使用python代码编写BP神经网络预测实例代码，加上注释

当然可以。以下是使用Python和`numpy`库编写简单BP（Backpropagation）神经网络进行线性回归预测的一个示例代码，包括了初始化权重、前向传播、反向传播以及训练过程。这里我们假设有一个简单的数据集，目标是预测y基于x。

import numpy as np

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

# 初始化权重函数（随机值）
def initialize_weights(input_nodes, hidden_nodes, output_nodes):
    weights_ih = np.random.uniform(size=(hidden_nodes, input_nodes)) - 0.5
    weights_ho = np.random.uniform(size=(output_nodes, hidden_nodes)) - 0.5
    return weights_ih, weights_ho

# 前向传播函数
def forward_propagation(weights_ih, weights_ho, inputs):
    hidden_layer = sigmoid(np.dot(weights_ih, inputs))
    outputs = sigmoid(np.dot(weights_ho, hidden_layer))
    return hidden_layer, outputs

# 反向传播函数
def back_propagation(inputs, targets, outputs, hidden_layer, weights_ih, weights_ho):
    d_weights_ho = np.dot(outputs - targets, hidden_layer.T)
    d_weights_ih = np.dot(hidden_layer.T, np.dot(weights_ho.T, (outputs - targets)))
    return d_weights_ih, d_weights_ho

# 神经网络训练
def train_network(data, targets, epochs, learning_rate, hidden_nodes):
    # 初始化权重
    weights_ih, weights_ho = initialize_weights(len(data[0]), hidden_nodes, 1)

    for epoch in range(epochs):
        # 前向传播
        hidden_layer, outputs = forward_propagation(weights_ih, weights_ho, data)
        
        # 计算误差
        error = outputs - targets
        
        # 反向传播更新权重
        d_weights_ih, d_weights_ho = back_propagation(data, targets, outputs, hidden_layer, weights_ih, weights_ho)
        
        # 更新权重
        weights_ih -= learning_rate * d_weights_ih
        weights_ho -= learning_rate * d_weights_ho
    
    return weights_ih, weights_ho

# 示例数据（你可以替换为你实际的数据）
data = np.array([[0], [1]])
targets = np.array([[0], [1]])

# 训练模型
hidden_nodes = 2  # 隐藏节点数
epochs = 1000  # 迭代次数
learning_rate = 0.1  # 学习率
weights_ih, weights_ho = train_network(data, targets, epochs, learning_rate, hidden_nodes)

# 测试模型
test_input = np.array([[2]])  # 测试数据点
hidden_layer_test, output_test = forward_propagation(weights_ih, weights_ho, test_input)
print(f"预测值: {output_test[0][0]}")