神经网络训练的简单示例以及代码

介绍

这是一个简单的三层全连接网络，所谓全连接网络，就是每一层的每一个节点均与前一层的每一个节点相连，如下所示：$h_1$h1​、$h_2$h2​均与$i_1$i1​、$i_2$i2​相连 ；$o_1$o1​、$o_2$o2​也均与$h_1$h1​、$h_2$h2​相连

计算步骤

初始状态：

前向计算：

根据初始状态图可得以下公式：

第一层到第二层的节点间计算：
$\left\{ \begin{array}{l} net_{h1}=w1\times i1+w2\times i2+b1\\ net_{h2}=w3\times i1+w4\times i2+b1\\ \end{array} \right. \${neth1​=w1×i1+w2×i2+b1neth2​=w3×i1+w4×i2+b1​ 
使用**函数得到$out_{h1}$outh1​、$out_{h2}$outh2​：
$\left\{ \begin{array}{l} out_{h1}=\frac{1}{1+e^{-net_{h1}}}\\ out_{h2}=\frac{1}{1+e^{-net_{h2}}}\\ \end{array} \right. \${outh1​=1+e−neth1​1​outh2​=1+e−neth2​1​​ 
第二层到第三层的节点间计算：
$\left\{ \begin{array}{l} net_{o1}=w5\times out_{h1}+w6\times out_{h2}+b2\\ net_{o1}=w7\times out_{h1}+w8\times out_{h2}+b2\\ \end{array} \right. \${neto1​=w5×outh1​+w6×outh2​+b2neto1​=w7×outh1​+w8×outh2​+b2​ 
使用**函数得到$out_{o1}$outo1​、$out_{o2}$outo2​：
$\left\{ \begin{array}{l} out_{o1}=\frac{1}{1+e^{-net_{o1}}}\\ out_{o2}=\frac{1}{1+e^{-net_{o2}}}\\ \end{array} \right. \${outo1​=1+e−neto1​1​outo2​=1+e−neto2​1​​ 

根据公式可得以下值：

以 $out_ {h1}$outh1​为例：
$\left\{ \begin{array}{l} net_{h1}=w1\times i1+w2\times i2+b1 =0.15\times 0.05+0.2\times 0.1+0.35 = 0.3775\\ out_{h1}=\frac{1}{1+e^{-0.3775}} = 0.593269992\\ \end{array} \right. \${neth1​=w1×i1+w2×i2+b1=0.15×0.05+0.2×0.1+0.35=0.3775outh1​=1+e−0.37751​=0.593269992​ 
同理可得 $out_ {h2}= 0.596884378$outh2​=0.596884378
以 $out_ {o1}$outo1​为例：
$\left\{ \begin{array}{l} net_{o1}=w5\times out_{h1}+w6\times out_{h2}+b2 =0.4\times 0.593269992+0.45\times 0.596884378+0.6 = 1.105905967\\ out_{h1}=\frac{1}{1+e^{-1.105905967}} = 0.75136507\\ \end{array} \right. \${neto1​=w5×outh1​+w6×outh2​+b2=0.4×0.593269992+0.45×0.596884378+0.6=1.105905967outh1​=1+e−1.1059059671​=0.75136507​ 
同理可得 $out_ {o2}= 0.772928465$outo2​=0.772928465
通过得到的结果可以看到输出的结果和预期输出的结果存在差异，则需要更新参数

反向计算（更新参数）：

更新参数$w_1$w1​、$w_3$w3​、$w_3$w3​、$w_4$w4​

一次反向传播完成，再次正向计算可得到新的实际输出
迭代上述运算过程10000次后，输出为[0.015914607588423745， 0.9840961824613796]，已经
非常接近期望输出

代码

import numpy as np

# i1 = 0.05; i2 = 0.10
# out1 = 0.01;out2 = 0.99
# w1 = 0.15 ; w2 = 0.20; w3 = 0.25; w4 = 0.30; b1 = 0.35
# w5 = 0.4 ; w6 = 0.45;  w7 = 0.5; w8 = 0.55; b2 = 0.6

def cacu():

    i1 = 0.05; i2 = 0.10;
    out1 = 0.01;out2 = 0.99
    w1 = 0.15; w2 = 0.20; w3 = 0.25; w4 = 0.30; b1 = 0.35
    w5 = 0.4; w6 = 0.45; w7 = 0.5; w8 = 0.55; b2 = 0.6
    for i in range(10000):
        net_h1 = w1 * i1 + w2 * i2 + b1
        net_h2 = w3 * i1 + w4 * i2 + b1
        h1 = 1 / ( np.exp(-net_h1)+1 )
        h2 = 1 / ( np.exp(-net_h2)+1 )
        net_o1 = w5 * h1 + w6 * h2 + b2
        net_o2 = w7 * h1 + w8 * h2 + h2
        o1 = 1 / ( np.exp(-net_o1)+1 )
        o2 = 1 / ( np.exp(-net_o2)+1 )
        print("o1:",o1,o2 )
        E_total = (o1-out1)*(o1-out1)*0.5 + (o2-out2)*(o2-out2)*0.5
        print("total:",E_total)
        Etotal_O1 = (out1 - o1) * -1
        Eo1_neto1 = o1 *( 1-o1 )
        neto1_w5 = h1
        change_w5 = Etotal_O1 * Eo1_neto1 * neto1_w5
        w5_new = w5 - change_w5 *0.5
        print("w5: ",w5_new)
        neto1_w6 = h2
        change_w6 = Etotal_O1 * Eo1_neto1 * neto1_w6
        w6_new = w6 - change_w6 * 0.5
        print("w6: ",w6_new)

        Etotal_O2 = (out2 - o2) * -1
        Eo2_neto2 = o2 * (1 - o2)
        neto2_w7 = h1
        change_w7 = Etotal_O2 * Eo2_neto2 * neto2_w7
        w7_new = w7 - change_w7 * 0.5
        print("w7: ",w7_new)
        neto2_w8 = h2
        change_w8 = Etotal_O2 * Eo2_neto2 * neto2_w8
        w8_new = w8 - change_w8 * 0.5
        print("w8: ",w8_new)

        #计算  Etotal_w1
        # 计算 Etotal_outh1
        # 计算 E01_outh1
        E01_outh1 = Etotal_O1 * Eo1_neto1 * w5
        print("E01_outh1 : ", E01_outh1)
        # 计算 E02_outh1
        E02_outh1 = Etotal_O2 * Eo2_neto2 * w7
        print("E02_outh1 : ", E02_outh1)
        Etotal_outh1 = E02_outh1 + E01_outh1
        print("Etotal_outh1 : ", Etotal_outh1)
        # 计算 outh1_neth1
        outh1_neth1  =   h1 * ( 1- h1 )
        print("out1_neth1: ",outh1_neth1)
        # 计算neth1_w1
        neth1_w1 = i1
        Etotal_w1 = Etotal_outh1 * outh1_neth1 * neth1_w1
        print("Etotal_w1:",Etotal_w1)
        w1_new = w1 - 0.5 *  Etotal_w1
        print("w1_new:",w1_new)

        # 计算neth1_w2
        neth1_w2 = i2
        Etotal_w2 = Etotal_outh1 * outh1_neth1 * neth1_w2
        print("Etotal_w1:", Etotal_w2)
        w2_new = w2 - 0.5 * Etotal_w2
        print("w2_new:", w2_new)

        # 计算  Etotal_w3
        # 计算 Etotal_outh2
        # 计算 E01_outh2
        E01_outh2 = Etotal_O1 * Eo1_neto1 * w6
        print("E01_outh1 : ", E01_outh1)
        # 计算 E02_outh2
        E02_outh2 = Etotal_O2 * Eo2_neto2 * w8
        print("E02_outh1 : ", E02_outh1)
        Etotal_outh1 = E02_outh2 + E01_outh2
        print("Etotal_outh1 : ", Etotal_outh1)
        # 计算 outh2_neth2
        outh2_neth2 = h2 * (1 - h2)
        print("out1_neth1: ", outh2_neth2)
        # 计算neth1_w3
        neth1_w3 = i1
        Etotal_w3 = Etotal_outh1 * outh1_neth1 * neth1_w3
        print("Etotal_w3:", Etotal_w3)
        w3_new = w3 - 0.5 * Etotal_w3
        print("w3_new:", w3_new)

        # 计算neth1_w4
        neth1_w4 = i2
        Etotal_w4 = Etotal_outh1 * outh1_neth1 * neth1_w4
        print("Etotal_w4:", Etotal_w4)
        w4_new = w4 - 0.5 * Etotal_w4
        print("w4_new:", w4_new)
        w1 = w1_new; w2 = w2_new; w3 = w3_new; w4 = w4_new;
        w5 = w5_new; w6 = w6_new; w7 = w7_new; w8 = w8_new;

    print("o1: ", o1 )
    print("o2: ", o2)

def main():
    cacu()

if __name__ == "__main__":
    main()

示例输出：