请帮我写段代码，BP神经网络交叉的代码

以下是一个简单的BP神经网络交叉的代码示例：

import numpy as np

# 定义sigmoid函数
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

# 定义BP神经网络类
class BPNeuralNetwork:
    def __init__(self, input_size, hidden_size, output_size):
        self.input_size = input_size
        self.hidden_size = hidden_size
        self.output_size = output_size
        
        # 初始化权重矩阵
        self.weights_input_hidden = np.random.randn(self.hidden_size, self.input_size)
        self.weights_hidden_output = np.random.randn(self.output_size, self.hidden_size)
        
        # 初始化偏置向量
        self.bias_hidden = np.random.randn(self.hidden_size, 1)
        self.bias_output = np.random.randn(self.output_size, 1)
        
    # 前向传播
    def forward(self, x):
        # 计算隐藏层输出
        hidden = sigmoid(np.dot(self.weights_input_hidden, x) + self.bias_hidden)
        
        # 计算输出层输出
        output = sigmoid(np.dot(self.weights_hidden_output, hidden) + self.bias_output)
        
        return output
    
    # 反向传播
    def backward(self, x, y, output):
        # 计算输出层误差
        error_output = output - y
        
        # 计算隐藏层误差
        error_hidden = np.dot(self.weights_hidden_output.T, error_output) * sigmoid(np.dot(self.weights_input_hidden, x) + self.bias_hidden) * (1 - sigmoid(np.dot(self.weights_input_hidden, x) + self.bias_hidden))
        
        # 更新权重矩阵和偏置向量
        self.weights_hidden_output -= 0.1 * np.dot(error_output, sigmoid(np.dot(self.weights_hidden_output, sigmoid(np.dot(self.weights_input_hidden, x) + self.bias_hidden)) * (1 - sigmoid(np.dot(self.weights_hidden_output, sigmoid(np.dot(self.weights_input_hidden, x) + self.bias_hidden))) * sigmoid(np.dot(self.weights_input_hidden, x) + self.bias_hidden)).T)
        self.weights_input_hidden -= 0.1 * np.dot(error_hidden, x.T)
        self.bias_output -= 0.1 * error_output
        self.bias_hidden -= 0.1 * error_hidden
        
    # 训练模型
    def train(self, x_train, y_train, epochs):
        for i in range(epochs):
            for j in range(len(x_train)):
                x = x_train[j].reshape(self.input_size, 1)
                y = y_train[j].reshape(self.output_size, 1)
                
                output = self.forward(x)
                self.backward(x, y, output)
                
    # 预测
    def predict(self, x):
        x = x.reshape(self.input_size, 1)
        output = self.forward(x)
        return output

注意：这只是一个简单的示例，实际应用中需要根据具体情况进行调整和优化。