A bare bones neural network implementation to describe the inner workings of backpropagation. https://iamtrask.github.io/2015/07/12/basic-python-network/

nn_demo.py

#!/usr/bin/env python

import numpy as np
X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ])
y = np.array([[0,1,1,0]]).T
alpha,hidden_dim = (0.5,4)
np.random.seed(1)
# randomly initialize our weights with mean 0
synapse_0 = 2*np.random.random((3,hidden_dim)) - 1
synapse_1 = 2*np.random.random((hidden_dim,1)) - 1
for j in xrange(60000):
    # Feed forward through layers 0, 1, and 2
    layer_1 = 1/(1+np.exp(-(np.dot(X,synapse_0))))
    layer_2 = 1/(1+np.exp(-(np.dot(layer_1,synapse_1))))
    # Backpropagation:
    ## output error, delta = - (y - a) f'(z)
    layer_2_error = - (y - layer_2)
    layer_2_delta = layer_2_error * (layer_2 * (1-layer_2))
    ## hidden layer error, delta(l) = { delta(l+1) .* syn(l) } f'(z^l)
    layer_1_error = layer_2_delta.dot(synapse_1.T)
    layer_1_delta = layer_1_error * (layer_1 * (1-layer_1))
    # update weights by gradient descent
    synapse_1 -= (alpha * layer_1.T.dot(layer_2_delta))
    synapse_0 -= (alpha * X.T.dot(layer_1_delta))
    if (j% 10000) == 0:
        print "Error:" + str(np.mean(np.abs(layer_2_error)))