## 2 Materials and Methods
### Model Calculation and Accuracy Evaluation  
#### The Mind Evolutionary Algorithm Back Propagation Neural Network  
The BP neural network (BPNN) is a multi-layer feedforward network with error back propagation, primarily consisting of an input layer, a hidden layer, and an output layer. The number of hidden layers is uncertain and can be 1 or more. The BPNN exhibits a good approximation ability for any nonlinear  
Figure 1: Experimental process for the remediation of heavy metal contaminated soil. (**a**) _Neyraudia reynaudiana_ under Cd stress; (**b**) _Neyraudia reynaudiana_ under Cr stress; (**c**) _Neyraudia reynaudiana_ under Pb stress. continuous function and is one of the most widely used neural network models [43]. Its structure is shown in Figure 2.  
The mind evolutionary algorithm (MEA) is used to optimize the initial weights and thresholds of the BP neural network, and to solve the BPNN shortcoming of being only partially optimal due to its randomly generated weights and thresholds, which further affects the prediction results. The MEA-BPNN proposes similartaxis and dissimilation strategies, which are coordinated with one another and not opposed. The similartaxis strategy is aimed at one segment of the population, while the dissimilation strategy is aimed at the entire population, such that it can improve the selection efficiency, yield better prediction ability, and work more quickly [39]. The specific steps of the MEA are as follows [44]:  
(1) Network initialization  
The BP neural network in this study has a three-layer network structure. The nodes in the hidden layer are determined by the empirical model. The hidden layer defaults to the 'tansig' function, and the output layer defaults to the 'purelin' function. The learning rate was 0.1, the training frequency was 1000, and the expected error was 0.0001. The initial weights \(w\) and thresholds \(\theta\) of the BP neural network are optimized by the mind evolutionary algorithm, which can significantly reduce the number of network iterations and model errors. Based on the BP network structure, the solution space is mapped to the encoding space, and each solution corresponds to the encoding, where the encoding length s is:  
\[s=s_{1}\times s_{2}+s_{2}\times s_{3}+s_{2}+s_{3}, \tag{1}\]  
\[s_{3}=\sqrt{s_{1}+s_{2}}+a \tag{2}\]  
where \(s_{1},\ s_{2},\ s_{3}\) are the number of input, output, and hidden layer nodes, \(\text{a}\in[1,10]\), respectively. (2) Generation of individuals and subpopulations  
The size of the mind evolutionary algorithm is set to 8, the number of best subpopulations and temporary subpopulations are each set to 5, and the number of iterations is set to 20. The strategy of the MEA-BPNN is to generate individuals for a certain number (N), sort the scores, and search for the superior individual, that is, the individual with the highest score. It also searches for other temporary individuals, that is, those individuals that have the highest score. If M and T are respectively the number of superior and temporary subpopulations, then the size of each subpopulation m is as follows:  
\[\text{m}=\frac{\text{N}}{\text{M}+\text{T}}, \tag{3}\]  
The derivative of the mean square error of the training set is selected as the score function val of each individual and subpopulation:  
Figure 2: Structure of the back propagation (BP) neural network. \[\text{val}=\frac{1}{\text{SE}}, \tag{4}\]  
\[\text{SE}=\text{mse}(\text{T}-\text{A2}), \tag{5}\]  
where val is the output value of the output layer after each iteration of the A2 bit; T is the expected output; and SE is the mean square error. (3) Similaraxis operation  
Constantly comparing the scores of each individual in a subpopulation, and searching for the individual with the highest score, the similartaxis operation then selects this optimal individual and uses it as the center for the generation of a new subpopulation. This process continues until a new optimal individual is no longer generated.