给一个python的BP神经网络算法

BP（反向传播）神经网络算法是一种常用的人工神经网络训练方法，其通过反向传播算法不断修正权值以提高模型的预测准确度。

以下是一个使用 Python 实现 BP 神经网络算法的示例代码：

import numpy as np

class NeuralNetwork:
    def __init__(self, layers, alpha=0.1):
        # 初始化每层的节点数
        self.layers = layers
        self.alpha = alpha
        
        # 初始化权重和偏移量
        self.weights = [np.random.randn(y, x) for x, y in zip(layers[:-1], layers[1:])]
        self.biases = [np.random.randn(y, 1) for y in layers[1:]]
        
    def sigmoid(self, x):
        return 1.0 / (1 + np.exp(-x))
    
    def sigmoid_derivative(self, x):
        return x * (1 - x)
    
    def fit(self, X, y, epochs=1000):
        for epoch in range(epochs):
            # 前向传播
            a = [X]
            for w, b in zip(self.weights, self.biases):
                z = np.dot(w, a[-1]) + b
                a.append(self.sigmoid(z))
                
            # 计算误差
            error = y - a[-1]
            deltas = [error * self.sigmoid_derivative(a[-1])]
            
            # 反向传播
            for i in range(len(a) - 2, 0, -1):
                delta = np.dot(self.weights[i].T, deltas[-1]) * self.sigmoid_derivative(a[i])
                deltas.append(delta)
                
            deltas.reverse()
            
            # 更新权重和偏移量
            for i in range(len(self.weights)):
                a_i = np.atleast_2d(a[i])
                delta_i = np.atleast_2d(deltas[i])
                self.weights[i] += self.alpha * np.dot(delta_i, a_i.T)
                self.biases[i] += self.alpha * delta_i.sum(axis=1