这是原生python中的反向传播算法。

这是一个如何使用numpy实现前馈神经网络的示例。首先导入numpy并指定输入和目标的尺寸。

import numpy as np

input_dim = 1000
target_dim = 10

我们现在将建立网络结构。正如Bishop出色的“模式识别和机器学习”中所建议的那样，您可以简单地将numpy矩阵的最后一行视为偏差权重，将激活的最后一列视为偏差神经元。然后，第一个/最后一个权重矩阵的输入/输出尺寸需要大1。

dimensions = [input_dim+1, 500, 500, target_dim+1]

weight_matrices = []
for i in range(len(dimensions)-1):
  weight_matrix = np.ones((dimensions[i], dimensions[i]))
  weight_matrices.append(weight_matrix)

如果您的输入存储在2d numpy矩阵中，其中每一行对应一个样本，列对应于样本的属性，则您可以像这样通过网络传播（假设逻辑Sigmoid函数作为激活函数）

def activate_network(inputs):
  activations = [] # we store the activations for each layer here
  a = np.ones((inputs.shape[0], inputs.shape[1]+1)) #add the bias to the inputs
  a[:,:-1] = inputs

  for w in weight_matrices:
    x = a.dot(w) # sum of weighted inputs
    a = 1. / (1. - np.exp(-x)) # apply logistic sigmoid activation
    a[:,-1] = 1. # bias for the next layer.
    activations.append(a)

  return activations

activations的最后一个元素将是网络的输出，但是要小心，您需要省略额外的偏差列，因此您的输出将是activations[-1][:,:-1] 。

要训练网络，您需要实施反向传播，这需要花费一些额外的代码行。基本上，您需要从activations的最后一个元素到第一个元素进行循环。在每个反向传播步骤之前，请确保将每一层的误差信号中的偏置列设置为零。

一个简单的实现

这是使用Python类的可读实现。此实现将效率与可理解性进行了权衡：

    import math
    import random

    BIAS = -1

    """
    To view the structure of the Neural Network, type
    print network_name
    """

    class Neuron:
        def __init__(self, n_inputs ):
            self.n_inputs = n_inputs
            self.set_weights( [random.uniform(0,1) for x in range(0,n_inputs+1)] ) # +1 for bias weight

        def sum(self, inputs ):
            # Does not include the bias
            return sum(val*self.weights[i] for i,val in enumerate(inputs))

        def set_weights(self, weights ):
            self.weights = weights

        def __str__(self):
            return 'Weights: %s, Bias: %s' % ( str(self.weights[:-1]),str(self.weights[-1]) )

    class NeuronLayer:
        def __init__(self, n_neurons, n_inputs):
            self.n_neurons = n_neurons
            self.neurons = [Neuron( n_inputs ) for _ in range(0,self.n_neurons)]

        def __str__(self):
            return 'Layer:\n\t'+'\n\t'.join([str(neuron) for neuron in self.neurons])+''

    class NeuralNetwork:
        def __init__(self, n_inputs, n_outputs, n_neurons_to_hl, n_hidden_layers):
            self.n_inputs = n_inputs
            self.n_outputs = n_outputs
            self.n_hidden_layers = n_hidden_layers
            self.n_neurons_to_hl = n_neurons_to_hl

            # Do not touch
            self._create_network()
            self._n_weights = None
            # end

        def _create_network(self):
            if self.n_hidden_layers>0:
                # create the first layer
                self.layers = [NeuronLayer( self.n_neurons_to_hl,self.n_inputs )]

                # create hidden layers
                self.layers += [NeuronLayer( self.n_neurons_to_hl,self.n_neurons_to_hl ) for _ in range(0,self.n_hidden_layers)]

                # hidden-to-output layer
                self.layers += [NeuronLayer( self.n_outputs,self.n_neurons_to_hl )]
            else:
                # If we don't require hidden layers
                self.layers = [NeuronLayer( self.n_outputs,self.n_inputs )]

        def get_weights(self):
            weights = []

            for layer in self.layers:
                for neuron in layer.neurons:
                    weights += neuron.weights

            return weights

        @property
        def n_weights(self):
            if not self._n_weights:
                self._n_weights = 0
                for layer in self.layers:
                    for neuron in layer.neurons:
                        self._n_weights += neuron.n_inputs+1 # +1 for bias weight
            return self._n_weights

        def set_weights(self, weights ):
            assert len(weights)==self.n_weights, "Incorrect amount of weights."

            stop = 0
            for layer in self.layers:
                for neuron in layer.neurons:
                    start, stop = stop, stop+(neuron.n_inputs+1)
                    neuron.set_weights( weights[start:stop] )
            return self

        def update(self, inputs ):
            assert len(inputs)==self.n_inputs, "Incorrect amount of inputs."

            for layer in self.layers:
                outputs = []
                for neuron in layer.neurons:
                    tot = neuron.sum(inputs) + neuron.weights[-1]*BIAS
                    outputs.append( self.sigmoid(tot) )
                inputs = outputs   
            return outputs

        def sigmoid(self, activation,response=1 ):
            # the activation function
            try:
                return 1/(1+math.e**(-activation/response))
            except OverflowError:
                return float("inf")

        def __str__(self):
            return '\n'.join([str(i+1)+' '+str(layer) for i,layer in enumerate(self.layers)])

更加有效的实施（通过学习）

如果您正在寻找带有学习（反向传播）的神经网络的更有效示例，请在此处查看我的神经网络Github存储库 。

嗯，这很棘手。我之前也遇到过同样的问题，但在很好的解释却很难使用的实现之间，我找不到任何东西。

像PyBrain这样的即用型实现的问题在于它们隐藏了细节，因此对学习如何实现ANN感兴趣的人还需要其他东西。阅读此类解决方案的代码也可能具有挑战性，因为它们经常使用启发式方法来提高性能，这使入门的代码更难遵循。

但是，您可以使用一些资源：

http://msdn.microsoft.com/zh-CN/magazine/jj658979.aspx

http://itee.uq.edu.au/~cogs2010/cmc/chapters/BackProp/

http://www.codeproject.com/Articles/19323/Image-Recognition-with-Neural-Networks

http://freedelta.free.fr/r/php-code-samples/artificial-intelligence-neural-network-backpropagation/