反向传播是利用函数的链式求导来进行推导的，目的是通过不断调整权重和偏置来不断减小误差，最终得到误差最小的神经网络。
下面是其python的实现，只是一个3层的全连接网络。

import numpy as np

def nonlin(x,deriv = False):
    if(deriv==True):
        return x*(1-x)
    return 1/(1+np.exp(-x))

input_layer = np.array([[0.35],[0.9]])
print(input_layer)
output = np.array([[0.5]])
print(output)

w0 = np.array([[0.1,0.8],[0.4,0.6]])
print(w0)
w1 = np.array([[0.3,0.9]])
print(w1)

for j in range(100):
    l0 = input_layer
    l1 = nonlin(np.dot(w0,l0))
    l2 = nonlin(np.dot(w1,l1))
    l2_error = output - l2
    error = 1/2.0*(output-l2)**2
    print("error")
    print(error)
    l2_delta = l2_error * nonlin(l2,deriv = True)
    l1_error = l2_delta * w1
    l1_delta = l1_error * nonlin(l1,deriv = True)
    w1 += l2_delta * l1.T
    w0 += l0.T.dot(l1_delta)
    print('w0')
    print(w0)
    print('w1')
    print(w1)

这是算法使用的全连接网络

最终输出为：

error
[[8.22616248e-06]]
w0
[[0.09815656 0.64379488]
 [0.39815656 0.44379488]]
w1
[[-0.29141427  0.32375987]]
error
[[7.40440534e-06]]
w0
[[0.09823763 0.64370482]
 [0.39823763 0.44370482]]
w1
[[-0.29203841  0.32315234]]
error
[[6.66477319e-06]]
w0
[[0.09831471 0.64361953]
 [0.39831471 0.44361953]]
w1
[[-0.29263055  0.32257597]]
error
[[5.99905368e-06]]
w0
[[0.09838799 0.64353875]
 [0.39838799 0.44353875]]
w1
[[-0.29319234  0.32202914]]
error
[[5.39985637e-06]]
w0
[[0.09845764 0.64346224]
 [0.39845764 0.44346224]]
w1
[[-0.29372533  0.32151034]]
error
[[4.86053034e-06]]
w0
[[0.09852385 0.64338977]
 [0.39852385 0.44338977]]
w1
[[-0.29423099  0.32101815]]
error
[[4.37509018e-06]]
w0
[[0.09858678 0.64332111]
 [0.39858678 0.44332111]]
w1
[[-0.29471073  0.32055118]]
error
[[3.93814926e-06]]
w0
[[0.09864658 0.64325607]
 [0.39864658 0.44325607]]
w1
[[-0.29516588  0.32010815]]
error
[[3.5448598e-06]]
w0
[[0.0987034  0.64319445]
 [0.3987034  0.44319445]]
w1
[[-0.29559771  0.31968782]]
error
[[3.19085892e-06]]
w0
[[0.09875739 0.64313606]
 [0.39875739 0.44313606]]
w1
[[-0.2960074   0.31928904]]
error
[[2.87222006e-06]]
w0
[[0.09880868 0.64308073]
 [0.39880868 0.44308073]]
w1
[[-0.29639609  0.3189107 ]]
error
[[2.58540935e-06]]
w0
[[0.09885741 0.6430283 ]
 [0.39885741 0.4430283 ]]
w1
[[-0.29676486  0.31855175]]
error
[[2.32724628e-06]]
w0
[[0.0989037  0.64297861]
 [0.3989037  0.44297861]]
w1
[[-0.29711474  0.31821119]]
error
[[2.09486834e-06]]
w0
[[0.09894768 0.64293151]
 [0.39894768 0.44293151]]
w1
[[-0.29744668  0.31788809]]
error
[[1.88569918e-06]]
w0
[[0.09898944 0.64288688]
 [0.39898944 0.44288688]]
w1
[[-0.29776161  0.31758154]]
error
[[1.69742002e-06]]
w0
[[0.09902911 0.64284457]
 [0.39902911 0.44284457]]
w1
[[-0.29806041  0.31729071]]
error
[[1.52794387e-06]]
w0
[[0.09906678 0.64280446]
 [0.39906678 0.44280446]]
w1
[[-0.2983439   0.31701477]]
error
[[1.37539232e-06]]
w0
[[0.09910256 0.64276645]
 [0.39910256 0.44276645]]
w1
[[-0.29861286  0.31675297]]
error
[[1.23807471e-06]]
w0
[[0.09913654 0.64273041]
 [0.39913654 0.44273041]]
w1
[[-0.29886803  0.31650459]]
error
[[1.11446931e-06]]
w0
[[0.0991688  0.64269624]
 [0.3991688  0.44269624]]
w1
[[-0.29911014  0.31626894]]
error
[[1.0032065e-06]]
w0
[[0.09919944 0.64266385]
 [0.39919944 0.44266385]]
w1
[[-0.29933984  0.31604536]]
error
[[9.03053486e-07]]
w0
[[0.09922853 0.64263314]
 [0.39922853 0.44263314]]
w1
[[-0.29955777  0.31583323]]
error
[[8.12900662e-07]]
w0
[[0.09925614 0.64260402]
 [0.39925614 0.44260402]]
w1
[[-0.29976454  0.31563197]]
error
[[7.31749283e-07]]
w0
[[0.09928237 0.64257641]
 [0.39928237 0.44257641]]
w1
[[-0.29996071  0.31544102]]
error
[[6.58700387e-07]]
w0
[[0.09930726 0.64255023]
 [0.39930726 0.44255023]]
w1
[[-0.30014683  0.31525986]]
error
[[5.92944818e-07]]
w0
[[0.0993309 0.6425254]
 [0.3993309 0.4425254]]
w1
[[-0.30032342  0.31508797]]

这些仅为部分输出，这便是bp算法的python实现。