本文完全利用numpy實現一個簡單的BP神經網絡，由于是做regression而不是classification，因此在這里輸出層選取的激勵函數就是f(x)=x。BP神經網絡的具體原理此處不再介紹。

import numpy as np  class NeuralNetwork(object):   def __init__(self, input_nodes, hidden_nodes, output_nodes, learning_rate):     # Set number of nodes in input, hidden and output layers.設定輸入層、隱藏層和輸出層的node數目     self.input_nodes = input_nodes     self.hidden_nodes = hidden_nodes     self.output_nodes = output_nodes      # Initialize weights，初始化權重和學習速率     self.weights_input_to_hidden = np.random.normal(0.0, self.hidden_nodes**-0.5,                      ( self.hidden_nodes, self.input_nodes))      self.weights_hidden_to_output = np.random.normal(0.0, self.output_nodes**-0.5,                      (self.output_nodes, self.hidden_nodes))     self.lr = learning_rate          # 隱藏層的激勵函數為sigmoid函數，Activation function is the sigmoid function     self.activation_function = (lambda x: 1/(1 + np.exp(-x)))      def train(self, inputs_list, targets_list):     # Convert inputs list to 2d array     inputs = np.array(inputs_list, ndmin=2).T  # 輸入向量的shape為 [feature_diemension, 1]     targets = np.array(targets_list, ndmin=2).T       # 向前傳播，Forward pass     # TODO: Hidden layer     hidden_inputs = np.dot(self.weights_input_to_hidden, inputs) # signals into hidden layer     hidden_outputs = self.activation_function(hidden_inputs) # signals from hidden layer           # 輸出層，輸出層的激勵函數就是 y = x     final_inputs = np.dot(self.weights_hidden_to_output, hidden_outputs) # signals into final output layer     final_outputs = final_inputs # signals from final output layer          ### 反向傳播 Backward pass，使用梯度下降對權重進行更新 ###          # 輸出誤差     # Output layer error is the difference between desired target and actual output.     output_errors = (targets_list-final_outputs)      # 反向傳播誤差 Backpropagated error     # errors propagated to the hidden layer     hidden_errors = np.dot(output_errors, self.weights_hidden_to_output)*(hidden_outputs*(1-hidden_outputs)).T      # 更新權重 Update the weights     # 更新隱藏層與輸出層之間的權重 update hidden-to-output weights with gradient descent step     self.weights_hidden_to_output += output_errors * hidden_outputs.T * self.lr     # 更新輸入層與隱藏層之間的權重 update input-to-hidden weights with gradient descent step     self.weights_input_to_hidden += (inputs * hidden_errors * self.lr).T     # 進行預測     def run(self, inputs_list):     # Run a forward pass through the network     inputs = np.array(inputs_list, ndmin=2).T          #### 實現向前傳播 Implement the forward pass here ####     # 隱藏層 Hidden layer     hidden_inputs = np.dot(self.weights_input_to_hidden, inputs) # signals into hidden layer     hidden_outputs = self.activation_function(hidden_inputs) # signals from hidden layer          # 輸出層 Output layer     final_inputs = np.dot(self.weights_hidden_to_output, hidden_outputs) # signals into final output layer     final_outputs = final_inputs # signals from final output layer           return final_outputs