# 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("------------")