In the era of big data, the field of artificial intelligence is making a splash, [1–8]. Accurate prediction plays a crucial role in our modern life, where the research of prediction methods based on machine learning, especially neural networks, has become increasingly popular, [9–12]. The back propagation (BP) neural network, a simple structured basic model, generally includes an input, implicit, and output layer. The BP neural network can simulate any complex nonlinear relationship by nonlinear elements in the three-layer structure. The method has been widely used in various fields because of its sound data processing and nonlinear mapping capabilities, [13–15]. Although BP neural networks have many advantages, they tend to fall into local optimum, have low learning efficiency, and converge slowly, [16].

In recent years, much research has been conducted on improving the convergence speed of traditional BP neural networks and avoiding convergence to local optimum. Many optimization methods have been proposed. Among them, intelligent optimization algorithms abstracted by simulating the evolutionary process or foraging behavior of biological populations, [17–20], such as genetic algorithm (GA), ant colony optimization (ACO) and particle swarm optimization (PSO), have been widely used to solve optimization problems due to the advantages of simple implementation and easy scalability. Optimizing BP neural networks using intelligent algorithms has become a research hotspot. Researchers applied standard PSO algorithms to BP neural networks to effectively reduce learning time and improve computational accuracy, [21,22]. Li et al. conducted virtual simulation experiments on the short-term power generation of photovoltaic power plants through three sets of models of BP, GA-BP, and PSO-BP. They verified that GA-BP and PSO-BP could effectively reduce errors, [23]. Mohamad et al. used PSO to optimize BP neural networks for predicting uniaxial compression strength (UCS) of rocks and conducted simulation experiments with laboratory datasets to demonstrate that PSO-BP models have good predictive performance, [24]. Zhu et al. used a GA to optimize BP neural network to obtain the GA-BP model to predict the risk coefficient of rainfall-induced landslides. The results showed that the GA-BP model could predict the landslide risk coefficient of large areas more effectively by testing 100 landslide data in Sichuan Province, China, [25]. Hu et al. proposed a prediction model using ACO optimized BP neural network to predict the production increase effect of oil field development accurately. The experimental results showed that the model effectively predicted the production increase effect of oil fields, [26]. Li et al. proposed a battery state of charge estimation based on gray wolf optimization (GWO) and BP neural network for the problem of inaccurate battery state of charge (SOC) estimation of lithium batteries model, which has a higher SOC estimation accuracy and a minor relative error compared with the traditional BP neural network, [27]. Wen et al. developed a novel BP neural network model based on PSO to forecast the carbon dioxide emissions and made some improvements to PSO to improve the accuracy of the forecast. Finally, the validity of the model was verified using panel data of the Chinese commercial sector from 1997 to 2017, [28].

However, these standard intelligent optimization algorithms decline in species diversity in the late iterations and tend to fall into local optima, which may fail to search for optimal weights and thresholds when guiding BP neural networks to adjust network parameters, thus failing to achieve the best prediction.

In 2020, a novel sparrow search algorithm (SSA) was proposed by Xue et al. [29]. The authors conducted comparative experiments with 19 sets of functions and demonstrated that SSA has high search accuracy and fast convergence. However, like other population intelligence optimization algorithms, SSA still suffers from the problem of reduced population diversity and tends to fall into local optimum when it iterates to a later stage. Based on the standard SSA algorithm, we design a dynamic discoverer strategy that can adjust the proportion of discoverers according to the number of iterations. We also introduce adaptive t-distribution changes to improve the algorithm’s global exploitation capability in the early stage and local search capability in the later stage. Meanwhile, after the sparrow search is completed, we employ a random wandering strategy to perturb the sparrow population and prevent the algorithm from falling into local optimum.

Based on the proposed algorithm, this paper designs the MSSA-BP model. In this part, we conduct simulation experiments on nine benchmark functions to verify the algorithm’s search performance and select the offshore wind farm dataset in the western Gulf of Mexico and the air quality dataset in Beijing, China, for the SSA-BP, MSSA-BP model, and DPSO-BP model to verify the effectiveness of the models.

Firstly, we determine the three-layer structure of the BP neural network, as shown in Fig. 1, and determine the number of input layer neurons and output layer neurons according to the number of inputs and outputs. Secondly, the range of the number of neurons in the hidden layer is determined according to the empirical Eq. (9). The optimal number of neurons in the hidden layer is finally determined by comparing the MSE of the predicted and actual values of the training set corresponding to different numbers of neurons in the hidden layer. (9)h=n1+n2+v

The MSE generated by each network training set is used as an approximate fitness function to calculate the fitness value, and the MSE is as in Eq. (10). (10)MSE=1n∑i=1n(Yi′−Yi)2

In Eq. (8), Yi and Yi^{\prime} are the target and predicted values, respectively. The smaller the MSE, the more accurate the model.

Fig. 2 shows the running process of MSSA in the BP neural network.

Step 1: Set initial parameters: population size, the maximum number of iterations, warning value ST, the proportion of scouts SD, etc.

Step 2: Initialize the population.

Step 3: The fitness fi of each sparrow is calculated by the fitness function and then sorted, select the current optimal fitness fg and its corresponding position Xb, and the current worst fitness fw and its corresponding position Xw.

Step 4: Determine the current PD value according to Eq. (4), select the sparrow with high adaptability as the discoverers, and the rest as followers, and update the positions of the discoverers and followers according to Eqs. (1) and (2). Randomly select a part of sparrows from the sparrow population for reconnaissance and early warning, and update their positions according to Eq. (3).

Step 5: After one iteration is completed, the fitness value fi of each sparrow is recalculated. According to Eq. (4), the sparrow is mutated. If the sparrow is better than the one before the mutation, the previous sparrow will be replaced by the mutated sparrow. Otherwise, it will remain unchanged.

Step 6: According to the current state of the sparrow population, update the optimal position Xb and its fitness fg experienced by the entire population, as well as the worst position Xw and its fitness fw.

Step 7: Perturbation of the optimal sparrow employing a random wandering strategy. If the scrambled sparrow is better than the previous one, replace the previous sparrow with the scrambled one and update fg. Otherwise, leave it unchanged.

Step 8: Determine if the algorithm has reached the maximum number of iterations. If the condition is met, the loop ends, and the optimization result is output; otherwise, return to step 5.

Step 9: The obtained optimal weights and thresholds are assigned to the BP neural network for training and learning.

This research proposes a BP neural network prediction method based on a multi-strategy improved sparrow search algorithm. Based on the standard SSA, we design a dynamic discoverers strategy, which uses adaptive t-distribution to mutate the sparrow, and uses a random walk strategy to sparrow perturbs and improve the algorithm’s performance. The MSSA-BP prediction model was also designed to improve the prediction performance based on the MSSA algorithm. The superiority-seeking ability of MSSA is demonstrated by testing nine different types of benchmark functions. The results of simulation experiments with two data sets show that the average MAE, average RMSE, and average MAPE of MSSA-BP are better than the comparison model, proving that MSSA-BP has better prediction accuracy and robustness. The contributions of our research to engineering practice are as follows: (1)

The MSSA proposed in this paper has the advantages of fast convergence speed, high convergence accuracy and good stability.

Of course, MSSA-BP still has some shortcomings, and its more complex structure leads to slower running time than SSA-BP, which is an area for improvement. Meanwhile, the intelligent algorithm inevitably encounters the optimal localization problem due to NP theory, which remains a challenge. Our future research may incorporate engineering practice problems.