RBNNA is a forward neural network with good performance. It maps nonlinear problems to linear space, transforms them into the adaptive algorithm processing, and maintains the high accuracy and efficiency of the algorithm. RBNNA is a three-layer feedforward neural network consisting of an input, hidden, and output layer (Wang et al., 2023). The input layer is the node that receives the original input data, and each node corresponds to the input feature. The hidden layer is a set of nodes for radial basis functions, which is used to measure the distance between input data and some centers. The output layer produces the final output, which is a linear layer that combines the outputs of the hidden layers. Figure 1 shows the prediction process of the RBNNA algorithm. The core of RBFNN lies in the radial basis function of the hidden layer, which is a Gaussian or other symmetric function. Gaussian is one of the radial basis functions, which is expressed by Equation 1:

In Equation 1,  G s ( ρ ) represents the output of the s-th basis function, o s represents the center of the basis function, and σ s 2 represents the width parameter.

The gray wolf algorithm is a meta-heuristic algorithm proposed by Mirjalili et al (Li et al., 2021), which is derived from gray wolves’ social hierarchy and hunting strategy. In this algorithm, the population is divided into levels A–D. Wolves A control all actions of the wolf pack, which are the supreme leader of the wolf pack. Wolves B assist wolf A in making some decisions, which are some experienced wolves (Xu et al., 2023). Wolves C are responsible for the reconnaissance of the wolf pack, which are responsible for guarding and caring cubs. Wolves D belong to the lowest level of gray wolves and obey the commands of gray wolves from other classes, which are accounting for the vast majority. The best wolves are A, B, and C. They help wolves D to find the favorable area (Verma et al., 2022). First, the wolves need to locate their prey and surround it. The process is shown in Equation 2:

In Equation 2, D represents the distance between the gray wolf and the prey, N represents the coefficient vector, W φ ( k ) represents the position vector of the prey, W ( k ) represents the position vector of the gray wolf, and k represents the number of iterations. The position of k +1 wolves is shown in Equation 3:

In Equation 3, M represents the coefficient vector; other parameters have the same meaning as Equation 2. N represents the calculation process of the coefficient vector sum, which is shown in Equations 4, 5:

In Equations 4, 5, ω represents the convergence factor, which decreases linearly from 2 to 0 as k increases; p 1   and p 2 represent the random number with a value range of (0,1). When the prey is surrounded, the wolves start hunting. The hunting process is carried out under the leadership of wolves A, B, and C. They guide wolves D to track the prey location. The calculation process is as shown in Equations 6–8:

In Equations 6–8, A ( D i s t ) , B ( D i s t ) , and C ( D i s t ) represent the distance between the three wolves and other individuals; W a , W b , and W c represent the current positions of the three wolves; N 1 , N 2 , and N 3 represent the random vectors; and W represents the current position of the gray wolf. The vectors of wolves D in the wolf pack moving toward wolves A, B, and C are represented by W 1 , W 2 , and W 3 . The calculation process is as shown in Equations 9–11:

In Equation 9, Equation 10, Equation 11, according to the calculation results of W 1 , W 2 , and W 3 , the final position of wolves D can be determined. The calculation process is shown in Equation 12:

Finally, the hunt is completed by attacking the prey when it cannot move. The processing of gray wolf algorithm is shown in Figure 2 .

In order to enhance the predicting accuracy of the RBNNA algorithm, this study proposes an IM-RBNNA algorithm. It uses the gray wolf algorithm to optimize the weights and thresholds of the RBNNA algorithm so that the weights and threshold are optimal. When the output result is different from the expected value, the principle of backpropagation is used to optimize. The threshold and weight of the gray wolf algorithm are used as the weight and threshold of the RBNNA algorithm (Liu and Wang, 2020). The relative error value between the predicted and true value of soil nutrient content is used as the fitness value. The continuous iterative update of the gray wolf algorithm is used to adjust the weights and thresholds of the RBNNA algorithm. The advantages with better global effects can improve the model’s prediction accuracy (Feng et al., 2023). Figure 3 shows the processing of the gray wolf algorithm for optimizing the RBNNA algorithm.

This study is conducted in Area 2 of a durian orchard in Penang, Malaysia. It is located in Sungai Pinang Balik Pulau, Penang, which coveres an area of 3,200 acres. The rows of planting density is 5.0 m × 4.0 m. Every acre has 30 plants. The durian trees in this area are all in the peak production period of 15–20 years. During this period, the durian trees have fully developed, so the canopy is dense, which can produce more durian fruits. This area has a tropical rainforest climate, with an average annual temperature of 28°C, an average annual precipitation of 2,525.3 mm, and an annual sunshine count of 2,076.9 h. The study area has significant spatial differences in the growth and yield of durian, which is suitable for the precise fertilization, so this site is chosen to study the precise fertilization. Four experiments are set up in the research area to verify the differences of soil fertility under different algorithms (Portela et al., 2011). From west to east are areas A, B, C, and D, which are used for experiments on different fertilization decision-making methods. Each area is 5 acres. In this plantation soil, the alkaline hydrolyzed nitrogen is 21.5 ± 3.0 mg/kg (low), the available phosphorus is 47.1 ± 0.6 mg/kg (high), the available potassium is 117.7 ± 20.9 mg/kg (low), and the pH is 7.6 ± 0.1 (alkaline). All plots are used for unified measures. Figure 4 shows the design of soil sampling locations for this experiment.

In order to get the nutrient information of durian soil, five samples are collected within 20 m of the sampling center in each sampling point. The corresponding network of the plot is determined by using manual measurement. The plum blossom sampling method is used in each grid (Zhou and Staver, 2019). Five points of the soil sampling samples are mixed into labeling bags, which have 5,000 sampling points and 150 plots. The sampling time is from February to August 2022. The sample depth is 0–20 cm. These samples are mixed and labeled as soil samples at that point. RTK (real-time kinematic) is used to collect and record the longitude and latitude of the sample point. After the soil samples are naturally air-dried and sieved, the pH is measured to use an electrode method with a water-to-soil volume ratio of 1:1; the organic matter is measured to use the dichromic acid. The potassium method is used to measure the total nitrogen; the copper sulfate digestion method is used to measure the total nitrogen; and the available phosphorus is measured to use the suitable method for neutral and calcareous soils. The soil is measured to use the sodium bicarbonate; the available potassium is measured to use the flame photometry. The fertilizer amount is calculated based on the soil nutrient data, which is obtained from laboratory tests (Liu and Feng, 2017) by using the fertilizer balance model of the target yield method. The physical and chemical properties of the soil are measured through the above laboratory methods to obtain the various nutrient data for each plot (Kim, 2018). Table 1 shows the nutrient data of some sampling points in the durian orchard.

The nutrient contents of alkaline hydrolyzable nitrogen, available phosphorus, and available potassium in the soil vary greatly. For example, the potassium is approximately 200 mg/kg, but the phosphorus is approximately 10 mg/kg. When the cluster analysis is performed, the impact of available phosphorus is almost negligible, which is difficult not to meet the requirements of the soil similarity calculation. In order to solve these problems, this study standardized the data by using the same standard. The normal standardization subtracts each attribute of each data object from the average value of the attribute and then divides it by the variance of the attribute (Mykhailenko et al., 2020). The data standardized by this method reach the standard normal distribution. The data have a mean of 0 and a variance of 1, which is shown in Equation 13.

In Equation 13, H i ¯ represents the i-th standardized data attribute value, H i represents the i-th data attribute value to be standardized, R represents the mean value of the attribute, and α h represents the variance of the attribute. The standardized value of the soil nutrient content is calculated. Table 2 shows the standardized results of the soil nutrient data in Table 1 .

The Inter@core i7–9700K processor was used in this study, the graphics card is NVIDIA Geforce GTX3080 32GB, the memory is 64GB, the operating system is Ubuntu19.04 64-bit, the deep learning framework is Pyotrch1.9.2, the programming language is Python3.7.1, the integrated development environment is PycharmCE2023, and the drawing tool is Matplotlib 3.1.0. In order to ensure that the data distribution is representative, this study trains 5,000 samples according to the validation set =7:3, which is divided in 3,500 training and 1,500 validation sets. The deep learning network algorithm needs to preset hyperparameters before training, so this study set the hyper parameters to batch after comparison. The number of samples is 6, the epoch is 100, the learning rate optimizer uses the SGD algorithm (Stochastic Gradient Descent) (Thuwajit et al., 2021) to update the weights, the initial learning rate is 0.01, the learning rate decay is 0.001, the activation function uses the Sigmoid function, and the model classifier uses SVM (support vector machine) (Dou et al., 2023).

This study uses the mean relative error (MAPE), the mean absolute error (MAE), and the coefficient of determination ( R 2 ) to evaluate the performance of the IM-RBNNA and other similar algorithms (Li et al., 2020). The MAPE is used to calculate the relative difference between the actual and predicted values. MAPE is used to calculate the percentage error of each observed value relative to the actual value and then averages it. The smaller the value of MAPE, the better the model’s performance. The MAE is used to calculate the average of the absolute differences between actual and predicted values. The R 2 measures how well a model fits the data and represents the model’s ability to explain the variation in the dependent variable. In these calculation formula, U j represents the actual value of the soil nitrogen content; U j ¯ represents the predicted average value of the soil nitrogen content; and U represents the predictive value of the soil nitrogen content. t represents the number of samples. The calculation formula is Equations 14–16, which are as follows:

The target yield is the key to durian fertilization recommendations. This study uses the multiple linear stepwise regression methods to determine the durian yield (Sardoei et al., 2023). It determines an initial set containing multiple independent variables and builds a multiple regression formula that does not include this factor. It will run until no more independent variables X can be introduced. The soil nutrient content of the alkaline hydrolyzable nitrogen, available phosphorus, and available potassium are important factors to affect the durian yield, which is relatively in line with the requirements of the above method. The calculation process of the model is shown in Equation 17:

In Equation 17, X represents the independent variable, Y represents the dependent variable, z represents the number of independent variables, and e represents the regression coefficient of each variable. The training of the multiple linear stepwise regression algorithm in this study is conducted in IBMSPSS Statistics 25.

In order to compare the performance of the IM-RBNNA algorithm, this study introduces three methods for comparison: the RBNNA, Backpropagation Neural Network (BPNN) (LI et al., 2019), and Nutrient Balance Calculation Algorithm (NUBCA) (Nannan et al., 2021). The NUBCA algorithm keeps balance between the plants receiving adequate nutrients and their nutrient needs with the available nutrients in the soil. The BPNN algorithm builds the relationship between the plant growth and soil conditions, which uses the backpropagation algorithm for model training and reduces prediction errors by adjusting the weights and biases. This experiment also measures the performance of the four algorithms from three indicators: mean absolute percentage error (MAPE), mean absolute error (MAE), and coefficient of determination( R 2 ).