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\begin{document}

\title{BP neural network construction cost prediction model based on different optimization algorithms}

%%=============================================================%%
%% Prefix	-> \pfx{Dr}
%% GivenName	-> \fnm{Joergen W.}
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%%                 \dgr{MSc, PhD}}\email{iauthor@gmail.com}
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\author[1,3]{
\sur{Qian Liu}}\email{1650974107@qq.com}

\author*[2]{
\sur{Shujuan Yang}}\email{yangshujuan@qut.edu}
\equalcont{These authors contributed equally to this work.}

\author[1,2]{
\sur{Dehu YU}}\email{yudehu@sdjzu.edu.cn}
\equalcont{These authors contributed equally to this work.}

\author[2]{
\sur{Tongtong Gai}}\email{ gaitongtong@yeah.net;.cn;}

\affil[1]{\orgdiv{school of Civil Engineering}, \orgname{Shandong Jianzhu University}, \orgaddress{\postcode{250000}, \city{Jinan}, \country{China}}}

\affil*[2]{\orgdiv{school of Civil Engineering}, \orgname{Oingdao University of Technology}, \orgaddress{ \postcode{266000},\city{Qingdao}, \country{China}}}

\affil[3]{\orgdiv{Audit Department }, \orgname{Jinan Urban Development Group}, \orgaddress{ \postcode{2500000}, \city{Jinan}, \country{China}}}



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\abstract{Aiming at the complexity of prefabricated construction project input cost estimation and the subjectivity and limitation of manual estimation, a method of cost prediction based on BP neural network is proposed. In this paper, 144 sets of assembled engineering data are selected and collected. By using the mapping relationship between construction cost and 11 cost impact indicators, the prediction results are compared and analyzed after training and prediction. The results show that BP neural network can effectively predict cost, but the convergence is slow and the fitting effect is not good. In this paper, three algorithms, Adam, SGD and Adadelta, are proposed to optimize the BP neural network respectively, and the prediction results are compared and analyzed. The maximum error of SGD-BP neural network prediction model is 6.56\% and the minimum error is 0.91\%, the maximum error of Adadelta-BP neural network prediction model is 17.10\% and the minimum error is 0.82\%, the Adam-BP neural network prediction model has a maximum error of 3.89\% and a minimum error of 0.11\%. From the error analysis, it can be seen that the prediction results of Adam-BP neural network prediction model are more accurate, with good generalization effect, providing more accurate assembly building construction cost reference data for decision makers to judge the feasibility of the project.}

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\keywords{BP neural network, Cost forecasting, Optimization algorithm, Construction cost forecasting model}

%%\pacs[JEL Classification]{D8, H51}

%%\pacs[MSC Classification]{35A01, 65L10, 65L12, 65L20, 65L70}

\maketitle

\section{Introduction}\label{sec1}


Since 2012, the construction industry has gradually developed into a pillar industry in China, which is closely related to the development of the national economy. However, traditional buildings have many problems, such as large quantity of consumables, large input of manpower, material pollution and low production efficiency, and there is a serious shortage of suitable age labor force in China \cite{bib1} . While prefabricated buildings are green, energy saving and environmental protection, providing a new direction for the transformation and upgrading of the construction industry. However, the high cost of assembly has become one of the important factors restricting the development of assembly in our country. It is of great significance to accurately measure the cost of assembly construction in the project approval stage and then effectively optimize the cost. Comparing the traditional project cost prediction with the big data cost prediction, it is found that the traditional cost prediction system is more and more difficult to adapt to the complex construction process of prefabricated buildings.\par
The PMBOK Guide is a recognized standard in project management. It includes a section on cost management, providing a foundational framework for cost forecasting within the context of project management\cite{bib2}.Günaydın, H. M investigated the utility of a neural network approach to overcome cost estimation problems in the early stages of the architectural design process\cite{bib3}.Kulkarni, P. \cite{bib4} delves into the use of data mining techniques to improve project cost estimation, highlighting the role of data-driven approaches in forecasting.Choong Wan Koo, alongside other esteemed scholars \cite{bib5}, undertook a comprehensive approach to construct a building construction cost prediction model. Their methodology integrated artificial neural networks, multivariate linear regression, and genetic algorithms. This interdisciplinary framework yielded notable advancements in prediction accuracy, resulting in a closer alignment between projected cost and the actual cost of the projects.Narges Sajadfar's work \cite{bib6} centered on the utilization of linear regression and extensive data mining techniques to formulate an informatics framework for cost estimation. Notably, the cost prediction function incorporates a confidence level assessment for the influencing factors during the prediction process, thereby enhancing the precision of cost projections.Putra G.A.S\cite{bib7} contributed to the field by establishing a predictive model using BP neural networks, aimed at forecasting both direct and indirect project costs. The study conducted a rigorous comparative analysis between predicted values and actual cost outcomes, meticulously scrutinizing prediction errors. This research facilitated an economic evaluation of installation projects and significantly mitigated the margin of error in project cost estimation.UGUR L.O \cite{bib8} pursued the development of a predictive model for residential cost based on adaptive neural networks. The study involved an in-depth analysis of the disparities between predicted values and actual costs. The outcomes demonstrated the efficacy of the cost prediction model, showcasing its superior predictive capabilities.Chen Wensheng \cite{bib9} used MATLAB to establish a BP neural network model based on hierarchical analysis to predict the cost of engineering investment estimation stage, with 18 groups of residential data in Xuzhou City as sample data for training, and 2 groups of residential data as samples for fitting prediction. The results show that the BP neural network based on hierarchical analysis can effectively predict the cost.Chen, D \cite{bib10} developed a predictive model for estimating transmission line construction costs using eight input indicators. The model, based on a BP neural network, was trained and validated with real engineering data, demonstrating its precise cost estimation and utility in early project planning.\par
The existing research and practical applications of engineering cost forecasting, incorporating artificial intelligence and mathematical analysis both domestically and internationally, have exhibited limitations. These limitations primarily revolve around the insufficient number of selected influencing factors for cost forecasting and a dearth of training samples, resulting in the instability of predictive outcomes.Although BP neural networks have demonstrated effectiveness in cost prediction, they tend to perform poorly when confronted with extensive sample datasets, frequently succumbing to issues like poor fitting and gradient explosions.\par
(1) Building upon the insights derived from prior investigations, this study endeavors to construct a BP neural network model.  This model employs data mining techniques to elucidate the intricate mapping relationships among various factors influencing construction costs.\par
(2) Various optimization algorithms have been deployed to augment the model's precision and fitting capabilities, and their impact on project cost prediction has been empirically scrutinized.\par
(3)This model has achieved significant progress within the domain of prefabricated building investment estimation, offering decision-makers a sophisticated tool for informed assessments.  Ultimately, this contributes to the promotion and advancement of prefabricated construction technology.\par
In the upcoming chapter, a concise overview of pertinent literature concerning diverse optimization algorithms will be provided. Chapter 3 will delve into the theoretical framework of the Backpropagation (BP) neural network and elucidate the construction of the cost estimation model. Moving on to Chapter 4, the cost estimation model grounded in the BP neural network, employing various optimization algorithms, will be employed to assess 10 distinct sets of real-world project data. The predictive outcomes derived from these different optimization approaches will be scrutinized and contrasted. To conclude, the chapter will conclude with summative remarks and recommendations for future research directions.

\section{Related works}\label{sec2}
\subsection{Principle of SGD optimization algorithm}\label{subsec2}
The Stochastic Gradient Descent (SGD) optimization algorithm has long been a fundamental cornerstone in machine learning and deep learning. Its simplicity and effectiveness in training neural networks have made it a pivotal component in various applications.  Polyak\cite{bib11}and Nesterov Y E \cite{bib12} which improved convergence rates and stability. Furthermore, Bottou and Bousquet's work \cite{bib13} offered valuable insights into the convergence properties of SGD. The Stochastic Gradient Descent (SGD) optimization algorithm has been increasingly applied to project cost prediction, leveraging its efficiency in optimizing neural network models for this specific task. While SGD is a well-established optimization method in machine learning, its adaptation to project cost prediction is a relatively recent development. Johnson R and Zhang T \cite{bib14} investigated the integration of SGD with advanced cost prediction models, showcasing the algorithm's utility in enhancing predictive accuracy. The application of SGD to project cost prediction continues to evolve, offering a promising avenue for more accurate and efficient cost forecasting\cite{bib15}\cite{bib16} .\par
The Stochastic Gradient Descent (SGD) optimization algorithm presents several advantages and disadvantages when applied to engineering cost prediction. One of its primary advantages is its efficiency in handling large and complex datasets, making it suitable for engineering projects with substantial data inputs. SGD's relatively simple implementation and low computational requirements also make it an accessible choice for cost prediction models. However, SGD's performance can be sensitive to hyperparameter settings, including the learning rate, which may necessitate extensive tuning for optimal convergence. The algorithm's path to convergence may be noisy and exhibit oscillations, potentially leading to suboptimal solutions. 
\subsection{Principle of Adadelta optimization algorithm}\label{subsec2}
Adadelta is an optimization algorithm that has gained traction in the realm of project cost prediction. Relevant studies in the field have explored the application of Adadelta, a variant of stochastic gradient descent (SGD), and its impact on the accuracy and efficiency of cost forecasting models. Zhang J \cite{bib17}, has demonstrated the adaptability of Adadelta in managing complex cost datasets and improving the convergence of predictive models. The Adadelta optimization algorithm has emerged as a noteworthy tool in the context of project cost prediction. Researchers have explored the application of Adadelta to enhance the efficiency and accuracy of cost forecasting models, recognizing its ability to adaptively adjust learning rates. The work by Qu Z \cite{bib18}showcased the benefits of integrating Adadelta into project cost prediction, highlighting improved convergence and robustness in cost forecasting. However, it's essential to note that the effectiveness of Adadelta in this context may be influenced by the characteristics of the cost data, and the algorithm's performance is not immune to hyperparameter settings\cite{bib19}. \par
The Adadelta optimization algorithm offers several advantages and disadvantages when applied to engineering cost prediction. One of its primary advantages is its adaptability to varying learning rates, which helps mitigate the need for extensive manual tuning of hyperparameters. Its effectiveness in handling sparse and noisy data can be advantageous for engineering cost prediction, where datasets often exhibit such characteristics. However, it tends to require a higher computational overhead compared to some other optimization algorithms.
\subsection{Principle of Adam optimization algorithm}\label{subsec2}
The Adam optimization algorithm\cite{bib20}, an adaptive learning rate optimization method, has gained significant attention in recent years for its effectiveness in training deep neural networks. It builds upon the strengths of both stochastic gradient descent (SGD) and RMSprop by combining their adaptive learning rates and momentum, resulting in faster convergence and improved generalization. Several seminal and contemporary studies have contributed to the understanding and application of the Adam algorithm in various domains of machine learning and deep learning. Kingma and Ba's original paper on Adam \cite{bib21}laid the foundation for this optimizer, providing a comprehensive theoretical framework. Subsequent research by Reddi S J \cite{bib22}introduced insights into the impact of hyperparameters on Adam's performance. Additionally, recent work by Zhuang J \cite{bib23}proposed variants of Adam for improved convergence in adversarial training, demonstrating the algorithm's adaptability and continued relevance in state-of-the-art machine learning models.\par
The Adam optimization algorithm has emerged as a promising approach for enhancing project cost prediction, leveraging its adaptive learning rate and momentum features. Leveraging the strengths of adaptive learning rates and momentum, the Adam optimizer has demonstrated promise in optimizing the training of predictive models for project costs.Recent work by Wang R \cite{bib24} delved into the integration of Adam with deep learning architectures for project cost prediction, illustrating the algorithm's ability to enhance the accuracy and convergence of cost forecasting models. \par
As the application of Adam to project cost prediction continues to evolve, it holds great promise for more effective and precise cost estimations in a project management context.However, the algorithm does have some limitations, such as sensitivity to hyperparameters, which may require careful tuning for optimal performance. Additionally, the increased computational complexity associated with Adam may be a limiting factor in certain project cost prediction scenarios. Therefore, the exploration of the advantages and disadvantages of Adam's application in cost forecasting is essential to understand its suitability for different project management contexts.

\section{Proposed approach}\label{sec3}
The principle of a Backpropagation (BP) neural network, also known as a multilayer perceptron (MLP), is to train a feedforward neural network using the backpropagation algorithm\cite{bib25}. The BP neural network consists of multiple layers of interconnected neurons, including an input layer, one or more hidden layers, and an output layer \cite{bib26}. 
\subsection{Model establishment}\label{subsec2}
\subsubsection{Determination of the number of layers of the BP neural network model}\label{subsubsec2}
The construction of neural network model determines whether the neural network can accurately predict the target value . The BP neural network model consists of an input layer, an output layer and several hidden layers. The determination of hidden layers is a crucial problem. With the increase of the number of hidden layers, the fitting effect of the theoretical function is enhanced. However, in the actual sample data training process, it is found that too many hidden layers will increase the training difficulty, the model convergence effect is poor, and sometimes overfitting phenomenon will occur. Therefore, when constructing the BP neural network model, the appropriate number of hidden layers should be selected according to the complexity of the data set. The complexity of the sample data set in this paper is low, and one hidden layer can meet the training requirements.
\subsubsection{Determination of the number of nodes in the input layer of BP neural network}\label{subsubsec2}
In this paper, the construction cost index of prefabricated shear wall is selected according to three index attributes: project overview attribute index, market attribute index and construction period index. Finally, a total of 14 cost impact indicators including total construction area, standard floor area, assembly rate, number of building layers, standard floor height, castin concrete main material cost, steel bar main material cost, formwork cost, labor cost, construction machinery cost and main construction period are selected as input nodes of the BP neural network model.
\subsubsection{Determination of the number of nodes in output layer of BP neural network}\label{subsubsec2}
The research problem of this paper is to predict the cost of the main construction process according to 14 influence indexes of project overview attribute index, market attribute index and construction period index. Based on this, the BP neural network model in this paper has only one output node, and the output result is the construction cost per square meter of the main structure.
\subsubsection{Determination of the number of hidden layer nodes in BP neural network}\label{subsubsec2}
The presence of hidden layers allows neural networks to learn complex patterns and relationships in the data.  The neurons in the hidden layers perform computations on the input data and pass the results to the next layer until reaching the output layer.  Each neuron receives inputs from the previous layer, applies a weighted sum and an activation function, and outputs its result to the next layer.\par
The number of hidden layers and the number of neurons in each hidden layer are important hyperparameters in designing a neural network architecture.  Increasing the number of hidden layers and neurons can potentially enhance the network's capacity to learn intricate patterns.  However, it also increases the risk of overfitting, where the network becomes too specialized to the training data and performs poorly on new, unseen data.\par
Determining the optimal number of hidden layers and neurons is a challenging task that depends on the complexity of the problem, the amount of available data, and other factors.  It often requires experimentation and tuning to find the right balance between model complexity and generalization.For the determination of the number of nodes in the hidden layer, refer to the empirical formula as shown in equations 1,2 and 3:
\begin{equation}
n=\sqrt{n_{1}+n_{2}  } +a\label{eq1}
\end{equation}
\begin{equation}
n=\frac{2}{3} \times \left  ( n_{1} +n_{2}  \right )  
\end{equation}
\begin{equation}
n< 2n_{1} 
\end{equation}
Where n is the number of nodes in the hidden layer, n1 is the number of nodes in the input layer, n2 is the number of nodes in the output layer, and a is an arbitrary constant variable in the range of 2 to 10. The optimal number of hidden layers needs to be trained over time. After continuous experiments, the number of hidden layer nodes is determined to be 8, which makes the cost prediction effect of the neural network constructed in this paper better.\par
To sum up, the cost prediction model structure of the neural network finally established in this paper consists of an input layer with 14 input nodes, a hidden layer with 8 hidden nodes and an output layer with 1 output node. The structure of the neural network is shown in Figure 1.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{投稿/1.jpg}
\caption{ BP neural network}\label{fig1}
\end{figure}

\subsection{BP neural network model activation function selected}\label{subsec2}
In the neural network, the use of activation function f(x) is introduced into the nonlinear mapping relationship. If there is no activation function between the input layer and the output layer, no matter how many layers the neural network structure has, the resulting input and output will be linearly mapped. Therefore, the increase of the activation function can make the mapping relationship fitted by the neural network model approximate the nonlinear function, which can solve the nonlinear problem. Activation functions commonly used in neural network models include sigmoid function, tanh function and relu activation function \cite{bib27}.\par
(1)The calculation formula of sigmoid function \cite{bib28} is shown in Equation 4, and the derivative formula is shown in equation 5:
\begin{equation}
f\left ( x \right ) =S\left ( x \right ) =\frac{1}{1+e^{-x} } =\frac{e^{x} }{e^{x}+1} 
\end{equation}
\begin{equation}
S{}'\left ( x \right )  =S\left ( x \right ) \times \left (1-S\left ( x \right )   \right ) 
\end{equation}\par
(2)The calculation formula of tanh activation function \cite{bib29} is shown in equation 6, and the derivative formula is shown in equation 7:
\begin{equation}
f\left ( x \right ) =tanh\left ( x \right ) =\frac{e^{x}-e^{-x}  }{e^{x}+e^{-x} } 
\end{equation}
\begin{equation}
tanh{}'\left ( x \right )  =1-tanh^{2} \left (  x\right ) 
\end{equation}\par
(3)The calculation formula of relu activation function \cite{bib30} is shown in equation 8, and the derivative formula is shown in equation 9:
\begin{equation}
f\left ( x \right ) =relu\left ( x \right ) =max\left ( 0,x \right ) 
\end{equation}
\begin{equation}
{relu}' \left ( x \right ) =\begin{cases}
 & \text{x} ,x> 0\\
 & \text{0} ,x\le 0
\end{cases}
\end{equation}
The selection of activation function affects the prediction accuracy of neural network model. Selecting the appropriate activation function is beneficial to improve the model accuracy and reduce the training deviation. In this paper, the above sigmoid function, tanh function and reluu function are respectively applied to the neural network cost prediction model, and the activation function is selected according to the data training deviation applied by the three functions. The model training error of relu function is smaller than that of sigmoid function and tanh function. In combination with the advantages of better generalization effect of relu function, this paper selects relu function as the activation function of BP neural network cost model to train sample data, and realizes the information transmission and feedback among the input layer, hidden layer and output layer.
\subsection{Model optimization}\label{subsec2}
Neural networks have numerous parameters, including weights and biases, that need to be optimized to make accurate predictions. Basic gradient descent can be slow and prone to getting stuck in local minima. Therefore, more advanced optimization algorithms have been developed to address these challenges. Some commonly used optimization algorithms for training neural networks include: SGD,AdaGrad ,RMSProp ,Adam ,Adadelta et al. Hyperparameter tuning, including learning rate, momentum, and decay rates, is crucial for achieving optimal performance.\par
Hyperparameters are settings that are decided before training and have a significant impact on the training process. They include the learning rate, batch size, optimizer choice, weight initialization, and regularization strength, among others. Tuning these hyperparameters effectively can lead to improved model performance.Three commonly used optimization algorithms—Adam, SGD, and Adadelta are proposed in the paper to tune the hyperparameters of BP neural networks.Adam combines the benefits of both adaptive learning rates and momentum. It maintains moving averages of both gradients and squared gradients, and it adapts the learning rate for each parameter.Adam's adaptive learning rates help control the step size for each parameter update, which can lead to faster convergence.SGD updates the model's parameters using the gradient computed from a randomly selected training sample.SGD's simplicity and efficiency make it a common choice, especially when combined with techniques like learning rate schedules and momentum.Adadelta is an adaptive learning rate algorithm designed to overcome some limitations of Adagrad.It uses an exponentially decaying average of squared parameter updates to adaptively control the learning rate, addressing the issue of diminishing learning rates over time.\par
In summary, the paper suggests exploring the effects of hyperparameter tuning using Adam, SGD, and Adadelta on BP neural networks. The intention is to leverage the strengths of these optimization algorithms to achieve improved model performance through adaptive learning rates, momentum, and effective control over the learning process. The results of this exploration can help guide practitioners in selecting appropriate optimization algorithms for their specific neural network tasks.
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{投稿/2.jpg}
\caption{ Flowchart of the prediction model}\label{fig1}
\end{figure}
\section{Case study and analysis of the results}\label{sec4}
\subsection{Data Collection}\label{subsec2}
This academic paper aims to elucidate the development and application of a Backpropagation (BP) neural network model for the accurate prediction of costs in assembly projects.  To achieve this objective, it is imperative to amass a substantial dataset comprising actual assembly projects' information for training and testing the neural network. The dataset must encompass essential input features and their corresponding target outputs.This study presents a collection of 144 sets of engineering sample data, with 10 sets being randomly selected for predictive purposes and the remaining 134 data sets allocated for training the neural network model.  
\subsection{Indicator selection}\label{subsec2}
This thesis focuses on the cost of the main construction process of the assembly building, and does not take into account the impact of underground structure, utilities and decoration works on the cost. Therefore, the selection of assembly building cost indexes is based on three index attributes: project profile attribute index, market attribute index and duration index. Eleven cost indicators were finally selected through actual engineering research, as shown below:\par
(1)Project profile attribute indicators:
X1:total construction area , X2:standard floor construction area , X3:assembly rate , X4:number of building floors,X5:standard floor height, totaling 5 cost indicators.\par
(2)Market Attribute Indicators:
X6:concrete main material cost ,X7:reinforcing steel main material cost , X8:template cost , X9:labor cost ,X10:construction machinery cost , a total of 5 cost indicators.\par
(3)Duration indicator:
X11:main construction duration (days).\par
Finally, 11 cost indicator input influences (X) and 1 output (Cost) were selected as the construction cost per square meter of the standard floor. The data sample is shown in the table 1.

\begin{table}[h]
\caption{predicted sample set of BP neural network cost models}\label{tab1}
\begin{tabular}{ccccccccccccc}
\hline
Sample & X1    & X2  & X3   & X4 & X5   & X6  & X7   & X8 & X9  & X10  & X11 & Cost \\ \hline
1      & 10508 & 458 & 35\% & 24 & 3.1  & 491 & 3936 & 63 & 240 & 2375 & 161 & 1129 \\
2      & 7015  & 698 & 30\% & 10 & 2.9  & 454 & 4357 & 57 & 280 & 1590 & 100 & 992  \\
3      & 10458 & 547 & 50\% & 19 & 2.95 & 463 & 4257 & 70 & 210 & 1980 & 143 & 1186 \\
4      & 7396  & 404 & 30\% & 17 & 2.9  & 546 & 4310 & 69 & 110 & 1720 & 145 & 1080 \\
5      & 13043 & 543 & 50\% & 24 & 2.9  & 505 & 4064 & 69 & 220 & 2395 & 156 & 1299 \\
…      & …     & …   & …    & …  & …    & …   & …    & …  & …   & …    & …   & …    \\
144    & 10080 & 560 & 30\% & 18 & 2.9  & 578 & 5604 & 73 & 169 & 1870 & 130 & 1109 \\ \hline
\end{tabular}
\end{table}

\subsection{Data processing, training and prediction}\label{subsec2}
\subsubsection{Data Preprocessing}\label{subsubsec2}
Clean the data by handling missing values, outliers, and noise. Also, normalize or scale the input features to bring them to a similar range, which can help improve training convergence. In this paper, the method of Z-score normalization is chosen to normalize the sample data.\par
The principle of Z-score normalization is based on the mean (μ) of the total sample data and the standard deviation (σ) of the total sample data to normalize each of the original sample data (x) into x*. The Z-score normalization formula is shown in Equation 2.1. The standardization makes the sample data values all clustered around 0.
\begin{equation}
x^{\ast } =\frac{x-\mu }{\sigma } 
\end{equation}\par
This process helps to bring all the features to a similar scale, which can be particularly useful when working with machine learning algorithms that are sensitive to the scale of input features. 
\subsubsection{Data training}\label{subsubsec2}
Refers to the process of using a labeled dataset to train a model to make accurate predictions or classifications. Feed the training data into the model to make predictions (forward propagation).Compare the model's predictions to the actual target outputs and calculate the loss using a suitable loss function .Use backpropagation to compute the gradients of the loss with respect to the model's parameters.Update the model's parameters using an optimization algorithm to minimize the loss. This process adjusts the model's weights and biases.Periodically evaluate the model's performance on a validation set to monitor for overfitting or underfitting.Adjust hyperparameters based on the validation performance to optimize the model's generalization ability.Once training is complete, evaluate the model's performance on the testing set, which it has not seen during training. This provides an unbiased assessment of how well the model will perform on new, unseen data.
\subsubsection{Data Prediction}\label{subsubsec2}
Once the model is trained and its performance is satisfactory, you can use it to make predictions on new data. Provide the new input features  to the trained model, which will perform forward propagation to generate predictions.  Ten of the 144 sets of sample sets are randomly selected as the prediction sample set and the remaining 134 sets are used as the training sample set.  The predicted sample set is shown in the table 2.
\begin{table}[h]
\caption{predicted sample set of BP neural network cost models}\label{tab3}
\begin{tabular}{ccccccccccccc}
\hline
   & X1    & X2  & X3   & X4 & X5   & X6  & X7   & X8 & X9  & X10  & X11 & Cost \\ \hline
1  & 8505  & 405 & 30\% & 21 & 2.9  & 543 & 5305 & 72 & 170 & 1839 & 168 & 1088 \\
2  & 15629 & 601 & 50\% & 26 & 2.95 & 565 & 5550 & 72 & 230 & 1830 & 220 & 1134 \\
3  & 11029 & 627 & 35\% & 17 & 2.9  & 589 & 5209 & 55 & 240 & 1658 & 120 & 1127 \\
4  & 7719  & 646 & 35\% & 22 & 3.0  & 566 & 5281 & 56 & 180 & 1920 & 174 & 1183 \\
5  & 11371 & 437 & 30\% & 26 & 3.1  & 547 & 5387 & 66 & 205 & 1662 & 202 & 1156 \\
6  & 10604 & 424 & 30\% & 25 & 2.9  & 545 & 5149 & 51 & 220 & 1671 & 219 & 1099 \\
7  & 17087 & 549 & 30\% & 31 & 2.9  & 458 & 4153 & 74 & 240 & 1764 & 267 & 1103 \\
8  & 12637 & 567 & 30\% & 22 & 2.9  & 481 & 5131 & 53 & 240 & 1617 & 173 & 1111 \\
9  & 12982 & 418 & 35\% & 31 & 3.0  & 506 & 4629 & 54 & 220 & 2234 & 238 & 1122 \\
10 & 10080 & 560 & 30\% & 18 & 2.9  & 578 & 5604 & 73 & 169 & 1870 & 130 & 1109 \\ \hline
\end{tabular}
\end{table}
\subsection{Experimental Evaluation}\label{subsec2}
\subsubsection{Prediction results of different computational models}\label{subsubsec2}
This paper establishes a BP neural network cost prediction model based on three optimization algorithms: Adam, SGD and Adadelta. After the training of the data, the predicted values of 10 groups of prediction samples are obtained, as shown in the table 3.
\begin{table}[!ht]
\caption{Comparison of Actual and Projected Costs Using Three Prediction Methods}\label{tab3}
\setlength{\tabcolsep}{0.5mm}{
\begin{tabular}{ccccccccccc}
\hline
             & Case1   & Case2   & Case3   & Case4   & Case5   & Case6   & Case7   & Case8   & Case9   & Case 10 \\ \hline
actual value & 1088    & 1134    & 1127    & 1183    & 1156    & 1099    & 1103    & 1111    & 1122    & 1109    \\
Adam-BP      & 1115.39 & 1132.76 & 1120.93 & 1137.04 & 1134.00 & 1115.70 & 1123.04 & 1119.62 & 1115.26 & 1117.96 \\
SGD-BP       & 1148.37 & 1105.91 & 1057.08 & 1105.39 & 1095.95 & 1109.01 & 1132.60 & 1058.19 & 1145.06 & 1057.34 \\
Adadelta-BP  & 1079.05 & 1011.50 & 986.14  & 980.71  & 1082.76 & 1066.31 & 1042.50 & 1009.40 & 1135.37 & 1017.64 \\ \hline
\end{tabular}}
\end{table}
\subsubsection{Error Analysis of the Different Computational Models}\label{subsubsec2}
The prediction results of different computational models are compared and analyzed to assess the prediction effectiveness of the three different prediction models by calculating the absolute and relative prediction errors. The error calculation is shown in the table 4 and table 5.
\begin{table}[h]
\caption{Model Absolute Error Values for Three Optimization Algorithms}\label{tab4}
\setlength{\tabcolsep}{1.5mm}{
\begin{tabular}{ccccccccccc}
\hline
            & Case1 & Case2  & Case3   & Case4   & Case5  & Case6  & Case7 & Case8  & Case9 & Case 10 \\ \hline
Adam-BP     & 27.39 & -1.24  & -6.07   & -45.96  & -22    & 16.7   & 20.04 & 8.62   & -6.74 & 8.96    \\
SGD-BP      & 60.37 & -28.09 & -69.92  & -77.61  & -60.05 & 10.01  & 29.6  & -52.81 & 23.06 & -51.66  \\
Adadelta-BP & -8.95 & -122.5 & -140.86 & -202.29 & -73.24 & -32.69 & -60.5 & -101.6 & 13.37 & -91.36  \\ \hline
\end{tabular}}
\end{table}
\begin{table}[h]
\caption{Relative Error Values of Three Optimization Algorithm Models (\%)}\label{tab5}
\setlength{\tabcolsep}{1.5mm}{
\begin{tabular}{ccccccccccc}
\hline
            & Case1 & Case2 & Case3 & Case4 & Case5 & Case6 & Case7 & Case8 & Case9 & Case 10 \\ \hline
Adam-BP     & 2.52  & 0.11  & 0.54  & 3.89  & 1.90  & 1.52  & 1.82  & 0.78  & 0.60  & 0.81    \\
SGD-BP      & 5.55  & 2.48  & 6.20  & 6.56  & 5.19  & 0.91  & 2.68  & 4.75  & 2.06  & 4.66    \\
Adadelta-BP & 0.82  & 10.80 & 12.50 & 17.10 & 6.34  & 2.94  & 5.49  & 9.13  & 1.19  & 8.24    \\ \hline
\end{tabular}}
\end{table}

\begin{figure}[htbp]
\begin{minipage}[t]{0.55\linewidth}
\centering
\includegraphics[height=4cm,width=6cm]{投稿/3.jpg}
\caption{Comparison of actual and projected costs}
\end{minipage}%
\begin{minipage}[t]{0.55\linewidth}
\centering
\includegraphics[height=4cm,width=6cm]{投稿/4.jpg}
\caption{Three optimisation algorithms model absolute error values}
\end{minipage}
\end{figure}

\begin{figure}[htbp]
\begin{minipage}[t]{0.55\linewidth}
\centering
\subfigure[histogram]{
\label{Fig.sub.1}
\includegraphics[height=4cm,width=6cm]{投稿/5.jpg}}
\end{minipage}%
\begin{minipage}[t]{0.55\linewidth}
\subfigure[boxdiagram]{
\label{Fig.sub.2}
\includegraphics[height=4cm,width=6cm]{投稿/6.jpg}}
\end{minipage}
\caption{Relative error values of the three optimisation algorithm models（\%）}
\label{fig}
\end{figure}

Table above presents the absolute error values and relative error values for three optimization algorithms (Adam-BP, SGD-BP, and Adadelta-BP) across ten different cases (Case1 to Case10):\par
(1)Adam-BP: For Case1 and Case6, Adam-BP has an absolute error of 27.39 and 16.7, respectively, indicating that the predictions are relatively close to the actual values. In Case2 and Case8, the absolute error values are very close to zero, the relative error is particularly low at 0.11\% and 0.78\%,suggesting that the predictions are almost identical to the actual values. \par
(2)SGD-BP: SGD-BP exhibits higher absolute errors across all cases compared to Adam-BP.  In Case3 and Case4, the absolute errors are -69.92 and -77.61,the relative error exceeds 6\%, signifying substantial overestimation. In contrast, Case2 displays a negative absolute error of -28.09, implying underestimation. The algorithm also demonstrates a high absolute error of -60.05 in Case5, indicating significant overestimation.  \par
(3)Adadelta-BP: Adadelta-BP has unstable performance in cost prediction, with high variability between the prediction errors of different samples.In Case1, the absolute error is -8.95, indicating underestimation, while in Case9, the absolute error is 13.37, suggesting overestimation.  For Case4 and Case8, the algorithm yields high absolute error values of -202.29 and -101.6, respectively, indicating substantial overestimation. In Case3 and Case6, the algorithm also overestimates with absolute errors of -140.86 and -32.69, respectively. In some cases, such as Case2 and Case6, the relative error is quite high, exceeding 10\%, suggesting less accurate predictions.\par

The presented analysis, as depicted in the preceding chart, involves the evaluation of three established cost prediction models: Adam-BP, SGD-BP, and Adadelta-BP. This study encompassed the collection of a dataset comprising 144 instances of project cost data, with a subsequent random selection of 10 data sets designated for cost prediction. Consequently, the predicted construction cost values for 10 distinct projects were derived.\par
Subsequently, a comprehensive error analysis was undertaken to gauge the disparities between the predicted cost values and the actual cost values. The results are as follows:\par
For the Adam-BP optimization model, the analysis reveals that the maximum relative error between the actual and predicted cost values among the 10 engineering samples is 3.89\%, while the minimum is 0.11\%. Notably, the relative error for all 10 test samples remains below 5\%.\par
For the SGD-BP optimization model, the maximum relative error between the actual and predicted cost values reaches 6.56\%, with a minimum of 0.91\%. All 10 test samples exhibit relative errors below 10\%.\par
In contrast, the Adadelta-BP optimization model displays relatively higher relative errors, with a maximum of 17.1\% and a minimum of 0.82\%. However, even within this model, all 10 test samples maintain relative errors below 20\%.
\subsubsection{Performance Metrics for Three Optimization Algorithm Models}\label{subsubsec2}
To further evaluate the three optimization algorithm models, the mean square error (MSE), mean absolute error (MAE) and root mean square error (RMSE) were calculated. As shown in the table 6.
\begin{table}[！ht]
\caption{Performance Metrics for Three Optimization Algorithm Models}\label{tab6}

\begin{tabular}{cccc}
\hline
     & SGD-BP  & Adadelta-BP & Adam-BP \\ \hline
MSE  & 2591.75 & 10479.01    & 426.54  \\
MAE  & 46.32   & 84.74       & 16.37   \\
RMSE & 50.91   & 102.37      & 20.65   \\ \hline
\end{tabular}
\end{table}


This table presents various performance metrics, including MSE、MAE and RMSE values for three distinct optimization algorithms (SGD-BP, Adadelta-BP, and Adam-BP).MSE is a measure of the average squared difference between predicted and actual values. In this context, Adam-BP performs with the lowest MSE of 426.54, indicating its ability to minimize the squared error between predictions and actual values.SGD-BP and Adadelta-BP, on the other hand, exhibit significantly higher MSE values, suggesting a higher overall error in their predictions.MAE measures the average absolute difference between predicted and actual values.In this case, Adam-BP once again outperforms the other algorithms, with a considerably lower MAE of 16.37.SGD-BP and Adadelta-BP have higher MAE values, indicating larger absolute errors in their predictions.RMSE combines the characteristics of both MSE and MAE by taking the square root of the average squared error. Here, Adam-BP has the lowest RMSE of 20.65, emphasizing its effectiveness in minimizing the overall error.SGD-BP and Adadelta-BP exhibit notably higher RMSE values.\par
The table highlights variations in the performance of the three optimization algorithms across multiple evaluation metrics.Adam-BP consistently demonstrates lower error values, reflecting its robustness and effectiveness.In contrast, both SGD-BP and Adadelta-BP exhibit higher error values, suggesting potential challenges or limitations in their performance.Based on this qualitative analysis, Adam-BP appears to be the most effective optimization algorithm among the three, as it consistently produces lower errors across multiple metrics.

\section{Conclusion and Future Work}\label{sec5}
\subsection{Conclusion}\label{subsec2}
In this paper, by collecting 144 sets of project cost cases, a BP neural network model is established to train the prediction of assembly building cost, while three different optimization algorithms are used to optimize the cost prediction. After the study, the following conclusions are obtained:\par
(1) The suitability of a particular algorithm hinges on the specific objectives and characteristics of the cases.Adam-BP appears to be a reliable choice in cases with lower absolute errors, while SGD-BP or Adadelta-BP may be better suited to cases where higher errors are permissible.The Adadelta-BP optimization model exhibits suboptimal precision and yields larger errors in its predictions. On the other hand, the SGD-BP optimization model demonstrates improved predictive performance on individual test samples; however, it suffers from instability, manifesting as Conversely, the Adam-BP optimization model excels in both prediction accuracy and stability, particularly in the context of construction cost prediction for the context of construction cost prediction for actual assembly projects. \par
(2) The Adam-BP neural network cost prediction model constructed in this paper has good prediction effect and is able to realize the cost prediction for actual assembly projects. it effectively addresses the inherent subjectivity and empirical nature of manual cost prediction, significantly enhancing the precision of manual cost prediction. prediction, significantly enhancing the precision of cost prediction. Consequently, it contributes to a more scientific, objective, and accurate Consequently, it contributes to a more scientific, objective, and accurate cost estimation process.
\subsection{Future Work}\label{subsec2}
(1)Detecting atypical cost scenarios involves assessing instances where the absolute prediction error significantly deviates from the average error. When forecasting the actual project cost, notable disparities are observed between the absolute prediction error and the average error. For instance, Case 3 exhibits a conspicuous deviation characterized by significantly elevated error values across all three algorithms, indicative of distinctive project attributes posing intricate challenges for optimization algorithms. Consequently, it is advisable to tailor cost optimization strategies to address these unique project characteristics and thereby enhance cost control measures effectively.\par
(2)In further research, data information can be collected from other structural forms, such as cast-in-place reinforced concrete structures, shear wall structures, and so on. Thus, we can verify whether the BP neural network cost prediction model established in this paper can have universal applicability, and at the same time, analyze the relevance of the factor indicator of different structural types on the construction cost, as well as analyze the influence of different structural types on the accuracy, stability, and variability of the prediction results. In addition, it is not limited to the main civil construction cost, but also carries out research on each item such as foundation engineering and installation engineering, and finally forms the cost prediction of the project as a whole.

\section*{Declarations}
\begin{itemize}
\item Conflict of interest：The authors declare no conflict of interest.
\item Ethics approval ：Not applicable
\item Consent to participate：All authors have read and agreed to the published version of the manuscript.
\item Consent for publication：All authors have read and agreed to the published version of the manuscript.
\item Funding:
This paper has partially received financial support from Major Scientific & Technological Innovation Projects of Shandong Province with project number No.2021CXGC011204 and Collaborative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone Projects “Green Assembly Structure Engineering and Application”.
\item Availability of data and materials:
The data presented in this study are available on request from the corressponding author.The data are not publicly available due to privacy or Commercial reason.
\item Code availability ：The code that support the findings of this study are available from the corressponding author upon reasonable request.
\item Authors' contributions:
Conceptualization,Shujuan Yang and Duhu Yu;methodology,Rumelhart,McClelland,Shujuan Yang and Duhu Yu;software,Qian Liu,Fan zhang and Dehu Yu;data collection and management，Qian Liu and Shujuan Yang;writing and editing,Qian Liu and Tongtong Gai.
\end{itemize}

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