Developing aquila optimization-based fuzzy system to predict the mechanical properties of the improved HPC

Abstract

High-performance concrete performs better than normal concrete because of using additional components than usual concrete components. Several artificially based analytics were used to evaluate the compressive strength (CS) of high-performance concrete (HPC) made with fly ash and blast furnace slag. In the present research, the Aquila optimizer (AO) was used to find the best values for the determinants of the adaptive neuro-fuzzy inference system (ANFIS), and radial basis function neural network (RBFNN) that may be changed to enhance performance. The suggested approaches were established using 1030 tests, eight inputs (a primary component of mix designs, admixtures, aggregates, and curing age), and the CS as the forecasting objective. The results of the outperformed model were then contrasted with those found in the existing scientific literature. Calculation results point to the potential benefit of combining AO-RBFNN and AO-ANFIS study. The AO-ANFIS demonstrated significantly higher R² (Train: 0.9862, Test: 0.9922) and lower error metrics (such as: RMSE at 2.1434 (train) and 1.2763 (Test)) when compared to the AO-RBFNN and previously published articles. In summation, the proposed method for determining the CS of HPC supplemented with blast furnace slag and fly ash may be established using the suggested AO-ANFIS analysis.

Keywords

High-performance concrete estimation;artificial intelligence ANFIS optimization algorithm

1 Introduction

High-performance concrete (HPC) components have been extensively used in long-span bridges, towering buildings, dams, etc. These ingredients often include fly ash, blast-furnace slag, and additional additives like super-plasticizer [1, 2]. The ratio of each ingredient may be changed to obtain preset goal efficiency and strengths [3 –6]. Because mixes are very nonhomogeneous, it is difficult to choose mixing percentages and thus predict the compressive strength of concrete (CS). The use of machine learning approaches has received a lot of attention in efforts to reduce the gap between predicted and observed outcomes [7 –10]. Over the last two decades, a variety of machine learning approaches have been applied to create accurate and effective solutions for the CS of HPC and other fileds [11 –13]. Artificial neural networks (ANNs) with fuzzy style [14], and multi-layer [15, 16] are the two most popular types, along NN with single layer [17] and multi-layer [18 –20].

A lab set of data from HPC with 1030/1133 individual tests was given by Yeh [21, 22], and the mixing percentages had eight dependent parameters and one independent parameter (CS). Further machine learning (ML) ideas that might be used to predict the capabilities of HPC include support vector machine (SVM) [23, 24], computational ensemble approaches including random forest (RF) [25], and boosting smooth transition regression trees [26]. A few authors have also combined ANNs with fuzzy logic [27, 28], regression analysis [29], or a range of algorithms comprising SVM, ANNs, and linear regression [30, 31]. Young et al. [32] used NN, gradient boosting (GB), RF, and SVM models to determine the CS of more than 10,000 specimens in accordance with actual mixtures and to take industrial relevance into consideration. More recent articles [33, 34] have information on ML models. They are “black box” techniques, which means they do not explain how inputs are combined to produce forecasts, but they do have benefits like good sensitivity, simplicity, and robustness. It is challenging to use since there is no obvious connection between the CS and the input variables. Numerous complex mathematical theories have been developed to indicate it. Yeh and Lien [35] developed a genetic operation tree approach that combined an operation tree with a genetic method to calculate the CS. Due of ANNs’ benefits, several researchers combined them with genetic programming (GEP) [36, 37] or fuzzy logic [38]. It should be noted that the detailed equations employed in the articles’ mathematical formulations may be laborious and challenging to comprehend. The correlation coefficients of percentages of each factor are used in the linear, non-linear, and metaheuristic regression approaches [39, 40] that may all be used to predict CS. Bharatkumar et al. [41] investigated how HPC’s water content and mineral additions influenced the mixture design method. The association between the W/C and silica fume (SF) substitution rates and the CS of SF concrete was discovered by Bhanja and Sengupta [42]. Namyong et al. [43] established a regression method for CS of conventional concrete. The goal of Zain and Abd’s [44] multiple non-linear regression approach was to forecast the HPC’s strength. Since the lab sample may include some errors in the combination percentages and examining procedure, it is also essential to account for unknown elements in the classification algorithm [32]. Some advantages associated with adaptive neuro-fuzzy inference system (ANFIS) can be flexibility, universal approximation, learning capability [45 –47], and related to radial basis function (RBF), nonlinear approximation, slability, and local approximation [48 –51]. ANFIS allows for flexible rule generation and modeling of complex systems with linguistic variables and if-then rules, making it suitable for domains where interpretability is important. ANFIS has the ability to approximate any continuous function to arbitrary accuracy. ANFIS utilizes a hybrid learning algorithm that combines gradient descent and least-squares estimation. This enables it to learn from data and optimize its parameters, making it suitable for both supervised and unsupervised learning tasks. RBF networks are known for their ability to approximate nonlinear functions effectively. RBF networks typically have fewer parameters compared to other methods, making them computationally efficient and scalable for large datasets. RBF networks use localized basis functions, which means they can focus on specific regions of the input space. This local approximation property allows them to capture local patterns and adapt to different regions of the data distribution. Novel optimization algorithms like AO employes advanced search strategies and mechanisms that improve both exploration and exploitation of the search space. Older evolutionary optimizers may struggle to handle complex optimization problems with high-dimensional search spaces, non-linear relationships, constraints, or multimodal landscapes. Novel optimization algorithms are designed to address these challenges. New algorithms often incorporate mechanisms to enhance robustness against noisy or uncertain problem environments. Many novel optimization methods offer a high degree of customizability and flexibility [52 –55].

The major goal of this study is to provide a practical method for thoroughly assessing how well machine learning algorithms work in the foretelling CS of HPC. Using the radial basis function (RBF) neural network and adaptive neuro-fuzzy inference system (ANFIS) approaches, we tried to create models for predicting the characteristics of HPC. This work implemented the Aquila optimizer (AO) method to discover important RBF and ANFIS methods’ variables that might be improved, named AOR and AOAN employing 1030 tests, 8 input parameters, and the CS as the prediction target. The results were then contrasted with those reported in the literature. Overall, the novelty of this research lies in the application of the Aquila optimizer, the evaluation of HPC with additional components, the comparison with existing literature, and the consideration of a large dataset with multiple input variables. These aspects contribute to advancing the understanding of predicting mechanical properties in high-performance concrete and provide a novel approach for optimizing the performance of predictive models.

2 Dataset description and applied methods

2.1 Data collection

In this work, algorithms were developed using 1030 HPC specimens that were exploited in previous publications [21 , 56–58]. Each sample was constructed using regular Portland cement, which was then normally cured. The HPC experiments described in the literature as of this writing employed samples of various dimensions and geometries.

The HPC’s CS is obtained as a function of eight inputs:

C: Contents of cement

BFS/C: blast furnace slag to cement ratio

FA/C: fly ash to cement ratio

W/C: water to cement ratio

SP/C: superplasticizer to cement ratio

CAG/C: coarse aggregate to cement ratio

FAG/C: fine aggregate to cement ratio

and AC: the HPC age

These components’ database ranges are shown in Table 1, and the training and testing dataset’s distribution graphs are shown in Fig. 1. Because there were 1030 instances in the dataset, they were split into two categories: the training collection, which consisted of seventy per cent of the records, and the testing collection, which included the remaining thirty per cent of the records [59, 60]. A normal distribution is used to ensure that these samples come from a random selection within the broader database. It would seem that the variation for each of the input variables is somewhat broad. Numerical research was carried out to establish the feasibility of these input parameters; as a result, there was not discovered to be any substantial cross-correlation in the eight-dimensional input space [21 , 56–58].

Table 1
The value of statistical indices of inputs and goal

Data Index Inputs Output

C

(kg/m³) $\frac{W}{C}$ $\frac{FA}{C}$ $\frac{BFS}{C}$ $\frac{CAG}{C}$ $\frac{FAG}{C}$ $\frac{AC}{(days)}$ $\frac{SP}{C}$ CS (MPa)

Train Minimum 102 0.267 0.0 0.0 1.552 1.226 3.0 0.0 2.332

Maximum 540 1.882 1 1.504 8.696 9.235 365.000 0.069 82.599

Standard dviation 101.84 0.298 0.306 0.446 1.549 1.323 67.717 0.02 17.718

Skewness 0.511 1.163 0.989 1.467 0.596 1.194 2.993 0.245 0.304

Kurtosis –0.692 2.104 –0.269 1.284 –0.127 2.572 10.189 –1.243 –0.549

Test Minimum 132 0.2 0.0 0.0 1.716 1.135 1.0 0.0 6.267

Maximum 540 1.694 1.430 1.584 7.148 5.993 360 0.125 74.987

Standard dviation 110.075 0.341 0.418 0.513 1.632 1.405 50.155 0.031 13.44

Skewness 0.533 0.554 1.043 1.036 0.580 0.556 4.229 0.873 0.499

Kurtosis –0.228 –1.068 –0.307 –0.314 –1.237 –0.956 20.129 0.190 0.214

Data	Index				Inputs					Output
Train	Minimum	102	0.267	0.0	0.0	1.552	1.226	3.0	0.0	2.332
	Maximum	540	1.882	1	1.504	8.696	9.235	365.000	0.069	82.599
	Standard dviation	101.84	0.298	0.306	0.446	1.549	1.323	67.717	0.02	17.718
	Skewness	0.511	1.163	0.989	1.467	0.596	1.194	2.993	0.245	0.304
	Kurtosis	–0.692	2.104	–0.269	1.284	–0.127	2.572	10.189	–1.243	–0.549
Test	Minimum	132	0.2	0.0	0.0	1.716	1.135	1.0	0.0	6.267
	Maximum	540	1.694	1.430	1.584	7.148	5.993	360	0.125	74.987
	Standard dviation	110.075	0.341	0.418	0.513	1.632	1.405	50.155	0.031	13.44
	Skewness	0.533	0.554	1.043	1.036	0.580	0.556	4.229	0.873	0.499
	Kurtosis	–0.228	–1.068	–0.307	–0.314	–1.237	–0.956	20.129	0.190	0.214

Fig. 1

The distribution plots of the variables: I_1-8) Inputs, O₁) Output.

2.2 Aquila optimizer (AO)

According to [61], AO begins by specifying the starting values for a group of N or solutions S using the formulas listed below (Fig. 2):

Fig. 2

Various treatment of the Aquila; a₁) high soar with the vertical stoop, a₂) contour flight with short glide attack, a₃) spiral shape, a₄) low flight with slow descent attack, a₅) walk and grab prey [61].

$\begin{matrix} S_{ij} = r_{1} \times (U_{j} - L_{j}) + L_{j}, i = 1, 2, \dots, N \\ j = 1, 2, \dots, D \end{matrix}$ (1)

Based on this equation, U_j and L_j show the upper bound and lower bound at j dimension, in that the D shows the dimension. r₁ shows an accidental number ∈ [0,1]. AO has two steps termed exploitation and exploration, which are similar to those of other optimization approaches. Typically, such steps are carried out in order to update the current answers. When t ⩽ (2/3) * T, the exploration step starts and consists of two methods. Equation (2) is used to identify the main method. $\begin{matrix} S_{1} (t + 1) + S_{best} (t) \times (\frac{1 - t}{T}) \\ + (S_{M} (t) - S_{best} (t) * rand) \end{matrix}$ (2)

Where T shows a number of iterations, S_b (t) stands for the optimal answer (individual), which is gained at t (the present iteration). Furthermore, ((1 - t)/T) is used in order to operate the explore accomplishment via the exploration step. Moreover, S_M (t) shows the individual mean, and the following equation is computed S_M (t): $S_{M} (t) = \frac{1}{N} \sum_{i = 1}^{N} S (t), \forall_{j} = 1, 2, \dots, D$ (3)

Levy flight diffusions are used in the second exploration step method using AO to update the present answers, as shown in Equation (4): $\begin{matrix} S_{2} (t + 1) = S_{best} (t) \times Levy \\ + S_{R} (t) + (x - y) * rand \end{matrix}$ (4)

In this equation, S_R shows an accidentally chosen individual, where Levy is associated with the Levy flight diffusion, that is defined as: $Levy = s \times \frac{u \times σ}{{| v |}^{\frac{1}{β}}}, σ = (\frac{Γ (1 + β) \times sine (\frac{π β}{2})}{Γ (\frac{1 + β}{2}) \times β \times 2^{(\frac{β - 1}{2})}})$ (5)

Based on this equation, β is equal to 1.5, and s is equal to 0.01. Furthermore, v and u show accidental numbers ∈ [0,1]. Moreover, based on Equation (4), x, y are used to model the helix form as presented in the equation below: $x = r \times cos (θ), y = r \times sin (θ)$ (6) $r = r_{1} + z \times D_{1}, θ = - ω \times D_{1} + θ_{1}, θ_{1} = \frac{3 \times π}{2}$ (7)

In these equations, r_1 ∈ [0,20], and following, ω and z are equal to 0.005 and 0.00565, respectively.

Furthermore, two techniques are used to mimic individuals’ capacity for exploitation through the search process. The first method uses the position mean of individuals (S_M) and the finest individual (X_best), as follows: $\begin{matrix} S_{3} (t + 1) = (S_{best} (t) - S_{M} (t)) \times a - rand \\ + ((U - L) \times rand + L) \times δ \end{matrix}$ (8)

In this equation, δ and a show the exploitation arrangement parameters.

The second one, which is recognized as a quality function, depends on s_best, Levy, and Q. $\begin{matrix} S_{4} (t + 1) = Q \times S_{best} (t) - (G_{1} \times S (t) \times rand) \\ - G_{2} \times Levy + rand \times G_{1}, \end{matrix}$ (9)

According to the definition of the following equation, the Q is used to sustain the explore procedure: $Q (t) = t^{\frac{2 \times rand () - 1}{{(1 - T)}^{2}}}$ (10)

G₁ highlights several movements made to follow the best answers, and it is stated: $G_{1} = 2 \times rand () - 1$ (11)

G₂ is lowered from 2 to 0 based on Equation (9) and it is able to be calculated as: $G_{2} = 2 \times (1 - \frac{t}{T})$ (12)

Based on this equation, rand shows an accidental value. Algorithm 1 presents the AO’s Pseudocode.

2.3 Radial basis function (RBF) neural network

A feed-forward network with a single input layer, hidden layer, and output layer is known as an RBF NN. Consequently, an RBF NN’s convergence pace rate is high [62]. The hidden layer nodes are shaped by a Gaussian activation function, which the input nodes use to move input parameters from the input layer to the hidden one. This kind of NN responds to input signals that are at the core of the fundamental function. The output layer, which mostly uses a basic linear function, receives the hidden layer’s final output [63]. The topology of the RBF NN is shown in Fig. 3, where t1, t2, …, t8 represent the network inputs and φ1, φ2, …, φq represent the base function’s centre in the hidden layer. As well, w0, w1 , … , wq are the weights (w0 is the output layer weight). The Gaussian function (φ) used is:

Fig. 3

Radial basis function structure.

$φ_{i} = \exp (- \frac{∥ t - c_{i} ∥}{σ_{i}^{2}})$ (13)

φ_i: Output of i^th node of hidden layer

c_i: Prototype center of i^th Gaussian function

σ: Spread rate parameter

∥t - c_i∥: Distance between input t and c_i

The finding of an RBF NN can be depicted via Equation (14): $Y = W^{T} φ = \sum_{i = 1}^{q} w_{i} φ (∥ t - c_{i} ∥)$ (14)

The spread rate and the number of neurons in the hidden layer are automatically dedicated by the customizable RBF NN algorithm. The ideal mix of neuron counts and spread rate is determined by calculating variables in the efficiency of RBF NN. The highest precise RBF NN is obtained using the combined AOR approach. The spread value and the number of hidden neurons are determined using the AO method to set the RBF structure.

2.4 Adaptive neuro-fuzzy inference system (ANFIS)

The ANFIS model is the foundational model used in this study. Utilizing a hybrid neuro-fuzzy method, the ANFIS network is renowned for simulating complicated systems [64, 65]. With ANFIS, the thinking type of a fuzzy system, which is analogous to a person thinking, is included using a set of fuzzy If-Then circumstances (rules). ANFIS models provide global estimation and explicable If-Then rules as global estimation methods [66].

To demonstrate how an ANFIS works, we merely take into account two inputs x and y, and one output, the f_out. The ANFIS structure employed in this study is shown in Fig. 4. An explanation of the node functions connected to every substrate may be found below.

Fig. 4

Architecture schematics for ANFIS.

Substrate 1: The nodes in this substrate are adaptable and capable of carrying out the following activities [66]: $O_{1, i} = μ_{A_{i}} (x) for i = 1, i = 2$ (15) $O_{1, i} = μ_{B_{i - 2}} (y) for i = 3, i = 4$ (16)

In the equations above, y (or x) shows the input of the node. A_i (Or B_j) serves as the linguistic label and μ(x)/μ(y) stands for the membership function. The most typical diffusion is a bell-shaped diffusion with a lower limit of 0 and an upper limit of 1.

Substrate 2: According to [66], the fixed nodes in the second substrate, denoted by the Π and circled in the diagram, will have their functions multiplied by input signals to produce an outcome. $O_{2, i} = μ_{A_{i}} (x) \cdot μ_{A_{i}} (x) = w_{i} for i = 1, 2$ (17)

The resulting signal w_i represents the firing power of a rule.

Substrate 3: This layer uses a node function to compute the ratio of every node’s firing power to the total of entire rules’ firing powers in order to standardize firing power, denoted by constant nodes that are shown by circles and called N [66]: $O_{3, i} = \frac{w_{i}}{\sum w_{i}} = \frac{w_{i}}{w_{1} + w_{2}} for i = 1, 2$ (18)

Substrate 4: This layer contains adaptive nodes, all of that is denoted by a square and has the following node functions [44]: $O_{3, i} = {\bar{w}}_{i} \cdot for i = 1, 2$ (19)

In this context, f₁, and f₂ are the fuzzy if-then rules according to [66]: $\begin{matrix} • Rule 1 : if (x = = A_{1} & y = = B_{1}) \\ f_{1} = p_{1} x + q_{1} y + r_{1} \\ • Rule 2 : if (x = = A_{2} & y = = B_{2}) \\ f_{2} = p_{2} x + q_{2} y + r_{2} \end{matrix} .$ (20)

[pi, qi, ri] are the variables that have been fixed in this situation.

Substrate 5: This substrate has four constant nodes that are every represented by a single circle, have a Sigma tag, and have a node function that determines the outcome as a whole [66]: $O_{5} = \sum {\bar{w}}_{i} \cdot f_{i} = f_{out} for i = 1, 2$ (21)

Utilizing backpropagation gradient descent, the fault signal in ANFIS is generated iteratively from the output to the input substrate. Feedforward neural networks use the identical backpropagation learning algorithm.

2.5 Evaluation metrics

To evaluate the usefulness of AO-RBF, and AO-ANFIS systems, four metrics were computed and compared. The bellow metrics were computed as appropriate metrics to gain this objective (Equations (22)–(25)):

Coefficient of determination (R²) $R^{2} = {(\frac{\sum_{p = 1}^{P} (t_{P} - \bar{t}) (y_{P} - \bar{y})}{\sqrt{[\sum_{p = 1}^{P} {(t_{P} - \bar{t})}^{2}] [\sum_{p = 1}^{P} {(y_{P} - \bar{y})}^{2}]}})}^{2}$ (22)

Root mean squared error (RMSE) $RMSE = \sqrt{\frac{1}{P} \sum_{p = 1}^{P} {(y_{p} - t_{p})}^{2}}$ (23)

Mean absolute error (MAE) $MAE = \frac{1}{P} \sum_{p = 1}^{P} | y_{p} - t_{p} |$ (24) $A_{20 - Index}$ $A_{20 - Index} = \frac{m_{20}}{M}$ (25) where, correspondingly, y_P, t_P, $\bar{t}$ , and $\bar{y}$ are the goal, forecasted values, and the mean of the goal and forecasted of them. Here, M is the sample count and m₂₀ is the sample count with Meas./Est.=0.8-1.2.

Algorithm 1. The AO’s Pseudocode [61]
1. Initiating phase
2. Fixed the S population parameter for the Aquila Optimization.
3. Fixed the Aquila optimization variables containing α, σ, and other things like these.
4. if the halting requirement is not met, do .
5. Calculate several suitable functions.
6. Reach the finest answer, S_best(t) taking into account the suitable values.
7. For i = 1, 2, 3, …, N do
8. Change the present answer S_z(t)’s average value.
9. Change the x, y, M₁, M₂, D, etc.
10. if t is equal or less than $\frac{2 T}{3}$ then
11. if M₁ ⩽ 0.5 then
12. 1. Increased exploitation (S₁)
13. Change the present location utilizing Equation (2). S_1(t+1)
14. if S_1(t+1) calculated less than S_(t) then
15. S_(t) calculated as equal to S_1(t+1)
16. if S_1(t+1) calculated less than S_best(t) then
17. S_best(t). calculated as equal to S_1(t+1)
18. end if
19. end if
20. else
21. 2. Limited exploitation (S₂)
22. Change the present answer utilizing Equation (4)S_2(t+1)
23. if S_2(t+1) calculated less than S_(t). then
24. S_(t). calculated as equal to S_2(t+1)
25. if S_2(t+1) calculated less than S_best(t) then
26. S_best(t). calculated as equal to S_2(t+1)
27. end if
28. end if
29. end if
30. else
31. if M₁ calculated equal to or less than 0.5, then
32. 3. Increased exploitation (S₃)
33. Change the present answer utilizing Equation (8)S_3(t+1)
34. if S_3(t+1) calculated less than S_(t) then
35. S_(t) calculated as equal to S_3(t+1)
36. if S_3(t+1) calculated less than S_best(t) then
37. S_best(t) calculated as equal to S_3(t+1)
38. end if
39. end if
40. else
41. 4. Limited exploitation (S₄)
42. Change the present location utilizing Equation (9)S_4(t+1)
43. if S_4(t+1) calculated less than S_(t) then
44. S_(t) calculated as equal to S_4(t+1)
45. if S_4(t+1) calculated less than S_best(t) then
46. S_best(t) calculated as equal to S_4(t+1)
47. end if
48. end if
49. end if
50. end if
51. end for
52. end while
53. Get back to the finest answer S_best(t)

3 Findings and explanation

In order to forecast the CS of the HPC enhanced with BFS and FA, the results of the AOAN and AOR designs are presented in this paper. If the crucial elements are combined in the right ratios, as previously mentioned, ANFIS and RBF performance will be maximized. The observed and calculated values of the CS of HPC during the training and testing phases of the generated AOAN and AOR systems are shown in Fig. 5. Additionally, when we plot the % error in CS concentration on a graph, we see a normal distribution of curves, with the centre of the distribution falling on the zero-error percentage line. The utility of the AOAN and AOR was assessed using the values of R², RMSE, MAE, and A_20-Index (Table 2). Both AOAN and AOR have a lot of promise for making precise forecasts of HPC’s CS.

Fig. 5

The conclusions of the AO-based models, a) Correlation plot, b) Error % distribution.

Table 2

Statistical errors of proposed models

Phase	Metric	This article	This article	Comparison
				[67]	[70]	[33]	[30]	[68]	[69]	[71]
		AOR	AOAN	GEP	MGGP	GPR	ANNs	SEM	GPR	XGB (All data)
Train	R ²	0.9787	0.9862	0.8224	0.7885	0.987	–	0.84	0.888	0.944
	MAE (MPa)	1.6155	1.3492	5.202	5.56	–	–	4.91	3.996	2.592
	RMSE (MPa)	2.6107	2.1434	–	7.36	–	–	6.3	5.59	3.878
	A _20-Index	0.9653	0.9792	–	–	0.9753	–	0.68	–	–
Test	R ²	0.9833	0.9922	0.8354	0.8046	0.8858	0.8649	0.8567	0.888	–
	MAE (MPa)	1.2127	0.9298	5.19	5.48	–	4.421	4.482	3.913	–
	RMSE (MPa)	1.7525	1.2763	–	7.31	–	6.329	5.968	5.597	–
	A _20-Index	0.9871	0.9871	–	–	0.757	–	0.752	–	–

A portion of this study examines the effectiveness of many iterations of the statistical identifiers (AOAN and AOR) created for published studies. The outcomes of this investigation have also been objectively evaluated to those of other published studies. The results show that the merged AOR and AOAN systems could estimate quite well, with R² values for the train and test portions of AOAN at 0.9862 and 0.9922 and for AOR at 0.9787 and 0.9833, respectively. To determine the best strategy, it is crucial to look into and assess the signals produced. In comparison to the AOR, the AOAN RMSE value decreased with training, going from 2.6107 MPa to 2.1434 MPa. Results of the testing portion indicated a modest drop from 1.7525 MPa to 1.2763 MPa. Similarly, the MAE measure produced the same outcomes as RMSE and demonstrated that the AOAN had superior capacity for CS estimation at MAE_Train=1.3492 MPa and MAE_Test=0.9298 MPa, as its values were less than those of the AOR at MAE_Train=1.6155 MPa and MAE_Test=1.2127 MPa. Comparable results were seen for the A_20-Index signal, which had an increase of 2% in the train section and a similar amount in the test portion for AOAN.

The estimates provided here were developed after comparing several different methods, including Gene Expression Programming (GEP) [67], Semi-Empirical Method (SEM) [68], Gaussian Process Regression (GPR) [69], Artificial Neural Networks (ANN) [30], Multi-Gene Genetic Programming (MGGP) [70], and Extreme Gradient Boosting (XGB) [71]. If you look at the table, you can see that our suggested AOAN outperformed the existing literature. SEM [46] performed poorly when compared to AOAN, with R² of 0.84 vs 0.9862, RMSE of 6.3 MPa vs 2.1434 MPa, MAE of 4.91 MPa vs 1.3492 MPa, and A_20-Index of 0.68 vs 0.9792. For instance, GEP [67] demonstrated a much higher MAE and a slightly lower R² when compared to AOAN (by 0.8224 and 5.202, respectively). The newest technique, XGB [71], came close to overtaking AOAN but eventually fell short. Other techniques, such MGGP [70] and ANNs [30], performed worse than AOAN, with R² values that are much lower than 0.9862, at 0.8046 and 0.8469, respectively. The AOAN structure, which was first created to represent HPC’s CS and then enhanced with FA and BFS, is recommended for usage in this situation.

Table 2. Statistical errors of proposed models

The Taylor diagram might be used to assess the effectiveness of the suggested AOR and AOAN concepts in further detail. The values for the root mean squared error (RMSE) are shown on the black dashed lines. As illustrated in Fig. 6, the training and testing sections of the results were divided into two groups. It is clear that algorithms act consistently both during training and during testing. The placement of AOAN was closer to the Reference point throughout training and testing but AOR produced worse performance. Consequently, it may be concluded by analyzing the findings of the Taylor diagram that the AOAN framework was more competent than others.

Fig. 6

Taylor diagram results.

4 Conclusion

The results show that the merged AOR and AOAN systems could estimate quite well, with R² values for the train and test portions of AOAN at 0.9862 and 0.9922 and for AOR at 0.9787 and 0.9833, respectively. In comparison to the AOR, the AOAN RMSE value decreased with training, going from 2.6107 MPa to 2.1434 MPa. Results of the testing portion indicated a modest drop from 1.7525 MPa to 1.2763 MPa. Similarly, the MAE measure produced the same outcomes as RMSE and demonstrated that the AOAN had a superior capacity for CS estimation. Comparable results were seen for the A_(20-Index) signal, which had an increase of 2% in the train section and a similar amount in the test portion for AOAN.

It was clear from the results that the proposed AOAN achieved the best results compared to the published literature.

It was clear from the Taylor diagram that algorithms act consistently both during training and during testing. The placement of AOAN was closer to the Reference point throughout training and testing but AOR produced worse performance. Consequently, it may be concluded by analyzing the findings of the Taylor diagram that the AOAN framework was more competent than others.

Finally, it was suggested to use of AOAN framework, which was designed to represent HPC’s CS and was later improved with FA and BFS.

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