本文完全利用numpy实现一个简单的BP神经网络，由于是做regression而不是classification，因此在这里输出层选取的激励函数就是f(x)=x。BP神经网络的具体原理此处不再介绍。

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.设定输入层、隐藏层和输出层的node数目 self.input_nodes = input_nodes self.hidden_nodes = hidden_nodes self.output_nodes = output_nodes # Initialize weights，初始化权重和学习速率 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 # 隐藏层的激励函数为sigmoid函数，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 # 输入向量的shape为 [feature_diemension, 1] targets = np.array(targets_list, ndmin=2).T # 向前传播，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 # 输出层，输出层的激励函数就是 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 ### 反向传播 Backward pass，使用梯度下降对权重进行更新 ### # 输出误差 # Output layer error is the difference between desired target and actual output. output_errors = (targets_list-final_outputs) # 反向传播误差 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 # 更新权重 Update the weights # 更新隐藏层与输出层之间的权重 update hidden-to-output weights with gradient descent step self.weights_hidden_to_output += output_errors * hidden_outputs.T * self.lr # 更新输入层与隐藏层之间的权重 update input-to-hidden weights with gradient descent step self.weights_input_to_hidden += (inputs * hidden_errors * self.lr).T # 进行预测 def run(self, inputs_list): # Run a forward pass through the network inputs = np.array(inputs_list, ndmin=2).T #### 实现向前传播 Implement the forward pass here #### # 隐藏层 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 # 输出层 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

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