前言

反向传播（英语：Backpropagation，缩写为BP）是“误差反向传播”的简称，是一种与最优化方法（如梯度下降法）结合使用的，用来训练人工神经网络的常见方法。 该方法对网络中所有权重计算损失函数的梯度。 这个梯度会反馈给最优化方法，用来更新权值以最小化损失函数。

输入值：x1, x2 = 0.5，0.3
输出值：y1, y2 =0.23， -0.07
激活函数：sigmoid
损失函数：MSE
初始权值：0.2 -0.4 0.5 0.6 0.1 -0.5 -0.3 0.8
目标：通过反向传播优化权值

一、反向传播1轮，检验PPT数值

=正向计算：h1, h2, o1 ,o2=
0.56 0.5 0.48 0.53
=损失函数：均方误差=
0.21
=反向传播：误差传给每个权值=
0.01 0.01 0.01 0.01 0.03 0.08 0.03 0.07
=更新前的权值=
0.2 -0.4 0.5 0.6 0.1 -0.5 -0.3 0.8
=更新后的权值=
0.19 -0.41 0.49 0.59 0.07 -0.58 -0.33 0.73

代码如下：

import numpy as np


def sigmoid(z):
    a = 1 / (1 + np.exp(-z))
    return a


if __name__ == "__main__":
    w1 = 0.2
    w2 = -0.4
    w3 = 0.5
    w4 = 0.6
    w5 = 0.1
    w6 = -0.5
    w7 = -0.3
    w8 = 0.8

    x1 = 0.5
    x2 = 0.3

    y1 = 0.23
    y2 = -0.07

    print("=====输入值：x1, x2；真实输出值：y1, y2=====")
    print(x1, x2, y1, y2)

    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    print("=====正向计算：h1, h2, o1 ,o2=====")
    print(round(out_h1, 2), round(out_h2, 2), round(out_o1, 2), round(out_o2, 2))

    error = (1 / 2) * (out_o1 - y1)**2 + (1 / 2) * (out_o2 - y2)**2

    print("=====损失函数：均方误差=====")
    print(round(error, 2))

    # 反向传播
    d_o1 = out_o1 - y1
    d_o2 = out_o2 - y2
    # print(round(d_o1, 2), round(d_o2, 2))

    d_w5 = d_o1 * out_o1 * (1 - out_o1) * out_h1
    d_w7 = d_o1 * out_o1 * (1 - out_o1) * out_h2
    # print(round(d_w5, 2), round(d_w7, 2))
    d_w6 = d_o2 * out_o2 * (1 - out_o2) * out_h1
    d_w8 = d_o2 * out_o2 * (1 - out_o2) * out_h2
    # print(round(d_w6, 2), round(d_w8, 2))

    d_w1 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x1
    d_w3 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x2
    # print(round(d_w1, 2), round(d_w3, 2))

    d_w2 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x1
    d_w4 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x2
    # print(round(d_w2, 2), round(d_w4, 2))
    print("=====反向传播：误差传给每个权值=====")
    print(round(d_w1, 2), round(d_w2, 2), round(d_w3, 2), round(d_w4, 2), round(d_w5, 2), round(d_w6, 2), round(d_w7, 2),
          round(d_w8, 2))

    print("=====更新前的权值=====")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    w1 = w1 - d_w1
    w2 = w2 - d_w2
    w3 = w3 - d_w3
    w4 = w4 - d_w4
    w5 = w5 - d_w5
    w6 = w6 - d_w6
    w7 = w7 - d_w7
    w8 = w8 - d_w8

    print("=====更新后的权值=====")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

输出结果：

二、增加到5轮，测试收敛

=第6轮=
正向计算：h1, h2, o1 ,o2
0.55 0.48 0.44 0.43
损失函数：均方误差
0.15

代码如下：

import numpy as np


def sigmoid(z):
    a = 1 / (1 + np.exp(-z))
    return a


def forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8):
    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    print("正向计算：h1, h2, o1 ,o2")
    print(round(out_h1, 2), round(out_h2, 2), round(out_o1, 2), round(out_o2, 2))

    error = (1 / 2) * (out_o1 - y1) ** 2 + (1 / 2) * (out_o2 - y2) ** 2

    print("损失函数：均方误差")
    print(round(error, 2))

    return out_o1, out_o2, out_h1, out_h2


def back_propagate(out_o1, out_o2, out_h1, out_h2):
    # 反向传播
    d_o1 = out_o1 - y1
    d_o2 = out_o2 - y2
    # print(round(d_o1, 2), round(d_o2, 2))

    d_w5 = d_o1 * out_o1 * (1 - out_o1) * out_h1
    d_w7 = d_o1 * out_o1 * (1 - out_o1) * out_h2
    # print(round(d_w5, 2), round(d_w7, 2))
    d_w6 = d_o2 * out_o2 * (1 - out_o2) * out_h1
    d_w8 = d_o2 * out_o2 * (1 - out_o2) * out_h2
    # print(round(d_w6, 2), round(d_w8, 2))

    d_w1 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x1
    d_w3 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x2
    # print(round(d_w1, 2), round(d_w3, 2))

    d_w2 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x1
    d_w4 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x2
    # print(round(d_w2, 2), round(d_w4, 2))
    print("反向传播：误差传给每个权值")
    print(round(d_w1, 2), round(d_w2, 2), round(d_w3, 2), round(d_w4, 2), round(d_w5, 2), round(d_w6, 2),
          round(d_w7, 2), round(d_w8, 2))

    return d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8


if __name__ == "__main__":
    w1 = 0.2
    w2 = -0.4
    w3 = 0.5
    w4 = 0.6
    w5 = 0.1
    w6 = -0.5
    w7 = -0.3
    w8 = 0.8
    x1 = 0.5
    x2 = 0.3
    y1 = 0.23
    y2 = -0.07
    print("=====输入值：x1, x2；真实输出值：y1, y2=====")
    print(x1, x2, y1, y2)
    print("=====更新前的权值=====")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)

    # 步长
    step = 1

    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8

    print("第1轮更新后的权值")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    print("=====第2轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8

    print("=====第3轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8

    print("=====第4轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8

    print("=====第5轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8

    print("=====第6轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    print("更新后的权值")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

输出结果：

三、改变步长（1变为50），看收敛速度

=第6轮=
正向计算：o1 ,o2
0.23 0.03
损失函数：均方误差
0.01

代码如下：

import numpy as np


def sigmoid(z):
    a = 1 / (1 + np.exp(-z))
    return a


def forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8):
    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    print("正向计算：o1 ,o2")
    print(round(out_o1, 2), round(out_o2, 2))

    error = (1 / 2) * (out_o1 - y1) ** 2 + (1 / 2) * (out_o2 - y2) ** 2

    print("损失函数：均方误差")
    print(round(error, 2))

    return out_o1, out_o2, out_h1, out_h2


def back_propagate(out_o1, out_o2, out_h1, out_h2):
    # 反向传播
    d_o1 = out_o1 - y1
    d_o2 = out_o2 - y2
    # print(round(d_o1, 2), round(d_o2, 2))

    d_w5 = d_o1 * out_o1 * (1 - out_o1) * out_h1
    d_w7 = d_o1 * out_o1 * (1 - out_o1) * out_h2
    # print(round(d_w5, 2), round(d_w7, 2))
    d_w6 = d_o2 * out_o2 * (1 - out_o2) * out_h1
    d_w8 = d_o2 * out_o2 * (1 - out_o2) * out_h2
    # print(round(d_w6, 2), round(d_w8, 2))

    d_w1 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x1
    d_w3 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x2
    # print(round(d_w1, 2), round(d_w3, 2))

    d_w2 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x1
    d_w4 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x2
    # print(round(d_w2, 2), round(d_w4, 2))
    print("反向传播：误差传给每个权值")
    print(round(d_w1, 2), round(d_w2, 2), round(d_w3, 2), round(d_w4, 2), round(d_w5, 2), round(d_w6, 2),
          round(d_w7, 2), round(d_w8, 2))

    return d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8


def update_w(w1, w2, w3, w4, w5, w6, w7, w8):
    # 步长
    step = 50
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8
    return w1, w2, w3, w4, w5, w6, w7, w8


if __name__ == "__main__":
    w1 = 0.2
    w2 = -0.4
    w3 = 0.5
    w4 = 0.6
    w5 = 0.1
    w6 = -0.5
    w7 = -0.3
    w8 = 0.8
    x1 = 0.5
    x2 = 0.3
    y1 = 0.23
    y2 = -0.07
    print("=====输入值：x1, x2；真实输出值：y1, y2=====")
    print(x1, x2, y1, y2)
    print("=====更新前的权值=====")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("第1轮更新后的权值")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    print("=====第2轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("=====第3轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("=====第4轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("=====第5轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
    w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("=====第6轮=====")
    out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
    print("更新后的权值")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

输出结果：

四、扩展到N轮，步长=5，训练N=1000次，查看效果

=第999轮=
正向计算：o1 ,o2
0.23038 0.00954
损失函数：均方误差
0.00316

代码如下：

import numpy as np


def sigmoid(z):
    a = 1 / (1 + np.exp(-z))
    return a


def forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8):
    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    print("正向计算：o1 ,o2")
    print(round(out_o1, 5), round(out_o2, 5))

    error = (1 / 2) * (out_o1 - y1) ** 2 + (1 / 2) * (out_o2 - y2) ** 2

    print("损失函数：均方误差")
    print(round(error, 5))

    return out_o1, out_o2, out_h1, out_h2


def back_propagate(out_o1, out_o2, out_h1, out_h2):
    # 反向传播
    d_o1 = out_o1 - y1
    d_o2 = out_o2 - y2
    # print(round(d_o1, 2), round(d_o2, 2))

    d_w5 = d_o1 * out_o1 * (1 - out_o1) * out_h1
    d_w7 = d_o1 * out_o1 * (1 - out_o1) * out_h2
    # print(round(d_w5, 2), round(d_w7, 2))
    d_w6 = d_o2 * out_o2 * (1 - out_o2) * out_h1
    d_w8 = d_o2 * out_o2 * (1 - out_o2) * out_h2
    # print(round(d_w6, 2), round(d_w8, 2))

    d_w1 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x1
    d_w3 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x2
    # print(round(d_w1, 2), round(d_w3, 2))

    d_w2 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x1
    d_w4 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x2
    # print(round(d_w2, 2), round(d_w4, 2))
    print("反向传播：误差传给每个权值")
    print(round(d_w1, 5), round(d_w2, 5), round(d_w3, 5), round(d_w4, 5), round(d_w5, 5), round(d_w6, 5),
          round(d_w7, 5), round(d_w8, 5))

    return d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8


def update_w(w1, w2, w3, w4, w5, w6, w7, w8):
    # 步长
    step = 5
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8
    return w1, w2, w3, w4, w5, w6, w7, w8


if __name__ == "__main__":
    w1, w2, w3, w4, w5, w6, w7, w8 = 0.2, -0.4, 0.5, 0.6, 0.1, -0.5, -0.3, 0.8
    x1, x2 = 0.5, 0.3
    y1, y2 = 0.23, -0.07
    print("=====输入值：x1, x2；真实输出值：y1, y2=====")
    print(x1, x2, y1, y2)
    print("=====更新前的权值=====")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    for i in range(1000):
        print("=====第" + str(i) + "轮=====")
        out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
        d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
        w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("更新后的权值")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

输出结果：

五、代码优化

修改输出值y2为正，收敛效果很好。
原因是：sigmoid，输出值应在（0,1）区间，所以最开始的假设 y2=-0.07，在这个模型里，无法很好的拟合。

优化后的代码：

import numpy as np
import matplotlib.pyplot as plt


def sigmoid(z):
    a = 1 / (1 + np.exp(-z))
    return a


def forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8): # 正向传播
    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    error = (1 / 2) * (out_o1 - y1) ** 2 + (1 / 2) * (out_o2 - y2) ** 2

    return out_o1, out_o2, out_h1, out_h2, error


def back_propagate(out_o1, out_o2, out_h1, out_h2):    # 反向传播
    d_o1 = out_o1 - y1
    d_o2 = out_o2 - y2

    d_w5 = d_o1 * out_o1 * (1 - out_o1) * out_h1
    d_w7 = d_o1 * out_o1 * (1 - out_o1) * out_h2
    d_w6 = d_o2 * out_o2 * (1 - out_o2) * out_h1
    d_w8 = d_o2 * out_o2 * (1 - out_o2) * out_h2

    d_w1 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x1
    d_w3 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x2
    d_w2 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x1
    d_w4 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x2

    return d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8


def update_w(step,w1, w2, w3, w4, w5, w6, w7, w8):    #梯度下降，更新权值
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8
    return w1, w2, w3, w4, w5, w6, w7, w8


if __name__ == "__main__":
    w1, w2, w3, w4, w5, w6, w7, w8 = 0.2, -0.4, 0.5, 0.6, 0.1, -0.5, -0.3, 0.8 # 可以给随机值，为配合PPT，给的指定值
    x1, x2 = 0.5, 0.3   # 输入值
    y1, y2 = 0.23, -0.07 # 正数可以准确收敛；负数不行。why? 因为用sigmoid输出，y1, y2 在 （0,1）范围内。
    N = 10             # 迭代次数
    step = 10           # 步长

    print("输入值：x1, x2；",x1, x2, "输出值：y1, y2:", y1, y2)
    eli = []
    lli = []
    for i in range(N):
        print("=====第" + str(i) + "轮=====")
        # 正向传播
        out_o1, out_o2, out_h1, out_h2, error = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
        print("正向传播:", round(out_o1, 5), round(out_o2, 5))
        print("损失函数：", round(error, 2))
        # 反向传播
        d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
        # 梯度下降，更新权值
        w1, w2, w3, w4, w5, w6, w7, w8 = update_w(step,w1, w2, w3, w4, w5, w6, w7, w8)
        eli.append(i)
        lli.append(error)


    plt.plot(eli, lli)
    plt.ylabel('Loss')
    plt.xlabel('w')
    plt.show()

输出结果：

总结

输出误差（某种形式）－>隐层（逐层）－>输入层 其主要目的是通过将输出误差反传，将误差分摊给各层所有单元，从而获得各层单元的误差信号，进而修正各单元的权值（其过程，是一个权值调整的过程）。
注2：权值调整的过程，也就是网络的学习训练过程（学习也就是这么的由来，权值调整）。
1）初始化
2）输入训练样本对，计算各层输出
3）计算网络输出误差
4）计算各层误差信号
5）调整各层权值
6）检查网络总误差是否达到精度要求
满足，则训练结束；不满足，则返回步骤2。

参考博客

【人工智能导论：模型与算法】MOOC 8.3 误差后向传播(BP) 例题 编程验证
误差反向传播算法