting different specifications andthen a new empirical formula considering pile compressionand modifying additional stress of the pile tip was proposed

by comparing with the settlement value of the large-scalecentrifugal model test Afterwards this formula was appliedto verify the settlement value of a super large-scale pile groupfoundation on the Nei-Kun Line and the calculation resultof which was relatively consistent with the measured dataHowever this formula was not suitable for the analysis ofsingle pile settlement which had certain limitations as well

Most of the forecasting methods mentioned above relyon a fixed knowledge framework and can only be adoptedunder certain preconditions which are rigid and not flexibleenough As a consequence an intelligent technology that candeal with various problems flexibly and has self-learningawareness will be needed Machine learning as a technologythat computers build models based on data to simulatehuman activities can meet this condition It possesses thestrong generalization ability and has been applied in dif-ferent aspects Methods of machine learning were used byAhmadi et al [8] to successively predict the solubility ofcarbon dioxide (CO2) in brines porosity and permeability ofpetroleum reservoirs the amount of dissolved calciumcarbonate concentration throughout oil field brines con-densate-to-gas ratio in retrograded condensate gas reser-voirs solubility of hydrogen sulfide (H2S) in ionic liquidsetc +e prediction of pile foundation depth belongs to ahighly nonlinear problem which can also be analyzed by thismethod +e artificial neural network (ANN) is a kind ofmachine learning which is a powerful intelligent learningtool with functions [9 10] such as mapping nonlinear re-lations information processing optimization calculationclassification and recognition It has been widely applied inthe fields of model prediction content prediction costcontrol fault diagnosis information processing construc-tion engineering mechanical engineering medicine etcand the results were great as expected Hamid et al [11] builtan ANN model based on the critical pressure (Pc) criticaltemperature (Tc) and molecular weight (Mw) of pure ionicliquids to predict the solubility of hydrogen sulfide (H2S) indifferent temperature pressure and concentration ranges

(a) (b) (c)

Figure 1 +e buried depth of piles in the site varied greatly

2 Advances in Civil Engineering

Moosavi et al established an ANN model based on 214 datarecords of published CO2-foam injection tests into oil-reservoir cores to predict the CO2-foam flooding perfor-mance for improving oil recovery Shang et al [12] proposedtwo kinds of ANN models to analyze the content and typesof heavy metals in the soil based on the complex permittivityof materials so as to determine whether the soil is con-taminated +e first ANNmodel was adopted to confirm thepresence of heavy metals in the site and the second modelwas employed to classify the presence of heavy metals in thesite Alias et al [13] completed the cost of the skeletal systemof a building project through the ANN model Sorsa et al[14] detected and diagnosed the faults in the testing processbased on three different ANN structures Jin et al [15]established an ANN model with water-cement ratio spec-imen shape and section size as input parameters based onthe size effect of concrete compressive strength then pre-dicted the compressive strength with different section sizesSuzuki [16] found it feasible to apply the improved ANN toreduce false positives in computer detection of pulmonarynodules in low-dose computed tomography images and theresults turned out to be great

+e emergence of ANN provides a more convenient andintelligent prediction way for many fields It no longer needsa large amount of statistical data to predict the future trendbut can achieve a good prediction effect based on limiteddata +erefore in this paper the highly nonlinear problemof the prediction of pile foundation burial depth can besolved by relying on ANN

Among them the backpropagating (BP) neural networkis the most widely used artificial neural network which is amultilayer feedforward neural network trained according tothe error backpropagation algorithm It has a certain abilityof summary and extension Some mathematical analysis hasdemonstrated its ability to deal with any nonlinear problemto solve complex internal mechanisms [17] However thereare still some shortcomings of the traditional BP network atpresent (1) It is a complex process for the BP neural network

to optimize the objective function by using the gradientdescending method and when the output of neurons is inthe vicinity of 0 and 1 the weight error only changes within asmall range +is phenomenon causes training to almoststop so that the efficiency of the BP neural network is lowand the speed of the BP neural network is at a slow pace [18](2) From the mathematical point of view the algorithm usedby the BP neural network is mainly to search the local areawhich will easily fall into the local extreme value +e finaltraining curve presented is almost going to be a straight lineas a result of which the training of the network will fail [19](3) +e prediction ability of the network is proportional tothe learning ability within a certain range but once beyondthis range the prediction ability of the BP neural networkwill decline with the improvement of the learning abilitywhich leads to the phenomenon of ldquooverfittingrdquo At thistime even if the network has learned a large sample itcannot directly and correctly reflect its rules [20] To sum upthere will be plenty of deficiency when the BP neural net-work is solely used for prediction as a result it is necessaryto apply some algorithms to optimize the BP neural networkand then establish a more accurate training model

In recent years several swarm intelligence optimizationalgorithms have emerged in an endless stream +is kind ofalgorithm is also one of the optimization algorithms thatscholars pay the most attention to which has the charac-teristics of simplicity and high efficiency when comparedwith others [21] So that they have been also widely adoptedin various fields +e swarm intelligence algorithm uses thegroup relations among some animals or individuals in thesociety such as interaction heredity variation cooperationand other behaviors to achieve the purpose of searching foroptimal solution

Fireworks algorithm (FWA) is a new swarm intelligenceoptimization algorithm proposed by Tan et al [22] in recentyears which simulates the mechanism of the simultaneousexplosion and diffusion of the firework explosion operatorIt introduces the idea of concentration suppression in the

(a) (b) (c)

Figure 2 +e section of the pile that has been cut off

Advances in Civil Engineering 3

immune algorithm and the mechanism of distributed in-formation sharing thus having stronger global search ca-pabilities [23] Compared with traditional algorithms thepopulation of FWA is more diverse and its characteristicshave also attracted the attention of many scholars Howeverthe FWA still has several shortcomings For example whenthe explosion point range is large and there are many ex-plosion operators the targets generated by the explosion willoverlap resulting in irrelevant searches +en it will greatlyaffect the optimization efficiency of FWA which is the majorcause of slower convergence speed and lower search accu-racy [24]

Particle swarm optimization (PSO) algorithm [25] asone of the most classic optimization algorithms is inspiredby the foraging behavior of birds It seeks the optimal valuein the stochastic solution of particle swarm through constantiteration Compared with FWA it has the advantages ofsimple operation and fewer parameters to be adjusted [26]Ahmadi et al [27] used the neural network model optimizedby the PSO algorithm to predict asphaltene precipitation dueto natural depletion Wang et al [28] predicted the me-chanical properties of hot rolled strip steel in materialprocessing based on the PSO-BP model Likewise this PSO-BPmodel was applied by Ismail et al [29] in the field of soil-structure composite interaction to predict the load-deformationcharacteristics of axially loaded piles as well Shafiei et al [30]predicted the solubility of hydrogen sulfide in differenttemperature pressure and concentration ranges in the sameway Ahmadi et al used the PSO-ANN model to predict thedew point pressure of condensate gas reservoir In anotherpaper by the same author estimation of efficiency ofchemical flooding in oil reservoirs was predicted Althoughthe above literature studies have achieved relatively goodprediction results the PSO algorithm still has plenty ofproblems and it has been proved that it is not a globallyconvergent algorithm [31] In the meantime it also has someproblems to be solved such as premature convergence lackof dynamic adjustment of velocity easy to fall into localextreme value lack of randomness in particle positionchange inability to effectively deal with discrete and com-binatorial optimization and limitation of search space [32]From the perspective of dynamics there is a point withpotential energy field in the search area that attracts theparticle swarm causing the surrounding particles to con-stantly approach this pointWhen the velocity decreases to 0the particles converge to this point as well +erefore themotion of each particle in the traditional PSO algorithm iscarried out along a fixed orbit the velocity of the particle isalways a finite value and the search area of its feasiblesolution is also small [33] In order to improve the globaloptimization capability of PSO this algorithm needs to beoptimized As a result the concept of quantum particleswarm algorithm (QPSO) was proposed by Sun [34]

Based on the traditional PSO algorithm the QPSO al-gorithm randomizes the velocity of the particle In thequantum space the state of the particle is not represented byposition and velocity vector any more but by wave functionIn this way within the feasible region the probability ofparticles appearing at a position is random and the motion

of particles is no longer along a fixed orbit +eir updatedposition in the next second has no correlation with theprevious position that is the search can be carried out in thewhole feasible solution region which improves the globaloptimization performance of particles Chen et al [35] tookthe gear reducer of belt conveyor as the research object andoptimized the parameters such as modulus tooth widthcoefficient and helical angle of the gear reducer based on theQPSO and PSO algorithm +e results showed that theoptimization effect of QPSO was obviously better than thatof PSO Genetic algorithm (GA) and ANN PSO and QPSOalgorithms were used by Lu et al [36] to predict the pa-rameters of the batch fermentation kinetic model +e re-sults demonstrated that the prediction effect of QPSO in allaspects was superior to that of other algorithms +ereforeon the basis of the BP neural network this paper used theQPSO algorithm to optimize the BP model and then pre-dicted the buried depth of pile foundation Finally threeerror analysis tools RMSE MAE and MAPE were re-spectively employed to analyze its reliability anduncertainty

Based on the above this paper provides the followingcontributions (1) In this paper the ANN in machinelearning was used to predict the buried depth of pilefoundation However there were very few researchstudies on this topic before as a result it can be applied asa new field in practical engineering (2) In this paper thesamples of piles were collected on the spot based onengineering examples +e relevant parameters of pilesamples in this area were sorted out and summarizedwhich were X-coordinate Y-coordinate Z-coordinatethickness of miscellaneous fill h1 thickness of silty clayh2 thickness of silt h3 thickness of fine sand h4 and pileburied depth H Some samples were selected as trainingmodels (3) In this paper the steps of predicting testobjects after optimizing the BP neural network by theQPSO algorithm were described in detail (4) +is paperused the QPSO algorithm to optimize the BP neuralnetwork for modeling training and then predicted theremaining samples in step (2) +e great global optimi-zation of the QPSO algorithm successfully made up forthe defect that the traditional PSO algorithm was easy tofall into the local extreme value and the prediction resultswere very close to the measured results indicating thatthis method had achieved a good prediction effect in theresearch objects (5) +is paper compared the errors ofthe QPSO algorithm with other robust models PSO al-gorithm FWA and BP neural network +e resultsshowed that QPSO had higher prediction accuracy

+is paper also introduces the following parts Section 2introduces training parameters based on project exampleSection 3 describes the concept of BP neural network and theoptimization methods of PSO and QPSO algorithms Sec-tion 4 is the error analysis after using different algorithms tooptimize the BP neural network for prediction Section 5 isthe conclusion of the above description and the analysis ofthe predicted results At the same time this paper also givesan overview of how to apply this method in engineeringexamples with similar soil propriety


2 Project Example

21 Project Profile +e project is located in the East Campusof Yangtze University in Jingzhou District Jingzhou CityHubei Province which was to build dormitory and canteenin this area According to the design document this in-vestigation site with pile location layout is shown inFigures 3ndash5 below

+e distribution of boreholes and piles can be obtainedfrom the figure Each long black dotted line such as ldquo11-11primerdquorepresents ldquo11-11prime sectionrdquo of boreholes from K64 to K67According to the section drawings the soil stratification ateach borehole fromK64 to K67 can be known First of all thepiles at the boreholes were selected as the data of the networkmodel X-coordinate Y-coordinate and Z-coordinate ofeach pile were taken according to the coordinate informa-tion provided by layout drawings and the length of thebearing stratum and the buried depth of pile were obtainedfrom the section drawings In order tomake the selected datarepresentative 43 piles were randomly selected as thetraining samples and 10 piles were randomly selected as theprediction samples from the boreholes of investigation in thefigure With the difference of the geographical location thefluctuation of the bearing stratum of the site will have acertain trend of change as well +e process of driving thepile into the bearing stratum needs to pass through differentsoil layers on the upper side However the thickness of eachsoil layer at the undetected coordinates is an uncertainunknown As the thickness of the soil layer is different thedepth of the bearing layer changes to another number whichwill affect the buried depth of the pile According to the fielddata the piles were all driven into the fine sand layer whichindicated that the fine sand layer was the bearing stratum+e depth of the sample pile into the first layer of soil calledmiscellaneous fill is h1 the depth of the sample pile into thesecond layer of soil called silty clay is h2 the depth of thesample pile into the third layer of soil called silt is h3 and thedepth of the sample pile into the fourth layer of soil calledfine sand is h4 which presents the bearing stratum H is thesum of h1 h2 h3 and h4 which represents the buried depthof pile +e depth of the sample pile into different layers ofsoil can be calculated by combining the geological profileand the data of pile buried depth H measured from actualengineering Based on the above the X-coordinate Y-co-ordinate and Z-coordinate of each pile were collected asinput parameters for model training In this training topredict the fluctuation of the bearing stratum is to predictthe depth of h4 to predict the buried depth of pile is topredict the depth ofH and the thickness of h4 is less than thethickness of fine sand layer In additionH h1+ h2+ h3+ h4H and h4 are output parameters +e schematic diagram isshown in Figure 6

22 Geological Overview +e terrain of the site is relativelyflat and the absolute elevation value of the ground is in therange of 315mndash3288m which belongs to the first-gradeterraced geomorphic unit on the north bank of the YangtzeRiver+ere is no adverse geological action such as landslide

soil collapse and debris flow According to the detailedinvestigation report of the site the area within this depthrange can be divided into artificial fill soil layer QuaternaryHolocene alluvium and Quaternary Upper Pleistocene al-luvium and diluvium according to its genetic type andsedimentary age [37]

According to their properties and composition thegeotechnical layers can be classified into the following partswhich are distributed as follows (1) artificial fill soil layer(Qml) miscellaneous fill brown moist and loose +e maincomponent is clay containing a small amount of plantrhizomes +is layer is distributed in the whole field and thesoil uniformity is poor +e thickness is 04mndash17m (2)Quaternary Holocene alluvium (Qal

4 ) silty clay yellowish-brown to grayish-brown soft to plastic saturated full-fielddistribution +is layer contains a small amount of ferro-manganese nodules and medium compressibility +ethickness is 45mndash148m (3) Quaternary Holocene allu-vium (Qal

4 ) silt gray slightly density to medium densitysaturated full-field distribution medium compressibility

+e thickness is 19mndash124m (4) Quaternary Holocenealluvium (Qal

4 ) fine sand gray medium density saturatedfull-field distribution mainly composed of quartz andfeldspar and low compressibility +e thickness is39mndash165m (5) Quaternary Upper Pleistocene alluviumand diluvium (Qal+pl

3 ) pebbles gray white and other colorsmedium dense to dense state low compressibility and full-field distribution +e main component is quartzite withgood roundness and poor sorting+e particle size is generally3sim5 cm and the larger particle size is greater than 7 cm ofwhich the particle size greater than 2 cm accounts for about51 +e filling material between pebbles is fine silty sand

It can be seen from the above data that the thickness anduniformity of each layer are greatly different

3 Optimization Algorithms for PileDepth Prediction

31 Implementation of BPNeural Network Algorithm As thename suggests the neural network is an artificial intelligencealgorithm to simulate the human brain nervous systemwhich has a strong self-learning ability and can deal withcomplex nonlinear models [9 10] +rough the connectionsof countless neurons it can carry out huge parallel pro-cessing and analysis on the information of the previous inputlayer and then pass it to the next layer A large amount oftraining can constantly update the weights of the neuronalconnections in the front and rear layers so as to achieve thegoal of reducing error and meeting peoplersquos expectations

X-coordinate Y-coordinate and Z-coordinate of pile wereregarded as input parameters for model training of the BPalgorithm Besides the depth of bearing stratum h4 and burieddepth of pile H were regarded as output parameters +e de-tailed process can be described as the following steps (1) Atraining model based on X-coordinate Y-coordinate Z-coor-dinate h4 andH of 43 training samples was established (2)+eh4 andH of 10 remaining sampleswere predicted (3)+e valuesof output parameters were compared with measured values (4)


+e error between predicted values and measured values wasanalyzed During the process of prediction there was no cor-relation between the input and output parameters which was anonlinear function As a result the three-layer network fornonlinear function can meet the training requirements in theBP algorithm [17] +e diagram is shown in Figure 7

+e S-type action function shown in (1) is its activationfunction [38]

f(x) 1

1 + eminus x (1)

where xk is the input parameter of the input layer vj is theoutput parameter of hidden layer yk is the output parameterof output layer wjk is the connection weight of neuronsbetween the input layer unit and the hidden layer unit andwij is the connection weight of neurons between hiddenlayer unit to output layer unit +e number of neurons ofinput layer hidden layer and output layer is respectively nm and l

+e training process was as follows

(1) Since the activation function of the neural network isa logarithmic S-type function it may have the

problem of convergence that is the infinite or in-finitesimal results appear in the calculation process+erefore the input data of X-coordinate Y-coor-dinate and Z-coordinate and output data h4 andH ofthe samples should be normalized first which was tomake these values vary from 0 to 1

(2) +e values randomly generated in the interval [minus1 1]were taken and the initial values were assigned to theweights

(3) +e independent variable parameters of the pro-cessed sample data were input at the correspondingnodes of the input layer and the output values of theBP neural network were calculated at the corre-sponding nodes of the output layer through theaction of weight and activation function

(4) +e output values of the BP neural network trainingwere compared with the expected values and thenthe error between them was calculated

(5) +e error obtained was propagated back from theoutput layer and the weight was corrected accordingto the gradient method when it reached the firstlayer and then step (3) was repeated andrecalculated

(6) +e above steps (1)ndash(5) were repeated until the errorfunction satisfied equation

E 12

1113944

m

i1yi minus oi( 1113857

2 le ε (2)

After the BP neural network had been trained accordingto the above steps the trained network model could be usedto predict the samples Based on the measured data fromengineering project 43 and 10 piles were respectively se-lected as input and test vectors and each pile was deter-mined by three parameters

+e BP algorithm with single hidden layer wasadopted in this paper It was difficult to determine thenumber of neurons in the hidden layer and the neuronsaffected the determination of accuracy to a certain degreeIf the number of neurons was too small the algorithm hadalmost no ability to train On the contrary if the numberof neurons was too large the training time would beextended which was easy to fall into the local optimalsolution As a result the normal predicted values werenot available to obtain Generally there are threemethods to identify hidden layer neurons [39] (1) ForFangfaGorman theory the relation between the numberof neurons S and the input parameter N is S log2N (2)for Kolmogorov theory the relation between the numberof neurons S and the input parameter N is S 2N+ 1 (3)the relation between the number of neurons S and theinput parameter N and the output parameter M is S sqrt(043MN+ 012NN+ 254M+ 077N+ 035) + 051 +einput parameterN was 3 and the output parameterM was2 in this BP training thus the calculated S using theabove three method was respectively 158 7 and 345Since the number of neurons needed to be an integer

8

75 6

7prime 5prime

6prime

8prime

K42233

23122327

2341

2351

23161231472313323120

23323

23314

23350 23361

2322923217

23203

2318323175

23450 2343823424

23269 23306

K43 K44 K45

K46

K38

K32

K39 K40

23 student dormitory

C13

C14

K41

K31

K37K24

K33 K34K35

K36K30

K25 K26 K27K28 K29

C12

C8

C9

C11

C10

Figure 4 +e investigation site with pile location layout of 23student dormitory

Dining hall

11

10

9 9prime

10prime

11primeK64 K65 K66 K67

K60 K61 K62 K63

K56 K57 K58 K59

ST160

ST151

ST148 ST139

ST2ST10 ST15

Figure 3 +e investigation site with pile location layout of dininghall


three values of 2 4 and 7 were selected for prediction andthe errors of them were compared in the later section

+e transfer function in hidden layer and output layerwas S-type tangent function and logarithmic function re-spectively +e network training function was ldquotraingdxrdquogradient descent method was used during learning processand the learning rate was adaptive

32 ImplementationofPSO-BPAlgorithm +eprocess of theBP algorithm optimized by PSO is shown as follows [40]

(1) Firstly the maximum number of iterations requiredthe number of independent variables required by theobjective function the maximum particle velocityand the position information of particles were set asthe whole search space In addition the velocity andposition coordinates were initialized randomly in thevelocity interval and search space and each particlewas given an initialized random flight velocity

(2) +e fitness function was defined and each particlewould have an extreme value which was the indi-vidual extreme value and the optimal solution of theunit particle+en a global value was found from theoptimal solution of all particles that was the globaloptimal solution Finally this optimal solution wasupdated after comparing with the global optimalsolution obtained in history

(3) +e updating velocity and position [41] were re-spectively shown in equations

Vi d ωVi d + C1random(0 1) Pi d minus Xi d( 1113857

+ C2random(0 1) Pg d minus Xi d1113872 1113873(3)

Xi d Xi d + Vi d (4)

where ω is inertial factor and a nonnegative valueWhen ω is large the ability to find the global optimalsolution is strong but the ability to find the local

optimal solution is weak When ω is small the abilityto find the global optimal solution is weak but theability to find the local optimal solution is strong+erefore the ability to find the global and localoptimal solution can be adjusted by different valuesof ω C1 and C2 are learning factors and currentresearch studies [42] have investigated that a bettersolution can be obtained when C1 and C2 are con-stants +e values of C1 and C2 are between [0 4]which are equal to 2 in general+e random (0 1) is arandom value on the interval [0 1] Pid is the in-dividual extremum of i-th variable at d-dimensionPgd is the global optimal solution at d-dimension

+e weights and thresholds optimized by PSO can beassigned as initial value for training and prediction of the BPalgorithm [40] +e detailed process is shown in Figure 8

+e PSO-BP algorithm can be realized in two methods (1)By combining the powerful global searching ability of the PSOalgorithm with the local searching ability of the BP neuralnetwork the global searching performance of the PSO algo-rithm is used to compensate for the topological structureweight and threshold of the BP neural network so as to op-timize the generalization and training ability and the overallsearching performance of the BP neural network (2) +e BPalgorithm is added to the PSO algorithm and the optimizationperformance of the PSO algorithm is improved through thepowerful training and learning skills of the neural networkwhich can reduce the huge requiredworkload and accelerate theconvergence of the PSO algorithm In this paper the firstmethod was adopted to obtain the optimal initial thresholdthrough the PSO algorithm and it was assigned to the BPalgorithm to improve the efficiency and accuracy

However in the PSO algorithm the convergence form ofthe particle is along the orbit and the maximum velocity ofthe particle is always a finite value which leads to certainlimitations in the search area of the PSO which cannotguarantee that it can search the whole feasible space and theglobal convergence will be affected [33]

33 Implementation of QPSO-BP Algorithm Based on thetraditional PSO algorithm the QPSO algorithm randomizesthe velocity of the particle In the quantum space the state ofthe particle is not represented by position and velocityvector but by wave function Due to the uncertaintyprinciple the probability of a particle appearing at a certainplace x is expressed by a probability density function and nolonger along a fixed orbit As a result the position of theparticle has no relationship with the previous position [43]+e evolution equations of each dimension of the particlestate are shown by the following equations

pi d(t) φi dPi d(t) + 1 minus φi d(t)( 1113857Pg d(t) (5)

Xi d(t + 1) pi d(t) plusmn12

Li d(t) times In1

ui d(t)1113890 1113891 (6)

where Pid is the attractor of the i-th particle in the evolu-tionary iteration Xid is the current position of the i-th


K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6



particle φid and uid are uniformly distributed randomnumbers on [0 1] and Lid represents the characteristiclength of the attractor and potential well which is used todescribe the search range of a particle and Lid can be shownin the following equation [43]

Li d(t) 2α times MBPi d(t) minus Xi d(t)1113868111386811138681113868

1113868111386811138681113868 (7)

where MBPi d(t) is the mean best position and α is thecompression-expansion factor

By substituting equations (7) into (6) the iterativeequation (8) below of quantum group evolution can beobtained

Xi d(t + 1) pi d(t) plusmn α times |MBPi d(t) minus Xi d(t) times In1

ui d(t)1113890 1113891

(8)

+e size of the particle swarm is set as M +e imple-mentation of its specific steps is as follows (1) Initialize theparticle swarm and set the maximum number of iterations

(2) Determine and initialize the individual optimal extre-mum and global optimal extremum of the particle swarm(3) +e fitness value of each particle is calculated (4) +eindividual optimal extremum of each particle and globaloptimal extremum of the particle swarm are updated (5)+e new position of the particle swarm is calculatedaccording to equation (8) and then the original particleswarm is updated (6) Repeat steps (2)ndash(5) until the fitnessvalues of the particle swarm meet the convergencecondition

+e above is the principle and realization of QPSO andthe method of using the QPSO algorithm to optimize the BPneural network is similar to that of PSO +e explanation inSection 32 above can be used as a reference

4 Analysis of Results

41 Prediction Results of Different Models +e original 43groups of data for training were shown in Table 1 whichwere prepared to form a database to predict the other 10

Miscellaneous fillSilty claySilt

Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl

Figure 6 Schematic diagram of h1 h2 h3 h4 and H


groups of samples to be evaluated where X-coordinate Y-coordinate and Z-coordinate were input parameters and h4and H were output parameters

+e original 10 groups of data to be evaluated are shownin Table 2

+e first was the prediction result of the BP neural networkAs mentioned above the number of hidden layer neurons wascalculated by three different methods which were 2 4 and 7respectively +erefore three different prediction results anderror comparison for h4 andH based on the number of neurons

in the hidden layer were obtained which are shown inFigures 9ndash12

It can be known from the above figures that differentnumbers of hidden layer neurons can affect the forecast resultsWhen the number of neurons in the hidden layer was 7 theerrors of the BP neural network in predicting h4 and H weresmaller than that of the other two neurons +is phenomenonindicated the forecast result of the secondmethodmentioned inSection 31 named Kolmogorov theorem was the best In ad-dition no matter how many neurons there were the errors

Input transform layer Input layer Output layer Output transform layer

Hide layer

Actual outputActual input

X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj

Figure 7 Schematic diagram of three-layer neural network transmission

Input data

Input preprocessing

Initialize the particle and velocity

Look for individual extremum and group

extremum

Update the velocity and location

Calculate the fitness value of particle

Update individual extremum and

group extremum

Meet the termination condition

N Y

Determine the topology structure of

network

Initialize the length of weight and threshold of

BP neural network

Get the optimal weight and threshold

Calculate the error

Update the weight and threshold

Meet the end condition

Obtain the experimental results by simulation and

prediction

Y

N

Regard the error obtained by BP neural network training

as fitness value

Figure 8 Flowchart of the PSO-BP algorithm


between the prediction results of the sample using the BP neuralnetwork algorithm and the actual values were still very largeEspecially for the prediction results of h4 the maximum errorwas up to about 41 when the number of neurons was 2 andwas up to about 31 when the number of neurons was 7Besides it was found in Figures 9 and 10 that the curves ofprediction results of the BP neural network were all relativelygentle and the basic trend was a straight line which proved thatthe BP network is easy to fall into the characteristics of searchingfor local optimal solution+erefore in order to compensate thelack of global search capacity of the BP network the QPSOalgorithm was going to be employed to optimize the BP net-work model when the number of neurons in the hidden layerwas 7

+e linear fitting formula for predicting h4 and H hadbeen calculated +e calculation results indicated that thedeviation between the prediction values and the actual valuesreached 588 which demonstrated the necessity of usingthe QPSO-BP algorithm In order to intuitively observe theerror comparison between different algorithms relativeerror was adopted to compare the accuracy of QPSO-BPPSO-BP and FWA-BP +e formula can be seen in thefollowing equation

ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)

where xp is the predicted value and xa is the actual value+e parameter settings of the BP neural network and

QPSO algorithm were as described below +e number ofiterations of the BP network was 1000 the training goal ofthe BP network was 001 the learning rate of the BP networkwas 0001 the population size of the QPSO algorithmwas 20the dimension of the QPSO algorithm was 30 and the it-eration termination error of QPSOwas 10minus7 Compared withother models QPSO was simple to operate and had fewerparameters to set

+e predicted curves of different models are shown inFigures 13 and 14 +e predicted error curves of differentmodels are shown in Figures 15 and 16

According to Figures 15 and 16 a conclusion can beconfirmed that the relative error of QPSO-BP was thesmallest compared with that of PSO-BP FWA-BP andlinear fitting In the process of predicting h4 the minimumrelative error was 94 the maximum relative error was only147 and all of the errors were basically around 11 in theprocess of predicting H the maximum relative error wasmerely 29 which confirmed the powerful predictionaccuracy of QPSO Furthermore the prediction curve of

Table 1 43 groups of data for training

No Pile number Borehole number X-coordinate Y-coordinate Z-coordinate h1 (m) h2 (m) h3 (m) h4 (m) H (m)1 3 K42 117510 1712880 319300 040 1260 380 660 23402 12 K43 322510 1742880 317000 070 1210 450 590 23203 27 K44 596510 1742880 319600 130 1260 310 670 23704 41 K45 824510 1756380 319200 080 1420 270 690 24605 51 K46 917000 1649380 319700 090 1270 330 760 2450⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮39 15 K66 530730 2465240 317400 140 1230 270 680 232040 160 K56 127330 2040270 316100 130 900 550 760 234041 151 K57 362730 2049410 315900 130 1190 300 710 233042 148 K58 530730 2035410 318600 120 1330 210 680 234043 139 K59 735230 2049410 319400 130 450 990 770 2340

Table 2 10 groups of data for prediction

No Pile number Borehole number X-coordinate Y-coordinate Z-coordinate h1 (m) h2 (m) h3 (m) h4 (m) H (m)1 314 K32 144510 1367880 319800 050 1100 490 810 24502 350 K33 250510 1244880 318700 120 1060 520 740 2440⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮9 306 K29 975713 1031992 320300 040 1430 390 590 245010 269 K28 740510 1025880 319100 110 1310 500 520 2440

Measured valueN = 7

N = 4N = 2

h 4 (m

)

1 2 3 4 5 6 7 8 9 100Sample serial number

3

4

5

6

7

8

9

10

Figure 9+e prediction curve of h4 when the number of BP neuralnetwork neurons was 2 4 and 7 respectively


QPSO-BP possessed the characteristic of fluctuation ratherthan an almost flat straight line like the BP network +isshowed that QPSO successfully compensated for the lack ofglobal search characteristics of BP and was not easy to fallinto the endless loop of finding local optimal solution Aftercomparison and analysis the accuracy of these models indescending order was QPSO-BPgtPSO-BPgt FWA-BPgt linear fitting

Figure 17 shows the convergence curves of QPSO-BPPSO-BP and FWA-BP It can be seen from the figure thatthe curve of decreasing fitness of QPSO-BP started to be verysmooth and tended to a fixed value after iterating for about25 times and then the program stopped iterating when thenumber of iterations was around 143 which presented theoptimal value of QPSO-BP had been discovered Besidesfrom the comparison of the iterative curves of the other two

algorithms it can be seen that the decline rate of QPSO inthe early stage was the fastest and it was the first of the threemodels to converge in the subsequent iterative processwhich demonstrated the capability of fast search and con-vergence of QPSO

+e above conclusions indicated that it was feasible touse the QPSO-BPmethod in machine learning to predict theburied depth of pile foundation and the fluctuation ofbearing stratum and the particle swarm optimization al-gorithm was already a relatively mature optimization al-gorithm compared with many other algorithms which wasnot difficult to implement QPSO-BP has the advantages offast search and fast convergence speed simple operationand high precision therefore it was more reasonable toapply this algorithm in this paper

Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)

Figure 10 +e prediction curve of H when the number of BPneural network neurons was 2 4 and 7 respectively

Net

wor

k er

ror (

)

N = 7N = 4N = 2

2 3 4 5 6 7 8 9 101Sample serial number

ndash40

ndash20

0

20

40

60

Figure 11+e prediction error curve of h4 when the number of BPneural network neurons was 2 4 and 7 respectively

N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30

Figure 12 +e prediction error curve ofH when the number of BPneural network neurons was 2 4 and 7 respectively

Measured valuePSOQPSO

FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12

Figure 13 +e prediction curve of h4 by using different models


42 Error Analysis of Different Models In order to furtherprove the powerful prediction accuracy of QPSO threestatistical test methods were respectively used [44ndash46]namely RMSE MAE andMAPE+e three formulas can beobtained from equations (10)ndash(12) and the error compar-ison of different algorithms can be seen in Table 3 in detail

RMSE is root mean square error which represents thesquare root of the ratio of the square deviation between theactual value and the predicted value to the number of testsets It evaluates the model by the following criteria thesmaller the value of RMSE the smaller the error of themodeland the higher the accuracy When the actual value iscompletely consistent with the predicted value it means thismodel is a perfect model

RMSE

1n

1113944

n

i1xai minus xpi1113872 1113873

2

11139741113972

(10)

where xai is the actual value xpi is the predicted value and nis the number of test sets

MAE is mean absolute error which represents the av-erage of the absolute values of the deviations of all predictedvalues and the arithmetic mean It evaluates themodel by thefollowing criteria the smaller the value of MAE the smallerthe error of the model and the higher the accuracy Similar toRMSE when the actual value is exactly the same as thepredicted value it is also a perfect model

MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)

MAPE is mean absolute percentage error which mea-sures the relative errors between the average predicted valueand the actual value on the test set [40] It evaluates themodel by the following criteria the smaller the value ofMAE the smaller the error of the model and the higher theaccuracy Similar to RMSE and MAE it is also an ideal

model when the actual value is consistent with the predictedvalue

MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)

According to the statistical tests the minimum RMSE ofQPSO-BP was only about 048 that of PSO-BP was about 099and that of FWAwas about 1 94 while themaximumRMSE ofBP reached 316 Similarly theMAE andMAPE values ofQPSOwere the minimum values compared with those of the otherthree models which confirmed our checking calculation aboveFurthermore by separately comparing the prediction results ofh4 and H the accuracy of these models was as the followingorders QPSO-BPgtPSO-BPgtFWA-BPgtBP which was alsoechoed above

Because of its high accuracy fast convergence and fewparameters QPSO successfully demonstrates its advantagesin practical application

43 Application of QPSO-BP in Practical Engineering In apractical project similar to the geological condition of theproject in Section 22 themethod proposed in this paper can beadopted to predict the distribution of bearing stratum and theburied depth of pile foundation +e specific implementationsteps are as follows (1) it is necessary and the most critical stepfor professional personnel to conduct geotechnical investiga-tion which determine whether the soil properties in the areameet the predicted conditions or not (2) +e designers shoulddetermine the specific location of each pile foundation inAutoCAD based on the pile foundation design drawings andgeotechnical investigation report and then sort out the X-co-ordinate Y-coordinate and Z-coordinate which must be basedon the geodetic origin (3)+e QPSO-BPmodel is employed topredict the indicators to be predicted with a method similar tothat in this paper (4) According to the prediction results thelength of the precast pile in the area with different bearing depthcan be determined and then the pile number and


FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)

Figure 14 +e prediction curve of H by using different models


PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60

Figure 15 +e prediction error curve of h4 by using different models

PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

Figure 16 +e prediction error curve of H by using different models

PSO-BPQPSO-BPFWA-BP

Fitn

ess

100 200 300 400 5000Number of iterations

0

20

40

60

80

100

120

140

Figure 17 Convergence curves of the QPSO-BP PSO-BP and FWA-BP models


corresponding coordinates are recorded Finally the piles will bedriven into the soil layer one by one in the actual engineering

5 Conclusion

It is of great significance to determine the fluctuation of bearingstratum and the buried depth of pile foundation before con-struction which can effectively reduce the project cost and avoidunnecessary losses +e QPSO-BP model was adopted to dealwith this highly nonlinear problem and the following part is asummary of the specific work completed in this paper

(1) Based on engineering examples the BP networkmodel was used to predict the fluctuation of bearingstratum and the buried depth of pile foundation inthis paper Besides when the number of neurons S inthe hidden layer and the input parameters N meetS 2N+ 1 the error of prediction results was theminimum

(2) +e prediction results indicated that the BP networkwould easily fall into the local optimal solution and theerror between the predicted value and the actual valuewas quite large the maximum error of which reachedabout 41when the number of neurons was 2 and 31when the number of neuronswas 7+erefore althoughthe predicted value of the BP neural network could beused as a reference its algorithm still had shortcomingsand disadvantages+erefore it needed to be optimized

(3) +e QPSO algorithm was adopted to optimize theBP network and then the model of QPSO-BP wasno longer trapped in the infinite loop of searchingfor local optimal solution +e relative error wasmerely 94 in predicting h4 and 29 in pre-dicting H Besides two other optimization algo-rithms (FWA and PSO) were used to optimize theBP model and the results demonstrated the highaccuracy of QPSO-BP +e error of QPSO-BP wasthe smallest of the three algorithms

(4) +ree different statistical tests (RMSE MAE andMAPE) were further employed to evaluate the accuracyof the three models +e calculation results of the threestatistical tests were consistent with the above and theaccuracy followed the order of QPSO-BPgtPSO-BPgtFWA-BP

(5) All the evidence demonstrated the superiority of theQPSO-BP model in engineering application

Data Availability

+e case analysis data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Science and TechnologyProject of Jingzhou (2019AC27) China

References

[1] M F Randolph ldquoScience and empiricism in pile foundationdesignrdquo Geotechnique vol 53 no 10 pp 847ndash876 2003

[2] J R Meyer Analysis and Design of Pile Foundations ASCEVirginia NV USA 2015

[3] Z M Zhang ldquoAchievements and problems of geotechnicalengineering investigation in chinardquo Journal of ZhejiangUniversity-Science vol 12 no 2 pp 87ndash102 2011

[4] W Fleming A J Weltman M F Randolph and K ElsonPiling Engineering CRC Press Boca Raton FL USA 3rdedition 2009

[5] K J Qi M J Xu and J M Zai ldquoGray prediction of ultimatebearing capacity of single pilerdquo Chinese Journal of RockMechanics and Engineering vol 23 no 12 p 2069 2004

[6] X J Gao and X R Zhu ldquoForecasting ultimate bearing ca-pacity of single squeezed branch pile by hyperbola methodrdquoRock and Soil Mechanics vol 27 no 9 pp 1596ndash1600 2006

[7] Y S Deng W M Gong and A M Yuan ldquoResearch oncalculating methods for settlement of extra-long large-di-ameter pile grouprdquo Journal of the China Railway Societyvol 29 no 4 pp 87ndash90 2007

[8] M Ali Ahmadi ldquoApplying a sophisticated approach to predictCO2 solubility in brines application to CO2 sequestrationrdquoInternational Journal of Low Carbon Technologies vol 11no 3 pp 325ndash332 2016

[9] E Masson and Y J Wang ldquoIntroduction to computation andlearning in artificial neural networksrdquo European Journal ofOperational Research vol 47 no 1 pp 1ndash28 2007

[10] M D Himmelblau ldquoAccounts of experiences in the appli-cation of artificial neural networks in chemical engineeringrdquoIndustrial amp Engineering Chemistry Research vol 47 no 16pp 5782ndash5796 2008

[11] H R Amedi A Baghban M A Ahmadi et al ldquoEvolvingmachine learning models to predict hydrogen sulfide solu-bility in the presence of various ionic liquidsrdquo Journal ofMolecular Liquids vol 216 pp 411ndash422 2016

[12] J Q Shang W Ding R K Rowe and L Josic ldquoDetectingheavy metal contamination in soil using complex permittivityand artificial neural networksrdquo Canadian GeotechnicalJournal vol 41 no 6 pp 1054ndash1067 2004

[13] M Alias R Dhanya and G Ramasamy ldquoStudy on factorsaffecting the performance of construction projects and de-veloping a cost prediction model usingrdquo Annals of Forestryvol 8 pp 2189ndash2194 2015

Table 3 Statistical tests of the different models

Model RMSE MAE MAPE ()

BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347


[14] T Sorsa and H N Koivo ldquoApplication of artificial neuralnetworks in process fault diagnosisrdquo IFAC Proceedings Vol-umes vol 24 no 6 pp 423ndash428 1991

[15] L Jin R Zhao and X L Du ldquoNeural network predictionmodel for size effect of concrete compressive strengthrdquoJournal of Beijing University of Technology vol 47 no 3pp 260ndash268 2021

[16] K Suzuk S G Armato F Li S Sone and K Doi ldquoMassivetraining artificial neural network (MTANN) for reduction of falsepositives in computerized detection of lung nodules in low-dosecomputed tomographyrdquo Medical Physics vol 30 no 7pp 1602ndash1617 2003

[17] S Ding H Li C Su J Yu and F Jin ldquoEvolutionary artificialneural networks a reviewrdquo Artificial Intelligence Reviewvol 39 no 3 pp 251ndash260 2013

[18] THe S Zheng P Zhang andM Zou ldquoInput values function forimproving generalization capability of BP neural networkrdquo inProceedings of theAsia-Pacific Conference on Wearable Com-puting Systems pp 228ndash231 Shenzhen China April 2010

[19] H Ai and S Guo ldquoBridge health evaluation system based onthe optimal BP neural networkrdquo International Journal ofControl amp Automation vol 7 no 1 pp 331ndash338 2014

[20] J H Wang J H Jiang and R Q Yu ldquoRobust back propagationalgorithm as a chemometric tool to prevent the overfitting tooutliersrdquo Chemometrics and Intelligent Laboratory Systemsvol 34 no 1 pp 109ndash115 1996

[21] C C Shi Y Y Zeng and S M Hou ldquoApplication of swarmintelligence algorithm in image segmentationrdquo ComputerEngineering and Applications vol 57 no 08 pp 36ndash47 2021

[22] Y Tan and Y C Zhu ldquoFireworks algorithm for optimiza-tionrdquo in Proceedings of the Advances in Swarm Intelligence(ICSI 2010) vol 6145 pp 355ndash364 Beijing China June 2010

[23] Y Tan FWA Application on Non-Negative Matrix Factor-ization Springer Berlin Germany 2015

[24] S Zheng A Janecek and Y Tan ldquoEnhanced fireworks al-gorithmrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation IEEE Cancun MX USA June 2013

[25] J Kennedy ldquoParticle swarm optimizationrdquo in Proceedings ofthe 1995 IEEE International Conference Neural NetworksPerth Australia December 2011

[26] A Khare and S Rangnekar ldquoA review of particle swarmoptimization and its applications in solar photovoltaic sys-temrdquo Applied Soft Computing Journal vol 13 no 5pp 2997ndash3006 2013

[27] M A Ahmadi and S R Shadizadeh ldquoNew approach for pre-diction of asphaltene precipitation due to natural depletion byusing evolutionary algorithm conceptrdquo Fuel vol 102 2012

[28] P Wang Z Y Huang M Y Zhang and X-W ZhaoildquoMechanical property prediction of strip model based onPSO-BP neural networkrdquo Journal of Iron amp Steel ResearchInternational vol 15 no 3 pp 87ndash91 2008

[29] A Ismail D-S Jeng and L L Zhang ldquoAn optimised product-unit neural network with a novel PSO-BP hybrid trainingalgorithm applications to load-deformation analysis of axiallyloaded pilesrdquo Engineering Applications of Artificial Intelli-gence vol 26 no 10 pp 2305ndash2314 2013

[30] A Shafiei M A Ahmadi S H Zaheri et al ldquoEstimatinghydrogen sulfide solubility in ionic liquids using a machinelearning approachrdquoFe Journal of Supercritical Fluids vol 95pp 525ndash534 2014

[31] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of ICNNrsquo95-International Conference on NeuralNetworks pp 1942ndash1948 Perth WA Australia December1995

[32] W Fang J Sun Y Ding X Wu and W Xu ldquoA review ofquantum-behaved particle swarm optimizationrdquo IETE Tech-nical Review vol 27 no 4 pp 336ndash348 2010

[33] J Li J Q Zhang C J Jiang and M Zhou ldquoCompositeparticle swarm optimizer with historical memory for functionoptimizationrdquo IEEE Transactions on Cybernetics vol 45no 10 pp 2350ndash2363 2017

[34] J Sun B Feng andW Xu ldquoParticle swarm optimization withparticles having quantum behaviorrdquo in Proceedings of the2004 Congress on Evolutionary Computation (IEEE Cat No04TH8753) pp 325ndash331 Portland OR USA June 2004

[35] Y T Chen and Q Zhang ldquoOptimization design and simu-lation of belt conveyor gear based on levy flight quantumparticle swarmsrdquo Machinery Design amp Manufacture vol 4no 4 pp 54ndash57 2020

[36] K Lu and RWang ldquoApplication of PSO and QPSO algorithmto estimate parameters from kinetic model of glutamic acidbatch fermetationrdquo in Proceedings of the 2008 7th WorldCongress on Intelligent Control and Automation pp 8968ndash8971 Chougqing China June 2008

[37] Q P Zuo and L Huang ldquoEvaluation factors of overburdenthickness and site type in Jingzhou urban areardquo ResourcesEnvironment amp Engineering vol 29 no 6 pp 940ndash944 2015

[38] Z B Xu R Zhang and W F Jing ldquoWhen does online BPtraining convergerdquo IEEE Transactions on Neural Networksvol 20 no 10 pp 1529ndash1539 2009

[39] S X Xu and L Chen ldquoA novel approach for determining theoptimal number of hidden layer neurons for FNNrsquos and itsapplication in data miningrdquo in Proceedings of the 5th Inter-national Conference on Information Technology and Appli-cations (ICITA) pp 683ndash686 Cairns Australia June 2008

[40] L J Liu D H Liu HWu and X YWang ldquo+e prediction ofmetro shield construction cost based on a backpropagationneural network improved by quantum particle swarm opti-mizationrdquo Advances in Civil Engineering vol 2020 Article ID6692130 15 pages 2020

[41] Y Shi and R Eberhart ldquoA modified particle swarm opti-mizerrdquo in Proceedings of the IEEE International Conference onEvolutionary Computation IEEE World Congress on Com-putational Intelligence (Cat No98TH8360) pp 69ndash73 An-chorage AK USA May 1998

[42] H Li Q Zhang and Y Zhang ldquoImprovement and applicationof particle swarm optimization algorithm based on the pa-rameters and the strategy of co-evolutionrdquo Applied Mathe-matics amp Information Sciences vol 9 no 3 pp 1355ndash13642015

[43] D Z Pan and Y J Chen ldquoQuantum-behaved particle swarmoptimization algorithm based on reverse random-weightedmean best positionrdquo Journal of ChinaWest Normal University(Natural Sciences) vol 33 no 3 pp 281ndash285 2012

[44] M Najafzadeh and G A Barani ldquoComparison of groupmethod of data handling based genetic programming andback propagation systems to predict scour depth aroundbridge piersrdquo Scientia Iranica vol 18 no 6 pp 1207ndash12132011

[45] M Najafzadeh and F Saberi-Movahed ldquoGMDH-GEP topredict free span expansion rates below pipelines underwavesrdquo Marine Georesources amp Geotechnology vol 37 no 3pp 1ndash18 2018

[46] F Saberi-Movahed M Najafzadeh and A Mehrpooya ldquoRe-ceiving more accurate predictions for longitudinal dispersioncoefficients in water pipelines training group method of datahandling using extreme learning machine conceptionsrdquo WaterResources Management vol 34 pp 529ndash561 2020


cannot continue going deeper At this time the pile head willextend too long from the soil and needs to be cut off whichcan be seen from Figure 2 below On the contrary in the areawhere the bearing stratum is deeply distributed the pilelength will be insufficient and needs to be supplemented+ereason for this situation is that the variety of pile length islimited especially the prefabricated pile If the constructionis carried out according to the designed pile length a largenumber of piles will be cut off or supplemented resulting inunnecessary waste in pile foundation engineering+ereforethis paper predicted the buried depth of pile foundation andbearing stratum then made targeted technical schemepreparation and construction deployment in practical en-gineering according to the prediction results

At present there are few research studies on predictingthe buried depth of pile foundation and the fluctuation ofbearing stratum however in the field of pile foundationengineering many scholars have made some achievementsIn the static load test of pile foundation finished by Qi et al[5] a first-order linear dynamic differential equation wasderived by studying the settlement of pile under variousloads On the basis of gray system theory the GM (1 1)model of load-settlement relationship of single pile wasestablished and employed to predict the ultimate bearingcapacity and the complete load-settlement relationshipDespite the accurate prediction results obtained by thismethod the uniformity of original data should be ensuredAccording to the measured data Gao et al [6] adopted thehyperbolic method to predict the bearing capacity ofsqueezed branch pile Although the error between thepredicted results and the measured curve was not great thevalues predicted by this method were generally too large andhad certain limitations Deng et al [7] used the superlongand large-diameter cast-in-place pile foundation of theSutong Yangtze River Bridge as an example to calculate itssettlement amount by adopting different specifications andthen a new empirical formula considering pile compressionand modifying additional stress of the pile tip was proposed

by comparing with the settlement value of the large-scalecentrifugal model test Afterwards this formula was appliedto verify the settlement value of a super large-scale pile groupfoundation on the Nei-Kun Line and the calculation resultof which was relatively consistent with the measured dataHowever this formula was not suitable for the analysis ofsingle pile settlement which had certain limitations as well

Most of the forecasting methods mentioned above relyon a fixed knowledge framework and can only be adoptedunder certain preconditions which are rigid and not flexibleenough As a consequence an intelligent technology that candeal with various problems flexibly and has self-learningawareness will be needed Machine learning as a technologythat computers build models based on data to simulatehuman activities can meet this condition It possesses thestrong generalization ability and has been applied in dif-ferent aspects Methods of machine learning were used byAhmadi et al [8] to successively predict the solubility ofcarbon dioxide (CO2) in brines porosity and permeability ofpetroleum reservoirs the amount of dissolved calciumcarbonate concentration throughout oil field brines con-densate-to-gas ratio in retrograded condensate gas reser-voirs solubility of hydrogen sulfide (H2S) in ionic liquidsetc +e prediction of pile foundation depth belongs to ahighly nonlinear problem which can also be analyzed by thismethod +e artificial neural network (ANN) is a kind ofmachine learning which is a powerful intelligent learningtool with functions [9 10] such as mapping nonlinear re-lations information processing optimization calculationclassification and recognition It has been widely applied inthe fields of model prediction content prediction costcontrol fault diagnosis information processing construc-tion engineering mechanical engineering medicine etcand the results were great as expected Hamid et al [11] builtan ANN model based on the critical pressure (Pc) criticaltemperature (Tc) and molecular weight (Mw) of pure ionicliquids to predict the solubility of hydrogen sulfide (H2S) indifferent temperature pressure and concentration ranges

(a) (b) (c)

Figure 1 +e buried depth of piles in the site varied greatly








(a) (b) (c)










2 Project Example


















f(x) 1

1 + eminus x (1)










E 12

1113944

m


2 le ε (2)



8

75 6

7prime 5prime

6prime

8prime

K42233

23122327

2341

2351

23161231472313323120

23323

23314

23350 23361

2322923217

23203

2318323175

23450 2343823424

23269 23306

K43 K44 K45

K46

K38

K32

K39 K40


C13

C14

K41

K31

K37K24

K33 K34K35

K36K30

K25 K26 K27K28 K29

C12

C8

C9

C11

C10


Dining hall

11

10

9 9prime

10prime


K60 K61 K62 K63

K56 K57 K58 K59

ST160

ST151

ST148 ST139

ST2ST10 ST15




















Li d(t) times In1

ui d(t)1113890 1113891 (6)



K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6





1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347










































(a) (b) (c)










2 Project Example


















f(x) 1

1 + eminus x (1)










E 12

1113944

m


2 le ε (2)



8

75 6

7prime 5prime

6prime

8prime

K42233

23122327

2341

2351

23161231472313323120

23323

23314

23350 23361

2322923217

23203

2318323175

23450 2343823424

23269 23306

K43 K44 K45

K46

K38

K32

K39 K40


C13

C14

K41

K31

K37K24

K33 K34K35

K36K30

K25 K26 K27K28 K29

C12

C8

C9

C11

C10


Dining hall

11

10

9 9prime

10prime


K60 K61 K62 K63

K56 K57 K58 K59

ST160

ST151

ST148 ST139

ST2ST10 ST15




















Li d(t) times In1

ui d(t)1113890 1113891 (6)



K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6





1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347











































2 Project Example


















f(x) 1

1 + eminus x (1)










E 12

1113944

m


2 le ε (2)



8

75 6

7prime 5prime

6prime

8prime

K42233

23122327

2341

2351

23161231472313323120

23323

23314

23350 23361

2322923217

23203

2318323175

23450 2343823424

23269 23306

K43 K44 K45

K46

K38

K32

K39 K40


C13

C14

K41

K31

K37K24

K33 K34K35

K36K30

K25 K26 K27K28 K29

C12

C8

C9

C11

C10


Dining hall

11

10

9 9prime

10prime


K60 K61 K62 K63

K56 K57 K58 K59

ST160

ST151

ST148 ST139

ST2ST10 ST15




















Li d(t) times In1

ui d(t)1113890 1113891 (6)



K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6





1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347




































2 Project Example


















f(x) 1

1 + eminus x (1)










E 12

1113944

m


2 le ε (2)



8

75 6

7prime 5prime

6prime

8prime

K42233

23122327

2341

2351

23161231472313323120

23323

23314

23350 23361

2322923217

23203

2318323175

23450 2343823424

23269 23306

K43 K44 K45

K46

K38

K32

K39 K40


C13

C14

K41

K31

K37K24

K33 K34K35

K36K30

K25 K26 K27K28 K29

C12

C8

C9

C11

C10


Dining hall

11

10

9 9prime

10prime


K60 K61 K62 K63

K56 K57 K58 K59

ST160

ST151

ST148 ST139

ST2ST10 ST15




















Li d(t) times In1

ui d(t)1113890 1113891 (6)



K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6





1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347






































f(x) 1

1 + eminus x (1)










E 12

1113944

m


2 le ε (2)



8

75 6

7prime 5prime

6prime

8prime

K42233

23122327

2341

2351

23161231472313323120

23323

23314

23350 23361

2322923217

23203

2318323175

23450 2343823424

23269 23306

K43 K44 K45

K46

K38

K32

K39 K40


C13

C14

K41

K31

K37K24

K33 K34K35

K36K30

K25 K26 K27K28 K29

C12

C8

C9

C11

C10


Dining hall

11

10

9 9prime

10prime


K60 K61 K62 K63

K56 K57 K58 K59

ST160

ST151

ST148 ST139

ST2ST10 ST15




















Li d(t) times In1

ui d(t)1113890 1113891 (6)



K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6





1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347





















































Li d(t) times In1

ui d(t)1113890 1113891 (6)



K19241

2412 2433 2450

24135

24148 2416224175

2457

24192

24182

24227

24314

2432824342

24459

2434424359

24471

24370

24377

C6

C7

C5

C3

C4

C2

C1

4

31 2

9

3prime 2prime

1prime


K15 K16 K17

K18

K14K8

K7

K1K9

K10 K11K12 K13

K2 K3K4 K5 K6





1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347






































1113868111386811138681113868 (7)




ui d(t)1113890 1113891

(8)







Fine sandPebble

H

Pile

h1h1

h2

h3

h4

h2

h3

h4

Qml

Miscellaneous fill1

Qal

Silty clay2 4

Qal

Silt3 4

Qal

Fine sand4 4

Qal

Pebble5 3

+pl









Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347










































Hide layer


X1

X2

Xk

Y1

Y2

Yk

Wjk Wij

vj


Input data

Input preprocessing



extremum




group extremum


N Y


network


BP neural network


Calculate the error




prediction

Y

N


as fitness value





ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347






































ER xp minus xa

11138681113868111386811138681113868

11138681113868111386811138681113868

xa

(9)









Measured valueN = 7

N = 4N = 2

h 4 (m

)


3

4

5

6

7

8

9

10







Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347








































Measured valueN = 7

N = 4N = 2


18

20

22

24

26

28

30

32

34H

(m)


Net

wor

k er

ror (

)

N = 7N = 4N = 2


ndash40

ndash20

0

20

40

60


N = 7N = 4N = 2

Rela

tive e

rror

()


ndash30

ndash20

ndash10

0

10

20

30



FWALinear fitting


h 4 (m

)

0

2

4

6

8

10

12





RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347






































RMSE

1n

1113944

n


2

11139741113972

(10)



MAE 1n

1113944

n

i1xai minus xpi

11138681113868111386811138681113868

11138681113868111386811138681113868 (11)



MAPE 100

n1113944

n

i1

xai minus xpi

xai

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (12)





FWALinear fitting


20

21

22

23

24

25

26

27

28

H (m

)



PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347




































PSOQPSO

FWALinear fitting

Rela

tive e

rror

()


ndash80

ndash60

ndash40

ndash20

0

20

40

60


PSOQPSO

FWALinear fitting


Rela

tive e

rror

()

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25


PSO-BPQPSO-BPFWA-BP

Fitn

ess


0

20

40

60

80

100

120

140




5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347





































5 Conclusion







Data Availability




Acknowledgments


References
















BP h4 1317045 1218072 216218H 3159338 3047166 124648

FWA-BP h4 1403003 1253222 205951H 1943137 1739423 71161

PSO-BP h4 1030985 0920521 151887H 0998472 093352 3819

QPSO-BP h4 0662284 065193 110622H 0478191 0423976 17347






































































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The Prediction of Pile Foundation Buried Depth Based on BP ...

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