# Recurrent Neural Networksimport copy, numpy as npnp.random.seed(0)# compute sigmoid nonlinearitydef sigmoid(x):  output = 1/(1+np.exp(-x))  return output# convert output of sigmoid function to its derivativedef sigmoid_output_to_derivative(output):  return output*(1-output)# training dataset generationint2binary = {}binary_dim = 8largest_number = pow(2,binary_dim)binary = np.unpackbits(  np.array([range(largest_number)],dtype=np.uint8).T,axis=1)for i in range(largest_number):  int2binary[i] = binary[i]# input variablesalpha = 0.1input_dim = 2hidden_dim = 16output_dim = 1# initialize neural network weightssynapse_0 = 2*np.random.random((input_dim,hidden_dim)) - 1synapse_1 = 2*np.random.random((hidden_dim,output_dim)) - 1synapse_h = 2*np.random.random((hidden_dim,hidden_dim)) - 1synapse_0_update = np.zeros_like(synapse_0)synapse_1_update = np.zeros_like(synapse_1)synapse_h_update = np.zeros_like(synapse_h)# training logicfor j in range(10000):  # generate a simple addition problem (a + b = c)  a_int = np.random.randint(largest_number/2) # int version  a = int2binary[a_int] # binary encoding  b_int = np.random.randint(largest_number/2) # int version  b = int2binary[b_int] # binary encoding  # true answer  c_int = a_int + b_int  c = int2binary[c_int]  # where we'll store our best guess (binary encoded)  d = np.zeros_like(c)  overallError = 0  layer_2_deltas = list()  layer_1_values = list()  layer_1_values.append(np.zeros(hidden_dim))  # moving along the positions in the binary encoding  for position in range(binary_dim):    # generate input and output    X = np.array([[a[binary_dim - position - 1],b[binary_dim - position - 1]]])    y = np.array([[c[binary_dim - position - 1]]]).T    # hidden layer (input ~+ prev_hidden)    layer_1 = sigmoid(np.dot(X,synapse_0) + np.dot(layer_1_values[-1],synapse_h))    # output layer (new binary representation)    layer_2 = sigmoid(np.dot(layer_1,synapse_1))    # did we miss?... if so, by how much?    layer_2_error = y - layer_2    layer_2_deltas.append((layer_2_error)*sigmoid_output_to_derivative(layer_2))    overallError += np.abs(layer_2_error[0])    # decode estimate so we can print(it out)    d[binary_dim - position - 1] = np.round(layer_2[0][0])    # store hidden layer so we can use it in the next timestep    layer_1_values.append(copy.deepcopy(layer_1))  future_layer_1_delta = np.zeros(hidden_dim)  for position in range(binary_dim):    X = np.array([[a[position],b[position]]])    layer_1 = layer_1_values[-position-1]    prev_layer_1 = layer_1_values[-position-2]    # error at output layer    layer_2_delta = layer_2_deltas[-position-1]    # error at hidden layer    layer_1_delta = (future_layer_1_delta.dot(synapse_h.T) + layer_2_delta.dot(synapse_1.T)) * sigmoid_output_to_derivative(layer_1)    # let's update all our weights so we can try again    synapse_1_update += np.atleast_2d(layer_1).T.dot(layer_2_delta)    synapse_h_update += np.atleast_2d(prev_layer_1).T.dot(layer_1_delta)    synapse_0_update += X.T.dot(layer_1_delta)    future_layer_1_delta = layer_1_delta  synapse_0 += synapse_0_update * alpha  synapse_1 += synapse_1_update * alpha  synapse_h += synapse_h_update * alpha  synapse_0_update *= 0  synapse_1_update *= 0  synapse_h_update *= 0  # print(out progress)  if j % 1000 == 0:    print("Error:" + str(overallError))    print("Pred:" + str(d))    print("True:" + str(c))    out = 0    for index,x in enumerate(reversed(d)):      out += x*pow(2,index)    print(str(a_int) + " + " + str(b_int) + " = " + str(out))    print("------------")

Error:[ 3.45638663]Pred:[0 0 0 0 0 0 0 1]True:[0 1 0 0 0 1 0 1]9 + 60 = 1------------Error:[ 3.63389116]Pred:[1 1 1 1 1 1 1 1]True:[0 0 1 1 1 1 1 1]28 + 35 = 255------------Error:[ 3.91366595]Pred:[0 1 0 0 1 0 0 0]True:[1 0 1 0 0 0 0 0]116 + 44 = 72------------Error:[ 3.72191702]Pred:[1 1 0 1 1 1 1 1]True:[0 1 0 0 1 1 0 1]4 + 73 = 223------------Error:[ 3.5852713]Pred:[0 0 0 0 1 0 0 0]True:[0 1 0 1 0 0 1 0]71 + 11 = 8------------Error:[ 2.53352328]Pred:[1 0 1 0 0 0 1 0]True:[1 1 0 0 0 0 1 0]81 + 113 = 162------------Error:[ 0.57691441]Pred:[0 1 0 1 0 0 0 1]True:[0 1 0 1 0 0 0 1]81 + 0 = 81------------Error:[ 1.42589952]Pred:[1 0 0 0 0 0 0 1]True:[1 0 0 0 0 0 0 1]4 + 125 = 129------------Error:[ 0.47477457]Pred:[0 0 1 1 1 0 0 0]True:[0 0 1 1 1 0 0 0]39 + 17 = 56------------Error:[ 0.21595037]Pred:[0 0 0 0 1 1 1 0]True:[0 0 0 0 1 1 1 0]11 + 3 = 14------------