深度學習是監督學習的一個分支。
簡單來說就是當線性模型無法解決問題時，引入的一種方法。
它綜合多種線行模型來從x空間——>學習到h空間，h空間為可用線行模型解決的空間

深度前饋網路（deep feedforward network)又叫多層感知機，是深度學習最典型的模型。

引入一個例子：
XOR異或問題

當 x1不變時，x2遞增，輸出結果的趨勢相反，即出現了遞增，也出現了遞減。這不是線性變化的，所以無法用線性模型來分類。

上圖兩個輸出空間，黑；綠
引出神經網路核心思想，多線性模型一起工作。

接下來使用python製作自己的神經網路：

構架：
1：初始化函式：設定輸入層節點，隱藏層節點和輸出層節點（上圖所示）
2：訓練：學習給定訓練集樣本後，優化權重
3：查詢：給定輸入，從輸出節點得到輸出結果

import numpy
# scipy.special for the sigmoid function expit()
import scipy.special

# neural network class definition
class neuralNetwork:
    
    
    # initialise the neural network
    def __init__(self, inputnodes, hiddennodes, outputnodes, learningrate):
        # set number of nodes in each input, hidden, output layer
 

        self.inodes = inputnodes
        self.hnodes = hiddennodes
        self.onodes = outputnodes
        
        # link weight matrices, wih and who
        # weights inside the arrays are w_i_j, where link is from node i to node j in the next layer
        # w11 w21
        # w12 w22 etc 
        self.
 
wih = numpy.random.normal(0.0, pow(self.inodes, -0.5), (self.hnodes, self.inodes))
        self.who = numpy.random.normal(0.0, pow(self.hnodes, -0.5), (self.onodes, self.hnodes))

        # learning rate
        self.lr = learningrate
        
        # activation function is the sigmoid function
        self.activation_function = lambda x: scipy.special.expit(x)
        
        pass

    
    # train the neural network
    def train(self, inputs_list, targets_list):
        # convert inputs list to 2d array
        inputs = numpy.array(inputs_list, ndmin=2).T
        targets = numpy.array(targets_list, ndmin=2).T
        
        # calculate signals into hidden layer
        hidden_inputs = numpy.dot(self.wih, inputs)
        # calculate the signals emerging from hidden layer
        hidden_outputs = self.activation_function(hidden_inputs)
        
        # calculate signals into final output layer
        final_inputs = numpy.dot(self.who, hidden_outputs)
        # calculate the signals emerging from final output layer
        final_outputs = self.activation_function(final_inputs)
        
        # output layer error is the (target - actual)
        output_errors = targets - final_outputs
        # hidden layer error is the output_errors, split by weights, recombined at hidden nodes
        hidden_errors = numpy.dot(self.who.T, output_errors) 
        
        # update the weights for the links between the hidden and output layers
        self.who += self.lr * numpy.dot((output_errors * final_outputs * (1.0 - final_outputs)), numpy.transpose(hidden_outputs))
        
        # update the weights for the links between the input and hidden layers
        self.wih += self.lr * numpy.dot((hidden_errors * hidden_outputs * (1.0 - hidden_outputs)), numpy.transpose(inputs))
        
        pass

    
    # query the neural network
    def query(self, inputs_list):
        # convert inputs list to 2d array
        inputs = numpy.array(inputs_list, ndmin=2).T
        
        # calculate signals into hidden layer
        hidden_inputs = numpy.dot(self.wih, inputs)
        # calculate the signals emerging from hidden layer
        hidden_outputs = self.activation_function(hidden_inputs)
        
        # calculate signals into final output layer
        final_inputs = numpy.dot(self.who, hidden_outputs)
        # calculate the signals emerging from final output layer
        final_outputs = self.activation_function(final_inputs)
        
        return final_outputs


if __name__ =="__main__":
    # number of input, hidden and output nodes
    input_nodes =2  
    hidden_nodes = 5
    output_nodes = 1

    # learning rate is 0.3
    learning_rate =  00.3
    n = neuralNetwork(input_nodes,hidden_nodes,output_nodes, learning_rate)
    print(n.query([1,1]))