找了很多资料，觉得就这篇博客讲的最好，原文如下，有非常详细的推导过程。http://www.hankcs.com/ml/back-propagation-neural-network.html ，转载一下。 
代码链接如下https://github.com/hankcs/neural_net/blob/master/bpnn.py 
作者讲解的十分详细了，这里就不多说。

# coding=utf-8
# 反向传播神经网络import math
import randomrandom.seed(0)def rand(a, b):"""创建一个满足 a <= rand < b 的随机数:param a::param b::return:"""return (b - a) * random.random() + adef makeMatrix(I, J, fill=0.0):"""创建一个矩阵（可以考虑用NumPy来加速）:param I: 行数:param J: 列数:param fill: 填充元素的值:return:"""m = []for i in range(I):m.append([fill] * J)return mdef randomizeMatrix(matrix, a, b):"""随机初始化矩阵:param matrix::param a::param b:"""for i in range(len(matrix)):for j in range(len(matrix[0])):matrix[i][j] = random.uniform(a, b)def sigmoid(x):"""sigmoid 函数，1/(1+e^-x):param x::return:"""return 1.0 / (1.0 + math.exp(-x))def dsigmoid(y):"""sigmoid 函数的导数:param y::return:"""return y * (1 - y)class NN:def __init__(self, ni, nh, no):# number of input, hidden, and output nodes"""构造神经网络:param ni:输入单元数量:param nh:隐藏单元数量:param no:输出单元数量"""self.ni = ni + 1  # +1 是为了偏置节点self.nh = nhself.no = no# 激活值（输出值）self.ai = [1.0] * self.niself.ah = [1.0] * self.nhself.ao = [1.0] * self.no# 权重矩阵self.wi = makeMatrix(self.ni, self.nh)  # 输入层到隐藏层self.wo = makeMatrix(self.nh, self.no)  # 隐藏层到输出层# 将权重矩阵随机化randomizeMatrix(self.wi, -0.2, 0.2)randomizeMatrix(self.wo, -2.0, 2.0)# 权重矩阵的上次梯度self.ci = makeMatrix(self.ni, self.nh)self.co = makeMatrix(self.nh, self.no)def runNN(self, inputs):"""前向传播进行分类:param inputs:输入:return:类别"""if len(inputs) != self.ni - 1:print 'incorrect number of inputs'for i in range(self.ni - 1):self.ai[i] = inputs[i]for j in range(self.nh):sum = 0.0for i in range(self.ni):sum += ( self.ai[i] * self.wi[i][j] )self.ah[j] = sigmoid(sum)for k in range(self.no):sum = 0.0for j in range(self.nh):sum += ( self.ah[j] * self.wo[j][k] )self.ao[k] = sigmoid(sum)return self.aodef backPropagate(self, targets, N, M):"""后向传播算法:param targets: 实例的类别:param N: 本次学习率:param M: 上次学习率:return: 最终的误差平方和的一半"""# http://www.youtube.com/watch?v=aVId8KMsdUU&feature=BFa&list=LLldMCkmXl4j9_v0HeKdNcRA# 计算输出层 deltas# dE/dw[j][k] = (t[k] - ao[k]) * s'( SUM( w[j][k]*ah[j] ) ) * ah[j]output_deltas = [0.0] * self.nofor k in range(self.no):error = targets[k] - self.ao[k]output_deltas[k] = error * dsigmoid(self.ao[k])# 更新输出层权值for j in range(self.nh):for k in range(self.no):# output_deltas[k] * self.ah[j] 才是 dError/dweight[j][k]change = output_deltas[k] * self.ah[j]#self.wo[j][k] += N * change + M * self.co[j][k] 这里没看懂，稍微修改了一下#self.co[j][k] = changeself.wo[j][k] += N*change# 计算隐藏层 deltashidden_deltas = [0.0] * self.nhfor j in range(self.nh):error = 0.0for k in range(self.no):error += output_deltas[k] * self.wo[j][k]hidden_deltas[j] = error * dsigmoid(self.ah[j])# 更新输入层权值for i in range(self.ni):for j in range(self.nh):change = hidden_deltas[j] * self.ai[i]# print 'activation',self.ai[i],'synapse',i,j,'change',change#self.wi[i][j] += N * change + M * self.ci[i][j]#self.ci[i][j] = changeself.wi[i][j] += N*change# 计算误差平方和# 1/2 是为了好看，**2 是平方error = 0.0for k in range(len(targets)):error = 0.5 * (targets[k] - self.ao[k]) ** 2return errordef weights(self):"""打印权值矩阵"""print 'Input weights:'for i in range(self.ni):print self.wi[i]printprint 'Output weights:'for j in range(self.nh):print self.wo[j]print ''def test(self, patterns):"""测试:param patterns:测试数据"""for p in patterns:inputs = p[0]print 'Inputs:', p[0], '-->', self.runNN(inputs), '\tTarget', p[1]def train(self, patterns, max_iterations=1000, N=0.5, M=0.1):"""训练:param patterns:训练集:param max_iterations:最大迭代次数:param N:本次学习率:param M:上次学习率"""for i in range(max_iterations):for p in patterns:inputs = p[0]targets = p[1]self.runNN(inputs)error = self.backPropagate(targets, N, M)if i % 50 == 0:print 'Combined error', errorself.test(patterns)def main():pat = [[[0, 0], [1]],[[0, 1], [1]],[[1, 0], [1]],[[1, 1], [0]]]myNN = NN(2, 2, 1)myNN.train(pat)if __name__ == "__main__":main()