The effect of ICA and PSO on ANN results in approximating elasticity modulus of rock material

Reliable determination/evaluation of the rock deformation can be useful prior any structural design application. Young’s modulus (E) affords great insight into the characteristics of the rock. However, its direct determination in the laboratory is costly and time-consuming. Therefore, rock deformation prediction through indirect techniques is greatly suggested. This paper describes hybrid particle swarm optimization (PSO)–artificial neural network (ANN) and imperialism competitive algorithm (ICA)–ANN to solve shortcomings of ANN itself. In fact, the influence of PSO and ICA on ANN results in predicting E was studied in this research. By investigating the related studies, the most important parameters of PSO and ICA were identified and a series of parametric studies for their determination were conducted. All models were built using three inputs (Schmidt hammer rebound number, point load index and p-wave velocity) and one output which is E. To have a fair comparison and to show the capability of the hybrid models, a pre-developed ANN model was also constructed to estimate E. Evaluation of the obtained results demonstrated that a higher ability of E prediction is received developing a hybrid ICA–ANN model. Coefficient of determination (R²) values of (0.952, 0.943 and 0.753) and (0.955, 0.949 and 0.712) were obtained for training and testing of ICA–ANN, PSO–ANN and ANN models, respectively. In addition, VAF values near to 100 (95.182 and 95.143 for train and test) were achieved for a developed ICA–ANN hybrid model. The results indicated that the proposed ICA–ANN model can be implemented better in improving performance capacity of ANN model compared to another implemented hybrid model.

     

A new design of evolutionary hybrid optimization of SVR model in predicting the blast-induced ground vibration

This study aims to identify the suitability of hybridizing the firefly algorithm (FA), genetic algorithm (GA), and particle swarm optimization (PSO) with two well-known data-driven models of support vector regression (SVR) and artificial neural network (ANN) to predict blast-induced ground vibration. Here, these combinations are abbreviated using FA–SVR, PSO–SVR, GA–SVR, FA–ANN, PSO–ANN, and GA–ANN models. In addition, a modified FA (MFA) combined with SVR model is also proposed in this study, namely, MFA–SVR. The feasibility of the proposed models is examined using a case study, located in Johor, Malaysia. Then, to provide an objective assessment of performances of the predictive models, their results were compared based on several well known and popular statistical criteria. According to the results, the MFA–SVR with the coefficient of determination (R²) of 0.984 and root mean square error (RMSE) of 0.614 was more accurate model to predict PPV than the PSO–SVR with R² = 0.977 and RMSE = 0.725, the FA–SVR with R² = 0.964 and RMSE = 0.923, the GA–SVR with R² = 0.957 and RMSE = 1.016, the GA–ANN with R² = 0.936 and RMSE = 1.252, the FA–ANN with R² = 0.925 and RMSE = 1.368, and the PSO–ANN with R² = 0.924 and RMSE = 1.366. Consequently, the MFA–SVR model can be sufficiently employed in estimating the ground vibration, and has the capacity to generalize.

     

Development of a hybrid PSO–ANN model for estimating glucose and xylose yields for microwave-assisted pretreatment and the enzymatic hydrolysis of lignocellulosic biomass

In this paper, two artificial intelligent systems, the artificial neural network (ANN) and particle swarm optimization (PSO), were combined to form a hybrid PSO–ANN model that was used to improve estimates of glucose and xylose yields from the microwave–acid pretreatment and enzymatic hydrolysis of lignocellulosic biomass based on pretreatment parameters. ANN is a powerful tool capable of determining the relationship between the desired input and output data while PSO was used as a robust population-based search algorithm to optimize the performance of the ANN model. Specifically, it was used to determine the optimum number of neurons in the hidden layer and the best value of the learning rate of the ANN model. The optimization method includes minimizing the fitness function mean absolute error that was found to be 0.0176. The PSO algorithm suggested an optimum number of neurons in the hidden layer as 15 and a learning rate of 0.761 these consequently used to construct the ANN model. After constructing the hybrid PSO–ANN model, the performance of the intelligent system was examined by determining the regression coefficient (R ²) for estimating the experimental values of glucose and xylose and compared to the results from a response surface methodology (RSM) model. The results of R ² of the hybrid PSO–ANN model for glucose and xylose were 0.9939 and 0.9479, respectively, while the RSM model results for the same sugars were 0.8901 and 0.8439. This analysis reveals that the hybrid PSO–ANN model offers a higher degree of accuracy in comparison with the more commonly used RSM model.

     

An optimized ANN model based on genetic algorithm for predicting ripping production

Due to the environmental constraints and the limitations on blasting, ripping as a ground loosening and breaking method has become more popular in both mining and civil engineering applications. As a result, a more applicable rippability model is required to predict ripping production (Q) before conducting such tests. In this research, a hybrid genetic algorithm (GA) optimized by artificial neural network (ANN) was developed to predict ripping production results obtained from three sites in Johor state, Malaysia. It should be noted that the mentioned hybrid model was first time applied in this field. In this regard, 74 ripping tests were investigated in the studied areas and the relevant parameters were also measured. A series of GA–ANN models were conducted in order to propose a hybrid model with a higher accuracy level. To demonstrate the performance capacity of the hybrid GA–ANN model, a pre-developed ANN model was also proposed and results of predictive models were compared using several performance indices. The results revealed higher accuracy of the proposed hybrid GA–ANN model in estimating Q compared to ANN technique. As an example, root-mean-square error values of 0.092 and 0.131 for testing datasets of GA–ANN and ANN techniques, respectively, express the superiority of the newly developed model in predicting ripping production.

     

Prediction of unconfined compressive strength of rock surrounding a roadway using artificial neural network

The unconfined compressive strength (UCS) of rocks is an important design parameter in rock engineering and geotechnics, which is required and determined for rock mechanical studies in mining and civil projects. This parameter is usually determined through a laboratory UCS test. Since the preparation of high-quality samples is difficult, expensive and time consuming for laboratory tests, development of predictive models for determining the mechanical properties of rocks seems to be essential in rock engineering. In this study, an attempt was made to develop an artificial neural network (ANN) and multivariable regression analysis (MVRA) models in order to predict UCS of rock surrounding a roadway. For this, a database of laboratory tests was prepared, which includes rock type, Schmidt hardness, density and porosity as input parameters and UCS as output parameter. To make a database (including 93 datasets), different rock samples, ranging from weak to very strong types, are used. To compare the performance of developed models, determination coefficient (R ²), variance account for (VAF), mean absolute error (E _a) and mean relative error (E _r) indices between predicted and measured values were calculated. Based on this comparison, it was concluded that performance of the ANN model is considerably better than the MVRA model. Further, a sensitivity analysis shows that rock density and Schmidt hardness were recognized as the most effective parameters, whereas porosity was considered as the least effective input parameter on the ANN model output (UCS) in this study.     

Design and implementation of a new tuned hybrid intelligent model to predict the uniaxial compressive strength of the rock using SFS-ANFIS

This study proposes a novel design to systematically optimize the parameters for the adaptive neuro-fuzzy inference system (ANFIS) model using stochastic fractal search (SFS) algorithm. To affirm the efficiency of the proposed SFS-ANFIS model, the predicting results were compared with ANFIS and three hybrid methodologies based on ANFIS combined with genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO). Accurate prediction of uniaxial compressive strength (UCS) is of great significance for all geotechnical projects such as tunnels and dams. Hence, this study proposes the use of SFS-ANFIS, GA-ANFIS, DE-ANFIS, PSO-ANFIS, and ANFIS models to predict UCS. In this regard, the fresh water tunnel of Pahang–Selangor located in Malaysia was considered and the requirement data samples were collected. Different statistical metrics such as coefficient of determination (R²) and mean absolute error were used to evaluate the models. Referring to the efficiency results of SFS-ANFIS, it can be found that the SFS-ANFIS (with the R² of 0.981) has higher ability than PSO-ANFIS, DE-ANFIS, GA-ANFIS, and ANFIS models in predicting the UCS.

     

Strength evaluation of granite block samples with different predictive models

Over the last decade, application of soft computing techniques has rapidly grown up in different scientific fields, especially in rock mechanics. One of these cases relates to indirect assessment of uniaxial compressive strength (UCS) of rock samples with different artificial intelligent-based methods. In fact, the main advantage of such systems is to readily remove some difficulties arising in direct assessment of UCS, such as time-consuming and costly UCS test procedure. This study puts an effort to propose four accurate and practical predictive models of UCS using artificial neural network (ANN), hybrid ANN with imperialism competitive algorithm (ICA–ANN), hybrid ANN with artificial bee colony (ABC–ANN) and genetic programming (GP) approaches. To reach the aim of the current study, an experimental database containing a total of 71 data sets was set up by performing a number of laboratory tests on the rock samples collected from a tunnel site in Malaysia. To construct the desired predictive models of UCS based on training and test patterns, a combination of several rock characteristics with the most influence on UCS has been used as input parameters, i.e. porosity (n), Schmidt hammer rebound number (R), p-wave velocity (V_p) and point load strength index (Is(50)). To evaluate and compare the prediction precision of the developed models, a series of statistical indices, such as root mean squared error (RMSE), determination coefficient (R²) and variance account for (VAF) are utilized. Based on the simulation results and the measured indices, it was observed that the proposed GP model with the training and test RMSE values 0.0726 and 0.0691, respectively, gives better performance as compared to the other proposed models with values of (0.0740 and 0.0885), (0.0785 and 0.0742), and (0.0746 and 0.0771) for ANN, ICA–ANN and ABC–ANN, respectively. Moreover, a parametric analysis is accomplished on the proposed GP model to further verify its generalization capability. Hence, this GP-based model can be considered as a new applicable equation to accurately estimate the uniaxial compressive strength of granite block samples.

     

Developing GEP tree-based,neuro-swarm,and whale optimization models for evaluation of bearing capacity of concrete-filled steel tube columns

The type of materials used in designing and constructing structures significantly affects the way the structures behave. The performance of concrete and steel, which are used as a composite in columns, has a considerable effect upon the structure behavior under different loading conditions. In this paper, several advanced methods were applied and developed to predict the bearing capacity of the concrete-filled steel tube (CFST) columns in two phases of prediction and optimization. In the prediction phase, bearing capacity values of CFST columns were estimated through developing gene expression programming (GEP)-based tree equation; then, the results were compared with the results obtained from a hybrid model of artificial neural network (ANN) and particle swarm optimization (PSO). In the modeling process, the outer diameter, concrete compressive strength, tensile yield stress of the steel column, thickness of steel cover, and the length of the samples were considered as the model inputs. After a series of analyses, the best predictive models were selected based on the coefficient of determination (R²) results. R² values of 0.928 and 0.939 for training and testing datasets of the selected GEP-based tree equation, respectively, demonstrated that GEP was able to provide higher performance capacity compared to PSO–ANN model with R² values of 0.910 and 0.904 and ANN with R² values of 0.895 and 0.881. In the optimization phase, whale optimization algorithm (WOA), which has not yet been applied in structural engineering, was selected and developed to maximize the results of the bearing capacity. Based on the obtained results, WOA, by increasing bearing capacity to 23436.63 kN, was able to maximize significantly the bearing capacity of CFST columns.

     

Rock strength assessment based on regression tree technique

Uniaxial compressive strength (UCS) is one of the most important parameters for investigation of rock behaviour in civil and mining engineering applications. The direct method to determine UCS is time consuming and expensive in the laboratory. Therefore, indirect estimation of UCS values using other rock index tests is of interest. In this study, extensive laboratory tests including density test, Schmidt hammer test, point load strength test and UCS test were conducted on 106 samples of sandstone which were taken from three sites in Malaysia. Based on the laboratory results, some new equations with acceptable reliability were developed to predict UCS using simple regression analysis. Additionally, results of simple regression analysis show that there is a need to propose UCS predictive models by multiple inputs. Therefore, considering the same laboratory results, multiple regression (MR) and regression tree (RT) models were also performed. To evaluate performance prediction of the developed models, several performance indices, i.e. coefficient of determination (R ²), variance account for and root mean squared error were examined. The results indicated that the RT model can predict UCS with higher performance capacity compared to MR technique. R ² values of 0.857 and 0.801 for training and testing datasets, respectively, suggests the superiority of the RT model in predicting UCS, while these values are obtained as 0.754 and 0.770 for MR model, respectively.     

Prediction of seismic slope stability through combination of particle swarm optimization and neural network

One of the main concerns in geotechnical engineering is slope stability prediction during the earthquake. In this study, two intelligent systems namely artificial neural network (ANN) and particle swarm optimization (PSO)–ANN models were developed to predict factor of safety (FOS) of homogeneous slopes. Geostudio program based on limit equilibrium method was utilized to obtain 699 FOS values with different conditions. The most influential factors on FOS such as slope height, gradient, cohesion, friction angle and peak ground acceleration were considered as model inputs in the present study. A series of sensitivity analyses were performed in modeling procedures of both intelligent systems. All 699 datasets were randomly selected to 5 different datasets based on training and testing. Considering some model performance indices, i.e., root mean square error, coefficient of determination (R ²) and value account for (VAF) and using simple ranking method, the best ANN and PSO–ANN models were selected. It was found that the PSO–ANN technique can predict FOS with higher performance capacities compared to ANN. R ² values of testing datasets equal to 0.915 and 0.986 for ANN and PSO–ANN techniques, respectively, suggest the superiority of the PSO–ANN technique.     

Rock brittleness prediction through two optimization algorithms namely particle swarm optimization and imperialism competitive algorithm

Brittleness index (BI) is a significant rock parameter when dealing with projects performed in rocks. The main goal of this research work is to propose the novel practical models to predict the BI through particle swarm optimization (PSO) and imperialism competitive algorithm (ICA). For this aim, two forms of equations, i.e., linear and power are considered and the weights of these equations are optimized by PSO and ICA. In the other words, four predictive models, namely ICA linear, ICA power, PSO linear, and PSO power models are developed to predict BI in this study. In the modeling of the predictive models, 79 datasets are used, so that Schmidt hammer rebound number, wave velocity, density, and Point Load Index (Is₅₀) are selected as the independent (input) parameters and the BI values are considered as the dependent (output) parameter. Then, the performances of the proposed predicting models are checked using two error indices, namely coefficient correlation (R²) and root mean squared error (RMSE). The results showed that the PSO power model has superior fitting specification for the prediction of the BI compared to the other prediction models and is quite practical for use. As a result, linear and power models of PSO received higher performance prediction compared to ICA. PSO power (with R² train = 0.937, R² test = 0.959, RMSE train = 0.377 and RMES test = 0.289) showed the most powerful technique to predict BI of the granite samples.

     

Uniaxial compressive strength prediction through a new technique based on gene expression programming

Proper estimation of rock strength is a critical task for evaluation and design of some geotechnical applications such as tunneling and excavation. Uniaxial compressive strength (UCS) test can be measured directly in the laboratory; nevertheless, the direct UCS determination is time-consuming and expensive. In this study, feasibility of gene expression programming (GEP) model in indirect determination of UCS values of sandstone rock samples is examined. In this regard, several laboratory tests including Brazilian test, density test, slake durability test and UCS test were conducted on 47 samples of sandstone which were collected from the Dengkil, Malaysia. Considering multiple inputs, several GEP models were constructed to estimate UCS of the rock and finally, the best GEP model was selected. In order to indicate capability of the proposed GEP model, linear multiple regression (LMR) was also performed. It was found that the GEP model is superior to LMR one in terms of applied performance indices. Based on coefficient of determination (R ²) of testing datasets, by proposing GEP model, it can be improved from 0.930 (which was obtained by LMR model) to 0.965. As a result, it is concluded that the proposed models in this study, could be utilized to estimate UCS of similar rock type in practice.

     

Forecasting of Software Reliability Using Neighborhood Fuzzy Particle Swarm Optimization Based Novel Neural Network

This paper proposes an artificial neural network (ANN) based software reliability model trained by novel particle swarm optimization (PSO) algorithm for enhanced forecasting of the reliability of software. The proposed ANN is developed considering the fault generation phenomenon during software testing with the fault complexity of different levels. We demonstrate the proposed model considering three types of faults residing in the software. We propose a neighborhood based fuzzy PSO algorithm for competent learning of the proposed ANN using software failure data. Fitting and prediction performances of the neighborhood fuzzy PSO based proposed neural network model are compared with the standard PSO based proposed neural network model and existing ANN based software reliability models in the literature through three real software failure data sets. We also compare the performance of the proposed PSO algorithm with the standard PSO algorithm through learning of the proposed ANN. Statistical analysis shows that the neighborhood fuzzy PSO based proposed neural network model has comparatively better fitting and predictive ability than the standard PSO based proposed neural network model and other ANN based software reliability models. Faster release of software is achievable by applying the proposed PSO based neural network model during the testing period.      

Application of several non-linear prediction tools for estimating uniaxial compressive strength of granitic rocks and comparison of their performances

Uniaxial compressive strength (UCS) of rock is crucial for any type of projects constructed in/on rock mass. The test that is conducted to measure the UCS of rock is expensive, time consuming and having sample restriction. For this reason, the UCS of rock may be estimated using simple rock tests such as point load index (I _s(50)), Schmidt hammer (R _n) and p-wave velocity (V _p) tests. To estimate the UCS of granitic rock as a function of relevant rock properties like R _n, p-wave and I _s(50), the rock cores were collected from the face of the Pahang–Selangor fresh water tunnel in Malaysia. Afterwards, 124 samples are prepared and tested in accordance with relevant standards and the dataset is obtained. Further an established dataset is used for estimating the UCS of rock via three-nonlinear prediction tools, namely non-linear multiple regression (NLMR), artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS). After conducting the mentioned models, considering several performance indices including coefficient of determination (R ²), variance account for and root mean squared error and also using simple ranking procedure, the models were examined and the best prediction model was selected. It is concluded that the R ² equal to 0.951 for testing dataset suggests the superiority of the ANFIS model, while these values are 0.651 and 0.886 for NLMR and ANN techniques, respectively. The results pointed out that the ANFIS model can be used for predicting UCS of rocks with higher capacity in comparison with others. However, the developed model may be useful at a preliminary stage of design; it should be used with caution and only for the specified rock types.     

Optimization of ANFIS with GA and PSO estimating α ratio in driven piles

This study aimed to optimize Adaptive Neuro-Fuzzy Inferences System (ANFIS) with two optimization algorithms, namely, Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) for the calculation friction capacity ratio (α) in driven shafts. Various studies are shown that both ANFIS are valuable methods for prediction of engineering problems. However, optimizing ANFIS with GA and PSO has not been used in the area of pile engineering. The training data set was collected from available full-scale results of the driven piles. The input parameters used in this study were pile diameter (m), pile length (m), relative density (I_d), embedment ratio (L/D), both of the pile end resistance (q_c) and base resistance at relatively 10% base settlement (q_b0.1) from CPT result, whereas the output was α. A learning fuzzy-based algorithm was used to train the ANFIS model in the MATLAB software. The system was optimized by changing the number of clusters in the FIS and then the output was used for the GA and PSO optimization algorithm. The prediction was compared with the real-monitoring field data. As a result, good agreement was attained representing reliability of all proposed models. The estimated results for the collected database were assessed based on several statistical indices such as R², RMSE, and VAF. According to R², RMSE, and VAF, values of (0.9439, 0.0123 and 99.91), (0.9872, 0.0117 and 99.99), and (0.9605, 0.0119 and 99.97) were obtained for testing data sets of the optimized ANFIS, GA–ANFIS, and PSO–ANFIS predictive models, respectively. This indicates higher reliability of the optimized GA–ANFIS model in estimating α ratio in driven shafts.

     

Feasibility of PSO–ANFIS model to estimate rock fragmentation produced by mine blasting

Desired rock fragmentation is the main goal of the blasting operation in surface mines, civil and tunneling works. Therefore, precise prediction of rock fragmentation is very important to achieve an economically successful outcome. The primary objective of this article is to propose a new model for forecasting the rock fragmentation using adaptive neuro-fuzzy inference system (ANFIS) in combination with particle swarm optimization (PSO). The proposed PSO–ANFIS model has been compared with support vector machines (SVM), ANFIS and nonlinear multiple regression (MR) models. To construct the predictive models, 72 blasting events were investigated, and the values of rock fragmentation as well as five effective parameters on rock fragmentation, i.e., specific charge, stemming, spacing, burden and maximum charge used per delay were measured. Based on several statistical functions [e.g., coefficient of correlation (R ²) and root-mean-square error (RMSE)], it was found that the PSO–ANFIS (with R ² = 0.89 and RMSE = 1.31) performs better than the SVM (with R ² = 0.83 and RMSE = 1.66), ANFIS (with R ² = 0.81 and RMSE = 1.78) and nonlinear MR (with R ² = 0.57 and RMSE = 3.93) models. Finally, the sensitivity analysis shows that the burden and maximum charge used per delay have the least and the most effects on the rock fragmentation, respectively.

     

Connectionist approaches for solubility prediction of n-alkanes in supercritical carbon dioxide

Carbon dioxide injection is a known promising and economical technology for improving oil recovery. Despite its immense effect on oil recovery, the application of this technique in modern recovery industry has been limited due to poor solubility of n-alkanes in supercritical CO₂. Therefore, it is very consequential to investigate the solubility of different n-alkanes in supercritical CO₂. Since experimental methods for measuring the solubility of n-alkanes in supercritical CO₂ at different temperatures and pressures are not economical and usually take a long time, feasibility of applying intelligent tools in the solubility prediction of different n-alkanes in supercritical CO₂ at pressures up to 45.9 MPa was conducted in this study. For this purpose, two models including an artificial neural network and an adaptive neuro-fuzzy interference system (ANFIS) both trained with particle swarm optimization (PSO) algorithm were used for simulating this process. Calculated mole fractions of n-alkanes in supercritical CO₂ from ANFIS–PSO model were excellently consistent with actual measured values. Moreover, comparison between these models and Chrastil semiempirical correlation show superiority and accuracy of the proposed ANFIS–PSO approach. Results of this study indicate that ANFIS–PSO method is a powerful technique for predicting solubility of n-alkanes in supercritical CO₂.

     

A novel artificial intelligence technique for analyzing slope stability using PSO-CA model

This study aims to develop a new artificial intelligence model for analyzing and evaluating slope stability in open-pit mines. Indeed, a novel hybrid intelligent technique based on an optimization of the cubist algorithm by an evolutionary method (i.e., PSO), namely PSO-CA technique, was developed for predicting the factor of safety (FS) in slope stability; 450 simulations from the Geostudio software for the FS of a quarry mine (Vietnam) were used as the datasets for this aim. Five factors include bench height, slope angle, angle of internal friction, cohesion, and unit weight were used as the input variables for estimating FS in this work. To clarify the performance of the proposed PSO-CA technique in slope stability analysis, SVM, CART, and kNN models were also developed and assessed. Three performance indices, such as mean absolute error (MAE), root-mean-squared error (RMSE), and determination coefficient (R²), were computed to evaluate the accuracy of the predictive models. The results clarified that the proposed PSO-CA technique was the most dominant accuracy with an MAE of 0.009, RMSE of 0.025, and R² of 0.981, in estimating the stability of slope. The remaining models (i.e., SVM, CART, kNN) obtained poorer performance with MAE from 0.014 to 0.038, RMSE 0.030–0.056, and R² 0.917–0.974.

     

Feasibility of PSO-ANN model for predicting surface settlement caused by tunneling

The potential surface settlement, especially in urban areas, is one of the most hazardous factors in subway and other infrastructure tunnel excavations. Therefore, accurate prediction of maximum surface settlement (MSS) is essential to minimize the possible risk of damage. This paper presents a new hybrid model of artificial neural network (ANN) optimized by particle swarm optimization (PSO) for prediction of MSS. Here, this combination is abbreviated using PSO-ANN. To indicate the performance capacity of the PSO-ANN model in predicting MSS, a pre-developed ANN model was also developed. To construct the mentioned models, horizontal to vertical stress ratio, cohesion and Young’s modulus were set as input parameters, whereas MSS was considered as system output. A database consisting of 143 data sets, obtained from the line No. 2 of Karaj subway, in Iran, was used to develop the predictive models. The performance of the predictive models was evaluated by comparing performance prediction parameters, including root mean square error (RMSE), variance account for (VAF) and coefficient correlation (R ²). The results indicate that the proposed PSO-ANN model is able to predict MSS with a higher degree of accuracy in comparison with the ANN results. In addition, the results of sensitivity analysis show that the horizontal to vertical stress ratio has slightly higher effect of MSS compared to other model inputs.     

An optimised product-unit neural network with a novel PSO–BP hybrid training algorithm: Applications to load–deformation analysis of axially loaded piles

In general, neural network training is a nonlinear multivariate optimisation problem. Unlike previous studies, in the present study, particle swarm optimisation (PSO) and back-propagation (BP) algorithms were coupled to develop a robust hybrid training algorithm with both local and global search capabilities. To demonstrate the capacity of the proposed model, we applied the model to the predictions of the load–deformation behaviour of axially loaded piles. This is a soil–structure interaction problem, involving a complex mechanism of load transfer from the pile to the supporting geologic medium. A database of full scale pile loading tests is used to train and validate the product-unit network. The results show that the proposed hybrid learning algorithm simulates the load–deformation curve of axially loaded piles more accurately than other BP, PSO, and existing PSO–BP hybrid methods. The network developed using the proposed algorithm also turns out to be more accurate than hyperbolic and $t?z$ models.