人工智能初级算法——误差反向传播算法(BP)

分手后的思念是犯贱 2023-10-11 13:44 8阅读 0赞

## 误差反向传播算法（BP） ##

*  误差反向传播算法（BP）
    
     *  一、神经网络基本概念
     *  二、神经元模型
        
         *  1. 单个神经元
            
             *  1> 感知器模型
             *  2> 输入与输出
             *  3> 激活函数
         *  2. 单个神经元的学习
            
             *  1> 识别
         *  3. 多层神经元模型
     *  三、误差反向传播算法(BP)
        
         *  1. 算法的推导
            
             *  1> 输出层权重偏导
             *  2> 隐藏层权重偏导
         *  3. 训练
            
             *  1> 调整单个权重 w。我们追求误差减小，故，有以下公式：
             *  2> 随机梯度下降
         *  4. 编码
     *  四、一个学习示例
        
         *  1. 性别预测
     *  五、 算法优化
        
         *  1. 指定网络结构
         *  2. 存储当前神经网络的训练数据

### 一、神经网络基本概念 ###

### 二、神经元模型 ###

**神经元细胞示意图**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70 1]

#### 1. 单个神经元 ####

##### 1> 感知器模型 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70 2]

##### 2> 输入与输出 #####

##### 3> 激活函数 #####

S i g m o i d ( x ) = 1 1 + e − x Sigmoid(x) = \\frac\{1\}\{1+e^\{-x\}\} Sigmoid(x)=1\+e−x1  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70 3]  
 S i g m o i d ′ ( x ) = S i g m o i d ( x ) ∗ ( 1 − S i g m o i d ( x ) ) Sigmoid'(x) = Sigmoid(x) \* (1 - Sigmoid(x)) Sigmoid′(x)=Sigmoid(x)∗(1−Sigmoid(x))

#### 2. 单个神经元的学习 ####

##### 1> 识别 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70 4]

#### 3. 多层神经元模型 ####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70 5]

### 三、误差反向传播算法(BP) ###

#### 1. 算法的推导 ####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3poaXlpa2VqaQ_size_16_color_FFFFFF_t_70 6]

**符号说明：**

*  `X`: 输入向量
 *   Y t r u e Y\_\{true\} Ytrue: 输入向量 `X` 对应的真实结果
 *   Y p r e d Y\_\{pred\} Ypred: 根据输入向量 `X` 预测的结果
 *  `Yi`: 向量的某个分量
 *  `h_in`: 隐藏层输入
 *  `h_out`: 隐藏层输出(经过激活函数处理)
 *  `h_w`: 隐藏层权重矩阵
 *  `output_in`: 输出层输入
 *  `output_out`: 输出层输出(经过激活函数处理)
 *  `output_w`: 输出层权重矩阵

##### 1> 输出层权重偏导 #####

已知对于某个训练样例 d 的误差函数：  
 E r r o r = 1 2 ∗ ∑ i ∈ o u t p u t ( Y i t r u e − Y i p r e d ) 2 Error = \\frac\{1\}\{2\} \* \\sum\_\{i \\in output\}(Yi\_\{true\}-Yi\_\{pred\})^2 Error=21∗i∈output∑(Yitrue−Yipred)2  
可得，总误差函数对输出层权重  w 35 w\_\{35\} w35 的偏导数：  
 ∂ E ∂ w 35 = ∂ E ∂ o u t p u t 5 \_ o u t ∗ ∂ o u t p u t 5 \_ o u t ∂ o u t p u t 5 \_ i n ∗ ∂ o u t p u t 5 \_ i n ∂ w 35 \\frac\{\\partial\{E\}\}\{\\partial\{w\_\{35\}\}\} = \\frac\{\\partial\{E\}\}\{\\partial\{output5\\\_out\}\} \* \\frac\{\\partial\{output5\\\_out\}\}\{\\partial\{output5\\\_in\}\} \* \\frac\{\\partial\{output5\\\_in\}\}\{\\partial\{w\_\{35\}\}\} ∂w35∂E=∂output5\_out∂E∗∂output5\_in∂output5\_out∗∂w35∂output5\_in  
各项展开得：  
 ∂ E ∂ o u t p u t 5 \_ o u t = − ( Y 5 t r u e − o u t p u t 5 \_ o u t ) \\frac\{\\partial\{E\}\}\{\\partial\{output5\\\_out\}\} = -(Y5\_\{true\} - output5\\\_out) ∂output5\_out∂E=−(Y5true−output5\_out)  
 ∂ o u t p u t 5 \_ o u t ∂ o u t p u t 5 \_ i n = S i g m o i d ( o u t p u t 5 \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t 5 \_ i n ) ) \\frac\{\\partial\{output5\\\_out\}\}\{\\partial\{output5\\\_in\}\} = Sigmoid(output5\\\_in)\*(1-Sigmoid(output5\\\_in)) ∂output5\_in∂output5\_out=Sigmoid(output5\_in)∗(1−Sigmoid(output5\_in))  
output5\_in = h3\_out \* w\_\{35\} + h4\_out \* w\_\{45\}  
 ∂ o u t p u t \_ i n ∂ w 35 = h 3 \_ o u t \\frac\{\\partial\{output\\\_in\}\}\{\\partial\{w\_\{35\}\}\} = h3\\\_out ∂w35∂output\_in=h3\_out

**最终偏导为：**  
 ∂ E ∂ w 35 = − ( Y 5 t r u e − o u t p u t 5 \_ o u t ) ∗ S i g m o i d ( o u t p u t 5 \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t 5 \_ i n ) ) ∗ h 3 \_ o u t \\frac\{\\partial\{E\}\}\{\\partial\{w35\}\} = -(Y5\_\{true\} - output5\\\_out) \* Sigmoid(output5\\\_in)\*(1-Sigmoid(output5\\\_in)) \* h3\\\_out ∂w35∂E=−(Y5true−output5\_out)∗Sigmoid(output5\_in)∗(1−Sigmoid(output5\_in))∗h3\_out

##### 2> 隐藏层权重偏导 #####

总体上，隐藏层权重偏导公式如下：  
 ∂ E ∂ w 13 = ∂ E ∂ h 3 \_ o u t ∗ ∂ h 3 \_ o u t ∂ h 3 \_ i n ∗ ∂ h 3 \_ i n ∂ w 13 \\frac\{\\partial\{E\}\}\{\\partial\{w\_\{13\}\}\} = \\frac\{\\partial\{E\}\}\{\\partial\{h3\\\_out\}\} \* \\frac\{\\partial\{h3\\\_out\}\}\{\\partial\{h3\\\_in\}\} \* \\frac\{\\partial\{h3\\\_in\}\}\{\\partial\{w\_\{13\}\}\} ∂w13∂E=∂h3\_out∂E∗∂h3\_in∂h3\_out∗∂w13∂h3\_in  
展开，得：  
 ∂ E ∂ w 13 = ∂ E ∂ h 3 \_ o u t ∗ S i g m o i d ( h 3 \_ i n ) ∗ ( 1 − S i g m o i d ( h 3 \_ i n ) ) ∗ x 1 \\frac\{\\partial\{E\}\}\{\\partial\{w\_\{13\}\}\} = \\frac\{\\partial\{E\}\}\{\\partial\{h3\\\_out\}\} \* Sigmoid(h3\\\_in) \* (1 - Sigmoid(h3\\\_in))\* x\_1 ∂w13∂E=∂h3\_out∂E∗Sigmoid(h3\_in)∗(1−Sigmoid(h3\_in))∗x1  
仔细分析，我们只需对权重  w 13 w\_\{13\} w13 直接影响的节点求导即可：  
 ∂ E ∂ h 3 \_ o u t = ∑ i ∈ D o w n s t r e a m ( h 3 ) ∂ E ∂ o u t p u t i \_ o u t ∗ ∂ o u t p u t i \_ o u t ∂ o u t p u t i \_ i n ∗ ∂ o u t p u t i \_ i n ∂ h 3 \_ o u t \\frac\{\\partial\{E\}\}\{\\partial\{h3\\\_out\}\} = \\sum\_\{i \\in Downstream(h3)\} \\frac\{\\partial\{E\}\}\{\\partial\{outputi\\\_out\}\} \* \\frac\{\\partial\{outputi\\\_out\}\}\{\\partial\{outputi\\\_in\}\} \* \\frac\{\\partial\{outputi\\\_in\}\}\{\\partial\{h3\\\_out\}\} ∂h3\_out∂E=i∈Downstream(h3)∑∂outputi\_out∂E∗∂outputi\_in∂outputi\_out∗∂h3\_out∂outputi\_in  
各项展开得：  
 ∂ E ∂ h 3 \_ o u t = ∑ i ∈ D o w n s t r e a m ( h 3 ) − ( Y i t r u e − o u t p u t i \_ o u t ) ∗ S i g m o i d ( o u t p u t i \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t i \_ i n ) ) ∗ w 3 i \\frac\{\\partial\{E\}\}\{\\partial\{h3\\\_out\}\} = \\sum\_\{i \\in Downstream(h3)\} -(Yi\_\{true\} - outputi\\\_out) \* Sigmoid(outputi\\\_in) \* (1 - Sigmoid(outputi\\\_in)) \* w\_\{3i\} ∂h3\_out∂E=i∈Downstream(h3)∑−(Yitrue−outputi\_out)∗Sigmoid(outputi\_in)∗(1−Sigmoid(outputi\_in))∗w3i

#### 3. 训练 ####

##### 1> 调整单个权重 w。我们追求误差减小，故，有以下公式： #####

w 35 = w 35 − Δ w 35 w\_\{35\} = w\_\{35\} - \\Delta w\_\{35\} w35=w35−Δw35  
定义学习速率： r r r  
 w 35 = w 35 − r ∗ ∂ E ∂ w 35 w\_\{35\} = w\_\{35\} - r \* \\frac\{\\partial\{E\}\}\{\\partial\{w\_\{35\}\}\} w35=w35−r∗∂w35∂E  
输出层权重调整：  
 w 35 = w 35 + r ∗ ( Y 5 t r u e − o u t p u t 5 \_ o u t ) ∗ S i g m o i d ( o u t p u t 5 \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t 5 \_ i n ) ) ∗ h 3 \_ o u t w\_\{35\} = w\_\{35\} + r \* (Y5\_\{true\} - output5\\\_out) \* Sigmoid(output5\\\_in)\*(1-Sigmoid(output5\\\_in)) \* h3\\\_out w35=w35\+r∗(Y5true−output5\_out)∗Sigmoid(output5\_in)∗(1−Sigmoid(output5\_in))∗h3\_out  
 w 45 = w 45 + r ∗ ( Y 5 t r u e − o u t p u t 5 \_ o u t ) ∗ S i g m o i d ( o u t p u t 5 \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t 5 \_ i n ) ) ∗ h 4 \_ o u t w\_\{45\} = w\_\{45\} + r \* (Y5\_\{true\} - output5\\\_out) \* Sigmoid(output5\\\_in)\*(1-Sigmoid(output5\\\_in)) \* h4\\\_out w45=w45\+r∗(Y5true−output5\_out)∗Sigmoid(output5\_in)∗(1−Sigmoid(output5\_in))∗h4\_out  
 w 36 = w 36 + r ∗ ( Y 6 t r u e − o u t p u t 6 \_ o u t ) ∗ S i g m o i d ( o u t p u t 6 \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t 6 \_ i n ) ) ∗ h 3 \_ o u t w\_\{36\} = w\_\{36\} + r \* (Y6\_\{true\} - output6\\\_out) \* Sigmoid(output6\\\_in)\*(1-Sigmoid(output6\\\_in)) \* h3\\\_out w36=w36\+r∗(Y6true−output6\_out)∗Sigmoid(output6\_in)∗(1−Sigmoid(output6\_in))∗h3\_out  
 w 46 = w 46 + r ∗ ( Y 6 t r u e − o u t p u t 6 \_ o u t ) ∗ S i g m o i d ( o u t p u t 6 \_ i n ) ∗ ( 1 − S i g m o i d ( o u t p u t 6 \_ i n ) ) ∗ h 4 \_ o u t w\_\{46\} = w\_\{46\} + r \* (Y6\_\{true\} - output6\\\_out) \* Sigmoid(output6\\\_in)\*(1-Sigmoid(output6\\\_in)) \* h4\\\_out w46=w46\+r∗(Y6true−output6\_out)∗Sigmoid(output6\_in)∗(1−Sigmoid(output6\_in))∗h4\_out  
利用矩阵运算简化  
 o w + = r ∗ ( Y 5 t r u e − Y 5 p r e d Y 6 t r u e − Y 6 p r e d ) ⋅ S i g m o i d ′ ( o u t p u t 5 \_ i n o u t p u t 6 \_ i n ) × ( h 3 \_ o u t h 4 \_ o u t ) ow += r \* \\left( \\begin\{array\}\{cc\} Y5\_\{true\} - Y5\_\{pred\} \\\\ Y6\_\{true\} - Y6\_\{pred\} \\end\{array\} \\right) \\cdot Sigmoid'\\left( \\begin\{array\}\{cc\} output5\\\_\{in\} \\\\ output6\\\_\{in\} \\end\{array\} \\right) \\times \\left( \\begin\{array\}\{c\} h3\\\_\{out\} & h4\\\_\{out\} \\end\{array\} \\right) ow\+=r∗(Y5true−Y5predY6true−Y6pred)⋅Sigmoid′(output5\_inoutput6\_in)×(h3\_outh4\_out)  
继续化简(默认向量为列向量)：  
 o w + = r ∗ ( Y t r u e − Y p r e d ) ⋅ S i g m o i d ′ ( o u t p u t \_ i n ) × h \_ o u t . T ow += r \* (Y\_\{true\} - Y\_\{pred\}) \\cdot Sigmoid'(output\\\_in) \\times h\\\_out.T ow\+=r∗(Ytrue−Ypred)⋅Sigmoid′(output\_in)×h\_out.T  
令：  
 o u t p u t \_ e r r o r s = ( Y t r u e − Y p r e d ) output\\\_errors = (Y\_\{true\} - Y\_\{pred\}) output\_errors=(Ytrue−Ypred)  
则：  
 h w + = r ∗ ( o u t p u t \_ e r r o r s ⋅ S i g m o i d ′ ( o u t p u t \_ i n ) ) × h \_ o u t . T hw += r \* (output\\\_errors \\cdot Sigmoid'(output\\\_in)) \\times h\\\_out.T hw\+=r∗(output\_errors⋅Sigmoid′(output\_in))×h\_out.T

隐藏层权重调整：  
 w 13 = w 13 + r ∗ S i g m o i d ′ ( h 3 \_ i n ) ∗ x 1 ∗ ∑ i ∈ D o w n s t r e a m ( h 3 ) ( Y i t r u e − o u t p u t i \_ o u t ) ∗ S i g m o i d ′ ( o u t p u t i \_ i n ) ∗ w 3 i w\_\{13\} = w\_\{13\} + r \* Sigmoid'(h3\\\_in) \* x\_1 \* \\sum\_\{i \\in Downstream(h3)\} (Yi\_\{true\} - outputi\\\_out) \* Sigmoid'(outputi\\\_in) \* w\_\{3i\} w13=w13\+r∗Sigmoid′(h3\_in)∗x1∗i∈Downstream(h3)∑(Yitrue−outputi\_out)∗Sigmoid′(outputi\_in)∗w3i

w 23 = w 23 + r ∗ S i g m o i d ′ ( h 3 \_ i n ) ∗ x 2 ∗ ∑ i ∈ D o w n s t r e a m ( h 3 ) ( Y i t r u e − o u t p u t i \_ o u t ) ∗ S i g m o i d ′ ( o u t p u t i \_ i n ) ∗ w 3 i w\_\{23\} = w\_\{23\} + r \* Sigmoid'(h3\\\_in) \* x\_2 \* \\sum\_\{i \\in Downstream(h3)\} (Yi\_\{true\} - outputi\\\_out) \* Sigmoid'(outputi\\\_in) \* w\_\{3i\} w23=w23\+r∗Sigmoid′(h3\_in)∗x2∗i∈Downstream(h3)∑(Yitrue−outputi\_out)∗Sigmoid′(outputi\_in)∗w3i

w 14 = w 14 + r ∗ S i g m o i d ′ ( h 4 \_ i n ) ∗ x 1 ∗ ∑ i ∈ D o w n s t r e a m ( h 4 ) ( Y i t r u e − o u t p u t i \_ o u t ) ∗ S i g m o i d ′ ( o u t p u t i \_ i n ) ∗ w 4 i w\_\{14\} = w\_\{14\} + r \* Sigmoid'(h4\\\_in) \* x\_1 \* \\sum\_\{i \\in Downstream(h4)\} (Yi\_\{true\} - outputi\\\_out) \* Sigmoid'(outputi\\\_in) \* w\_\{4i\} w14=w14\+r∗Sigmoid′(h4\_in)∗x1∗i∈Downstream(h4)∑(Yitrue−outputi\_out)∗Sigmoid′(outputi\_in)∗w4i

w 24 = w 24 + r ∗ S i g m o i d ′ ( h 4 \_ i n ) ∗ x 2 ∗ ∑ i ∈ D o w n s t r e a m ( h 4 ) ( Y i t r u e − o u t p u t i \_ o u t ) ∗ S i g m o i d ′ ( o u t p u t i \_ i n ) ∗ w 4 i w\_\{24\} = w\_\{24\} + r \* Sigmoid'(h4\\\_in) \* x\_2 \* \\sum\_\{i \\in Downstream(h4)\} (Yi\_\{true\} - outputi\\\_out) \* Sigmoid'(outputi\\\_in) \* w\_\{4i\} w24=w24\+r∗Sigmoid′(h4\_in)∗x2∗i∈Downstream(h4)∑(Yitrue−outputi\_out)∗Sigmoid′(outputi\_in)∗w4i  
利用矩阵简化：  
 h w i j + = r ∗ S i g m o i d ′ ( h j \_ i n h j \_ i n ) ⋅ ( x 1 x 2 ) × ∑ k ∈ D o w n s t r e a m ( h j ) ( Y k t r u e − Y k p r e d ) ⋅ S i g m o i d ′ ( o u t p u t k \_ i n ) × w j k hw\_\{ij\} += r \* Sigmoid'\\left(\\begin\{array\}\{c\} hj\\\_in \\\\ hj\\\_in \\end\{array\}\\right) \\cdot \\left(\\begin\{array\}\{c\} x\_1 \\\\ x\_2 \\end\{array\}\\right) \\times \\sum\_\{k \\in Downstream(h\_j)\} ( Yk\_\{true\} - Yk\_\{pred\} ) \\cdot Sigmoid'( outputk\\\_\{in\} ) \\times w\_\{jk\} hwij\+=r∗Sigmoid′(hj\_inhj\_in)⋅(x1x2)×k∈Downstream(hj)∑(Yktrue−Ykpred)⋅Sigmoid′(outputk\_in)×wjk  
令(默认向量为列向量):  
 h i d d e n \_ e r r o r s = o u t p u t \_ w . T × ( ( Y k t r u e − Y k p r e d ) ⋅ S i g m o i d ′ ( o u t p u t \_ i n ) ) hidden\\\_errors = output\\\_w.T \\times ((Yk\_\{true\} - Yk\_\{pred\} ) \\cdot Sigmoid'( output\\\_\{in\}) ) hidden\_errors=output\_w.T×((Yktrue−Ykpred)⋅Sigmoid′(output\_in))  
带入得(默认向量为列向量)：  
 h w + = r ∗ ( h i d d e n \_ e r r o r s ⋅ S i g m o i d ′ ( h \_ i n ) ) × X . T hw += r \* (hidden\\\_errors \\cdot Sigmoid'(h\\\_in)) \\times X.T hw\+=r∗(hidden\_errors⋅Sigmoid′(h\_in))×X.T

##### 2> 随机梯度下降 #####

1.  对训练样本中的每一个数据，进行一次训练
2.  对整个样本进行多次(例如：100次)训练

#### 4. 编码 ####

#_author :NineSun
    #data: 2019-10-12
    
    import  numpy as np
    
    def activate(x):
        return 1/(1+np.exp(-x))
    def activate_de(x):
        return activate(x)*(1-activate(x))
    class NeutualNetWork:
        # 构造方法，用于构造神经网络
        def __init__(self):
            # 隐藏层权重矩阵
            self.hw=np.random.normal(np.zeros((2,2)))
            # 输出层权重矩阵
            self.ow=np.random.normal(np.zeros((2,2)))
            # 学习速率
            self.r=0.1
            # 训练遍数
            self.epoch=100
    
        # 预测函数，根据输入数据预测结果
        # x:输入向量
        def prdict(self,x):
            # 将输入向量转化为列向量
            x=np.array(x,ndmin=2).T
            # 计算隐藏层输入
            h_in=np.dot(self.hw,x)
            # 计算隐藏层输出
            h_out = activate(h_in)
            # 计算输出层输入 output_in
            output_in=np.dot(self.ow,h_out)
            # 计算输出层输出
            output_out=activate(output_in)
            return output_out
    
        # 训练神经网络
        # x_data:输入数据集
        # ytrue_data:输出数据集
        def train(self,x_data,ytrue_data):
            # 训练epoch遍
            for i in range(self.epoch):
                # 针对每一个样本训练一遍
                for data in zip(x_data,ytrue_data):
                    self.train_once(data[0],data[1])
    
    
        # 针对一个样本（x,ytrue）进行训练
        def train_once(self,x,ytrue):
            '''
            :param x: 输入行向量
            :param ytrue: 输出行向量
            :return:
            '''
            # 将输入输出向量转化为列向量
            x = np.array(x, ndmin=2).T
            ytrue = np.array(ytrue, ndmin=2).T
    
    
            # 根据输入计算 Ypred,output_out
            h_in=np.dot(self.hw,x)
            h_out=activate(h_in)
            output_in=np.dot(self.ow,h_out)
            output_out=activate(output_in)
    
            # 计算输出误差
            output_errors=ytrue-output_out
            cal_tmp=output_errors*activate_de(output_in)
            # 更新输出层权重
            self.ow+=self.r*np.dot(cal_tmp,h_out.T)
    
            # 计算隐藏层误差
            hidden_errors=np.dot(self.ow.T,cal_tmp)
    
            # 更新隐藏层权重
            self.hw+=self.r*np.dot(hidden_errors*activate_de(h_in),x.T)
    
        # 1.准备数据
    in_data = [
            [165, 55],
            [160, 53],
            [175, 55],
            [163, 55],
            [173, 49],
            [163, 56],
            [180, 77],
            [155, 54],
            [176, 79],
            [161, 49],
            [180, 60],
            [168, 57],
            [172, 69],
            [166, 50],
            [172, 90],
            [163, 51],
            [175, 70],
            [164, 53],
            [160, 45],
            [160, 56],
            [182, 70],
            [180, 74],
            [185, 73],
            [165, 55],
            [185, 75]
        ]
    out_data = [
                [1,0],
                [0,1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1],
                [1, 0],
                [0, 1]
        ]
        # 测试数据
    in_test=[
            [165, 55],
            [160, 53],
            [175, 55],
            [163, 55],
            [173, 49]
        ]
    out_test=[
            [0,1],
            [0, 1],
            [1, 0],
            [0, 1],
            [1, 0],
        ]
    
    nn = NeutualNetWork()
        # 3.训练
    nn.train(in_data,out_data)
        # 4.预测
    for test in zip(in_test,out_test):
            ypred=nn.prdict(test[0])
            print("in:",test[0],"yture:",test[1])
            print("predit:",ypred)

这串代码可以直接运行

### 五、 算法优化 ###

#### 1. 指定网络结构 ####

可随意指定：输入层，隐藏层，输出层的神经元个数

#### 2. 存储当前神经网络的训练数据 ####

将神经网络结构，包括：学习速率，训练次数和所有权重矩阵存成文件，以便以后可以不必训练，加载数据后可直接进行预测。

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