## 1 Introduction  
Heavy-metal pollution in soil threatens human health and severely damages the human ecological environment [1]. The regional soil heavy-metal pollution detection system in China is constantly improving, developing toward \"digitization\", \"precision\", and \"intelligence\" [2]. A high-level mathematical model for the analysis of heavy-metal pollution in soil is required.  
There is a relatively complete set of mathematical analysis methods for ecological environments, including correlation analysis, principal component analysis, and regression analysis in statistical analysis, and classification and clustering methods in data mining [3]. However, the premise for validating these methods is the availability of complete sample data. In soil sample collection, missing sample data can often result from outdated sampling equipment, contaminated samples, and other issues. Collecting and analyzing soil samples is costly and time-consuming, making resampling impractical. Ecological environment researchers have been seeking an effective solution for missing values [4]; however, there is currently no mature method to address missing values. Traditional missing soil value processing methods are too simple, using only mean filling, mode filling, and ignoring tuples to interpolate missing data without considering the correlation between soil sample data, with poor interpolation results for missing values.  
Soil heavy-metal pollution control and restoration have long cycles and high costs; thus, current soil prevention and control measures focus mainly on prevention; restoration is auxiliary. There is limited research on soil pollution prediction; most research is based on time-series water and air pollution prediction [5; 6]. Unlike water and air pollution, heavy metals have a strong sedimentary type in soil; soil heavy-metal content does not change significantly without human intervention over a long period of time. Thus, it is not feasible to predict soil heavy-metal pollution content using time-series prediction methods. There have been many studies worldwide on qualitative analysis, prevention and control, and soil restoration with regard to soil pollution; however, there have been few studies on the quantitative analysis of soil pollution. It is becoming extremely important to accurately predict the heavy-metal content in soil.  
The main contributions of this study are as follows:  
1. A new SA-FOA-BP neural network model was constructed to predict missing values of heavy metals in soil. This algorithm model primarily combines the simulated annealing algorithm with the fruit fly optimization algorithm to optimize the parameters in place of traditional methods for parameter optimization in the BP neural network, thereby addressing the shortcomings of the traditional BP neural network.
2. A spatial information fusion graph convolutional network prediction model, SE-GCN, was proposed. It establishes a spatial information encoder capable of perceiving spatial contextual information and embeds it with spatial autocorrelation, serving as auxiliary learning to predict the heavy-metal content in soil.
3. The comparative experiments were conducted on the prediction of missing values and contents of heavy metals in soil using two neural network models, and their application scenarios and applicability were analyzed. The experiment showed that, compared to the traditional BP neural network, the SA-FOA-BP neural network model demonstrated improvements in error evaluation metrics, indicating that SA-FOA effectively optimized the parameters. The SE-GCN model achieved more accurate predictions of soil heavy-metal contents compared to existing methods, suggesting that the spatial encoder effectively extracted spatial information.