# Notebook 12: Identifying Phases in the 2D Ising Model with TensorFlow¶

## Learning Goal¶

The goal of this notebook is to familiarize the reader with the nuts and bolts on using the TensorFlow package for building Deep Neural Networks.

## Overview¶

In this notebook, we show how one can use deep neural nets to classify the states of the 2D Ising model according to their phase. This should be compared with the use of logistic-regression, Random Forests and XG Boost on the same dataset in the previous Notebooks 6 and 9.

The Hamiltonian for the classical Ising model is given by

$$H = -J\sum_{\langle ij\rangle}S_{i}S_j,\qquad \qquad S_j\in\{\pm 1\}$$

where the lattice site indices $i,j$ run over all nearest neighbors of a 2D square lattice, and $J$ is some arbitrary interaction energy scale. We adopt periodic boundary conditions. Onsager proved that this model undergoes a phase transition in the thermodynamic limit from an ordered ferromagnet with all spins aligned to a disordered phase at the critical temperature $T_c/J=2/\log(1+\sqrt{2})\approx 2.26$. For any finite system size, this critical point is expanded to a critical region around $T_c$.

In [1]:
# -*- coding: utf-8 -*-
from __future__ import absolute_import, division, print_function
import numpy as np
seed=12
np.random.seed(seed)
import sys, os, argparse
import tensorflow as tf
from tensorflow.python.framework import dtypes
# suppress tflow compilation warnings
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'

tf.set_random_seed(seed)


## Structure of the Procedure¶

Constructing a Deep Neural Network to solve ML problems is a multiple-stage process. Quite generally, one can identify the key steps as follows:

• step 1: Load and process the data
• step 2: Define the model and its architecture
• step 3: Choose the optimizer and the cost function
• step 4: Train the model
• step 5: Evaluate the model performance on the unseen test data
• step 6: Modify the hyperparameters to optimise performance for the specific data set

Below, we sometimes combine some of these steps together for convenience.

Notice that we take a rather different approach, compared to the simpler MNIST Keras notebook. We first define a set of classes and functions and run the actual computation only in the very end.

### Step 1: Load and Process the Data¶

We begin by writing a DataSet class and two functions read_data_sets and load_data to process the 2D Ising data.

The DataSet class performs checks on the data shape and casts the data into the correct data type for the calculation. It contains a function method called next_batch which shuffles the data and returns a mini-batch of a pre-defined size. This structure is particularly useful for the training procedure in TensorFlow.

In [2]:
class DataSet(object):

def __init__(self, data_X, data_Y, dtype=dtypes.float32):
"""Checks data and casts it into correct data type. """

dtype = dtypes.as_dtype(dtype).base_dtype
if dtype not in (dtypes.uint8, dtypes.float32):
raise TypeError('Invalid dtype %r, expected uint8 or float32' % dtype)

assert data_X.shape[0] == data_Y.shape[0], ('data_X.shape: %s data_Y.shape: %s' % (data_X.shape, data_Y.shape))
self.num_examples = data_X.shape[0]

if dtype == dtypes.float32:
data_X = data_X.astype(np.float32)
self.data_X = data_X
self.data_Y = data_Y

self.epochs_completed = 0
self.index_in_epoch = 0

def next_batch(self, batch_size, seed=None):
"""Return the next batch_size examples from this data set."""

if seed:
np.random.seed(seed)

start = self.index_in_epoch
self.index_in_epoch += batch_size
if self.index_in_epoch > self.num_examples:
# Finished epoch
self.epochs_completed += 1
# Shuffle the data
perm = np.arange(self.num_examples)
np.random.shuffle(perm)
self.data_X = self.data_X[perm]
self.data_Y = self.data_Y[perm]
# Start next epoch
start = 0
self.index_in_epoch = batch_size
assert batch_size <= self.num_examples
end = self.index_in_epoch

return self.data_X[start:end], self.data_Y[start:end]


The data itself are being processed in the function read_data_sets, which loads the Ising dataset, and splits it into three subsets: ordered, critical and disordered, depending on the temperature which sets the distribution they are drawn from. Once again, we use the ordered and disordered data to create a training and a test data set for the problem. Classifying the states in the critical region is expected to be harder and we only use this data to test the performance of our model in the end.

In [3]:
import pickle, os
os.environ['KMP_DUPLICATE_LIB_OK']='True'
from urllib.request import urlopen

# path to data directory (for testing)
#path_to_data=os.path.expanduser('~')+'/Dropbox/MachineLearningReview/Datasets/isingMC/'

url_main = 'https://physics.bu.edu/~pankajm/ML-Review-Datasets/isingMC/';

# The data consists of 16*10000 samples taken in T=np.arange(0.25,4.0001,0.25):
data_file_name = "Ising2DFM_reSample_L40_T=All.pkl"
# The labels are obtained from the following file:
label_file_name = "Ising2DFM_reSample_L40_T=All_labels.pkl"

#DATA
data = pickle.load(urlopen(url_main + data_file_name)) # pickle reads the file and returns the Python object (1D array, compressed bits)
data = np.unpackbits(data).reshape(-1, 1600) # Decompress array and reshape for convenience
data=data.astype('int')
data[np.where(data==0)]=-1 # map 0 state to -1 (Ising variable can take values +/-1)

#LABELS (convention is 1 for ordered states and 0 for disordered states)
labels = pickle.load(urlopen(url_main + label_file_name)) # pickle reads the file and returns the Python object (here just a 1D array with the binary labels)

return data, labels

In [4]:
import pickle
from sklearn.model_selection import train_test_split
from keras.utils import to_categorical

def prepare_data(data, labels, dtype=dtypes.float32, test_size=0.2, validation_size=5000):

L=40 # linear system size

# divide data into ordered, critical and disordered
X_ordered=data[:70000,:]
Y_ordered=labels[:70000]

X_critical=data[70000:100000,:]
Y_critical=labels[70000:100000]

X_disordered=data[100000:,:]
Y_disordered=labels[100000:]

# define training and test data sets
X=np.concatenate((X_ordered,X_disordered)) #np.concatenate((X_ordered,X_critical,X_disordered))
Y=np.concatenate((Y_ordered,Y_disordered)) #np.concatenate((Y_ordered,Y_critical,Y_disordered))

# pick random data points from ordered and disordered states to create the training and test sets
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=test_size, train_size=1.0-test_size)

# make data categorical (i.e [0,1] or [1,0])
Y_train=to_categorical(Y_train)
Y_test=to_categorical(Y_test)
Y_critical=to_categorical(Y_critical)

if not 0 <= validation_size <= len(X_train):
raise ValueError('Validation size should be between 0 and {}. Received: {}.'.format(len(X_train), validation_size))

X_validation = X_train[:validation_size]
Y_validation = Y_train[:validation_size]
X_train = X_train[validation_size:]
Y_train = Y_train[validation_size:]

# create data sets
dataset = {
'train':DataSet(X_train, Y_train),
'test':DataSet(X_test, Y_test),
'critical':DataSet(X_critical, Y_critical),
'validation':DataSet(X_validation, Y_validation)
}

return dataset

Using TensorFlow backend.


The DataSet class and the read_data_sets function are wrapped in another function: load_data. To call the latter, one specifies the sizes for the training, test and validation data sets. This function also contains the local path to the file with the Ising data.

In [5]:
def prepare_Ising_DNN():
return prepare_data(data, labels, test_size=0.2, validation_size=5000)


### Steps 2+3: Define the Neural Net and its Architecture, Choose the Optimizer and the Cost Function¶

We can now move on to construct our deep neural net using TensorFlow. To do this, we create a class called model. This class contains many useful function methods which break down the construction of the DNN. Unique for TensorFlow is creating placeholders for the variables of the model, such as the feed-in data self.X and self.Y or the dropout probability self.dropout_keepprob (which has to be set to unity explicitly during testing). Another peculiarity is using the with scope to give names to the most important operators. While we do not discuss this here, TensorFlow also allows one to visualise the computational graph for the model (see package documentation on https://www.tensorflow.org/).

To classify whether a given spin configuration is in the ordered or disordered phase, we construct a minimalistic model for a DNN with a single hidden layer containing $N_\mathrm{neurons}$ (which is kept variable so we can try out the performance of different sizes for the hidden layer).

First, we define two private functions: _weight_variable and _bias_variable, which we use to set up the precise DNN architecture in the function create_DNN. The network architecture thus includes a ReLU-activated input layer, the hidden layer, and the softmax output layer. Notice that the softmax layer is not part of the create_DNN function.

Instead, the softmax layer is part of the function create_loss which, as the name suggests, defines the cross entropy loss function, predefined in TensorFlow's nn module. We minimize the cost function using the SGD optimizer from the train module in the function create_optimiser. The latter accepts a dictionary opt_kwargs with optimizer arguments to be set externally when defining the DNN.

Last, the function create_accuracy evaluates the model performance.

All these function are called in the __init__ of our model class which sets up the DNN. It accepts the number of hidden neurons $N_\mathrm{neurons}$ and a dictionary with the optimizer arguments as input, as we shall study the performance of the DNN as a function of these parameters.

In [6]:
class model(object):
def __init__(self, N_neurons, opt_kwargs):
"""Builds the TFlow graph for the DNN.

N_neurons: number of neurons in the hidden layer
opt_kwargs: optimizer's arguments

"""

# define global step for checkpointing
self.global_step=tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')

self.L=40 # system linear size
self.n_feats=self.L**2 # 40x40 square lattice
self.n_categories=2 # 2 Ising phases: ordered and disordered

# create placeholders for input X and label Y
self.create_placeholders()
# create weight and bias, initialized to 0 and construct DNN to predict Y from X
self.deep_layer_neurons=N_neurons
self.create_DNN()
# define loss function
self.create_loss()
# use gradient descent to minimize loss
self.create_optimiser(opt_kwargs)
#  create accuracy
self.create_accuracy()

def create_placeholders(self):
with tf.name_scope('data'):
# input layer
self.X=tf.placeholder(tf.float32, shape=(None, self.n_feats), name="X_data")
# target
self.Y=tf.placeholder(tf.float32, shape=(None, self.n_categories), name="Y_data")
# p
self.dropout_keepprob=tf.placeholder(tf.float32, name="keep_probability")

def _weight_variable(self, shape, name='', dtype=tf.float32):
"""weight_variable generates a weight variable of a given shape."""
# weights are drawn from a normal distribution with std 0.1 and mean 0.
initial = tf.truncated_normal(shape, stddev=0.1)
return tf.Variable(initial, dtype=dtype, name=name)

def _bias_variable(self, shape, name='', dtype=tf.float32):
"""bias_variable generates a bias variable of a given shape."""
initial = tf.constant(0.1, shape=shape)
return tf.Variable(initial, dtype=dtype, name=name)

def create_DNN(self):
with tf.name_scope('DNN'):

# Fully connected layer
W_fc1 = self._weight_variable([self.n_feats, self.deep_layer_neurons],name='fc1',dtype=tf.float32)
b_fc1 = self._bias_variable([self.deep_layer_neurons],name='fc1',dtype=tf.float32)

a_fc1 = tf.nn.relu(tf.matmul(self.X, W_fc1) + b_fc1)

# Softmax layer (see loss function)
W_fc2 = self._weight_variable([self.deep_layer_neurons, self.n_categories],name='fc2',dtype=tf.float32)
b_fc2 = self._bias_variable([self.n_categories],name='fc2',dtype=tf.float32)

self.Y_predicted = tf.matmul(a_fc1, W_fc2) + b_fc2

def create_loss(self):
with tf.name_scope('loss'):
self.loss = tf.reduce_mean(
tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.Y_predicted)
)
# no need to use tf.stop_gradient() on labels because labels are placeholders and contain no params
# to be optimized. Backprop will be applied only to the logits.

def create_optimiser(self,opt_kwargs):
with tf.name_scope('optimiser'):

def create_accuracy(self):
with tf.name_scope('accuracy'):
correct_prediction = tf.equal(tf.argmax(self.Y, 1), tf.argmax(self.Y_predicted, 1))
correct_prediction = tf.cast(correct_prediction, tf.float64) # change data type
self.accuracy = tf.reduce_mean(correct_prediction)



### Steps 4+5: Train the Model and Evaluate its Performance¶

We want to evaluate the performance of our model over a set of different learning rates, and a set of different hidden neurons, i.e. we consider a variable size of the hidden layer. Therefore, we create a function evaluate_model which trains and evaluates the performance of our DNN for a fixed number of hidden neurons and a fixed SGD learning rate lr, and returns the final loss and accuracy for the three data sets of interest.

Apart from the number of neurons and the learning rate lr, evaluate_model accepts the data Ising_Data. This is done for convenience: loading the data is computationally expensive and we only need to do this once.

We train our DNN using mini-batches of size $100$ over a total of $100$ epochs, which we define first. We then set up the optimizer parameter dictionary opt_params, and use it to create a DNN model.

Running TensorFlow requires opening up a Session which we abbreviate as sess for short. All operations are performed in this session by calling the run method. First, we initialize the global variables in TensorFlow's computational graph by running the global_variables_initializer. To train the DNN, we loop over the number of epochs. In each fix epoch, we use the next_batch function of the DataSet class we defined above to create a mini-batch. The forward and backward passes through the weights are performed by running the DNN.loss and DNN.optimizer methods. To pass the mini-batch as well as any other external parameters, we use the feed_dict dictionary. Similarly, we evaluate the model performance, by running the DNN.accuracy function on the same minibatch data. Note that the dropout probability for testing is set to unity.

Once we have exhausted all training epochs, we test the final performance on the entire training, test and critical data sets. This is done in the same way as above.

Last, we return the loss and accuracy for each of the training, test and critical data sets.

In [7]:
def evaluate_model(neurons, lr, Ising_Data, verbose):
"""This function trains a DNN to solve the Ising classification problem

neurons: number of hidden neurons
lr: SGD learning rate
Ising_Data: Ising data set
verbose (bool): toggles output during the calculation

"""

training_epochs=100
batch_size=100

# SGD learning params
opt_params=dict(learning_rate=lr)

# create DNN
DNN=model(neurons,opt_params)

with tf.Session() as sess:

# initialize the necessary variables, in this case, w and b
sess.run(tf.global_variables_initializer())

# train the DNN
for epoch in range(training_epochs):

batch_X, batch_Y = Ising_Data['train'].next_batch(batch_size,seed=seed)

loss_batch, _ = sess.run([DNN.loss,DNN.optimizer],
feed_dict={DNN.X: batch_X,
DNN.Y: batch_Y,
DNN.dropout_keepprob: 0.5} )
accuracy = sess.run(DNN.accuracy,
feed_dict={DNN.X: batch_X,
DNN.Y: batch_Y,
DNN.dropout_keepprob: 1.0} )

# count training step
step = sess.run(DNN.global_step)

# test DNN performance on entire train test and critical data sets
train_loss, train_accuracy = sess.run([DNN.loss, DNN.accuracy],
feed_dict={DNN.X: Ising_Data['train'].data_X,
DNN.Y: Ising_Data['train'].data_Y,
DNN.dropout_keepprob: 0.5}
)
if verbose: print("train loss/accuracy:", train_loss, train_accuracy)

test_loss, test_accuracy = sess.run([DNN.loss, DNN.accuracy],
feed_dict={DNN.X: Ising_Data['test'].data_X,
DNN.Y: Ising_Data['test'].data_Y,
DNN.dropout_keepprob: 1.0}
)

if verbose: print("test loss/accuracy:", test_loss, test_accuracy)

critical_loss, critical_accuracy = sess.run([DNN.loss, DNN.accuracy],
feed_dict={DNN.X: Ising_Data['critical'].data_X,
DNN.Y: Ising_Data['critical'].data_Y,
DNN.dropout_keepprob: 1.0}
)
if verbose: print("crtitical loss/accuracy:", critical_loss, critical_accuracy)

return train_loss,train_accuracy,test_loss,test_accuracy,critical_loss,critical_accuracy


### Step 6: Modify the Hyperparameters to Optimize Performance of the Model¶

To study the dependence of our DNN on some of the hyperparameters, we do a grid search over the number of neurons in the hidden layer, and different SGD learning rates. As we explained in Sec. IX, these searches are best done over logarithmically-spaced points.

Since we created the evaluate_model function with this in hindsight, below we simply loop over the grid values and call evaluate_model.

In [8]:
def grid_search(verbose):
"""This function performs a grid search over a set of different learning rates
and a number of hidden layer neurons."""

Ising_Data = prepare_Ising_DNN()

# perform grid search over learnign rate and number of hidden neurons
N_neurons=np.logspace(0,3,4).astype('int') # check number of neurons over multiple decades
learning_rates=np.logspace(-6,-1,6)

# pre-alocate variables to store accuracy and loss data
train_loss=np.zeros((len(N_neurons),len(learning_rates)),dtype=np.float64)
train_accuracy=np.zeros_like(train_loss)
test_loss=np.zeros_like(train_loss)
test_accuracy=np.zeros_like(train_loss)
critical_loss=np.zeros_like(train_loss)
critical_accuracy=np.zeros_like(train_loss)

# do grid search
for i, neurons in enumerate(N_neurons):
for j, lr in enumerate(learning_rates):

print("training DNN with %4d neurons and SGD lr=%0.6f." %(neurons,lr) )

train_loss[i,j],train_accuracy[i,j],\
test_loss[i,j],test_accuracy[i,j],\
critical_loss[i,j],critical_accuracy[i,j] = evaluate_model(neurons,lr,Ising_Data,verbose)

plot_data(learning_rates,N_neurons,train_accuracy, 'training')
plot_data(learning_rates,N_neurons,test_accuracy, 'testing')
plot_data(learning_rates,N_neurons,critical_accuracy, 'critical')


To visualize the data, we used the function plot_data, defined below.

In [9]:
%matplotlib notebook
import matplotlib.pyplot as plt

def plot_data(x,y,data,title=None):

# plot results
fontsize=16

fig = plt.figure()
cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)

cbar=fig.colorbar(cax)
cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)
cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])
cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])

# put text on matrix elements
for i, x_val in enumerate(np.arange(len(x))):
for j, y_val in enumerate(np.arange(len(y))):
c = "${0:.1f}\\%$".format( 100*data[j,i])
ax.text(x_val, y_val, c, va='center', ha='center')

# convert axis vaues to to string labels
x=[str(i) for i in x]
y=[str(i) for i in y]

ax.set_xticklabels(['']+x)
ax.set_yticklabels(['']+y)

ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize)
ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize)
if title is not None:
ax.set_title(title)

plt.tight_layout()

plt.show()


## Run Code¶

As we mentioned in the beginning of the notebook, all functions and classes discussed above only specify the procedure but do not actually perform any computations. This allows us to re-use them for different problems.

Actually running the training and testing for every point in the grid search is done below.

In [10]:
verbose=False
grid_search(verbose)

Finished loading data
training DNN with    1 neurons and SGD lr=0.000001.
training DNN with    1 neurons and SGD lr=0.000010.
training DNN with    1 neurons and SGD lr=0.000100.
training DNN with    1 neurons and SGD lr=0.001000.
training DNN with    1 neurons and SGD lr=0.010000.
training DNN with    1 neurons and SGD lr=0.100000.
training DNN with   10 neurons and SGD lr=0.000001.
training DNN with   10 neurons and SGD lr=0.000010.
training DNN with   10 neurons and SGD lr=0.000100.
training DNN with   10 neurons and SGD lr=0.001000.
training DNN with   10 neurons and SGD lr=0.010000.
training DNN with   10 neurons and SGD lr=0.100000.
training DNN with  100 neurons and SGD lr=0.000001.
training DNN with  100 neurons and SGD lr=0.000010.
training DNN with  100 neurons and SGD lr=0.000100.
training DNN with  100 neurons and SGD lr=0.001000.
training DNN with  100 neurons and SGD lr=0.010000.
training DNN with  100 neurons and SGD lr=0.100000.
training DNN with 1000 neurons and SGD lr=0.000001.
training DNN with 1000 neurons and SGD lr=0.000010.
training DNN with 1000 neurons and SGD lr=0.000100.
training DNN with 1000 neurons and SGD lr=0.001000.
training DNN with 1000 neurons and SGD lr=0.010000.
training DNN with 1000 neurons and SGD lr=0.100000.


## Creating Convolutional Neural Nets with TensorFlow¶

We have so far considered each 2D-Ising state as a $(40\times 40,)$-long 1D vector. This approach neglects any spatial structure of the spin configuration. On the other hand, we do know that in every one of the 2D-Ising states there are local spatial correlations between the spins, which we would like to take advantage of to improve the accuracy of our classification model.

To this end, all we need to do is modify the class model to include convolutional layers. The code below is analogous to the DNN model class we discussed in detail above, except for the following noticeable discrepancies:

• note the auxiliary variable X_reshaped in the function create_CNN which casts the 2D-Ising configuration data as a 2D array, in order to bring out the spatial correlations.
• new below are the conv2d and max_pool_2x2 functions which help create the convolutional structures of the network.
In [11]:
class model(object):
# build the graph for the CNN
def __init__(self,opt_kwargs):

# define global step for checkpointing
self.global_step=tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')

self.L=40
self.n_feats= self.L**2
self.n_categories=2

# create placeholders for input X and label Y
self.create_placeholders()
# create weight and bias, initialized to 0 and construct CNN to predict Y from X
self.create_CNN()
# define loss function
self.create_loss()
# use gradient descent to minimize loss
self.create_optimiser(opt_kwargs)
# create accuracy
self.create_accuracy()

print("finished creating CNN")

def create_placeholders(self):
with tf.name_scope('data'):
self.X=tf.placeholder(tf.float32, shape=(None,self.n_feats), name="X_data")
self.Y=tf.placeholder(tf.float32, shape=(None,self.n_categories), name="Y_data")
self.dropout_keepprob=tf.placeholder(tf.float32, name="keep_probability")

def create_CNN(self, N_filters=10):
with tf.name_scope('CNN'):
# conv layer 1, 5x5 kernel, 1 input 10 output channels
W_conv1 = self.weight_variable([5, 5, 1, N_filters],name='conv1',dtype=tf.float32)
b_conv1 = self.bias_variable([N_filters],name='conv1',dtype=tf.float32)
X_reshaped = tf.reshape(self.X, [-1, self.L, self.L, 1])
h_conv1 = tf.nn.relu(self.conv2d(X_reshaped, W_conv1, name='conv1') + b_conv1)

# Pooling layer - downsamples by 2X.
h_pool1 = self.max_pool_2x2(h_conv1,name='pool1')
# conv layer 2, 5x5 kernel, 10 input 20 output channels
W_conv2 = self.weight_variable([5, 5, 10, 20],name='conv2',dtype=tf.float32)
b_conv2 = self.bias_variable([20],name='conv2',dtype=tf.float32)
h_conv2 = tf.nn.relu(self.conv2d(h_pool1, W_conv2, name='conv2') + b_conv2)

# Dropout - controls the complexity of the CNN, prevents co-adaptation of features.
h_conv2_drop = tf.nn.dropout(h_conv2, self.dropout_keepprob,name='conv2_dropout')

# Second pooling layer.
h_pool2 = self.max_pool_2x2(h_conv2_drop,name='pool2')

# Fully connected layer 1 -- after second round of downsampling, our 40x40 image
# is down to 7x7x20 feature maps -- maps this to 50 features.
h_pool2_flat = tf.reshape(h_pool2, [-1, 7*7*20])

W_fc1 = self.weight_variable([7*7*20, 50],name='fc1',dtype=tf.float32)
b_fc1 = self.bias_variable([50],name='fc1',dtype=tf.float32)

h_fc1 = tf.nn.relu(tf.matmul(h_pool2_flat, W_fc1) + b_fc1)

# Dropout - controls the complexity of the CNN, prevents co-adaptation of features.
h_fc1_drop = tf.nn.dropout(h_fc1, self.dropout_keepprob,name='fc1_dropout')

# Map the 50 features to 2 classes, one for each phase
W_fc2 = self.weight_variable([50, self.n_categories],name='fc12',dtype=tf.float32)
b_fc2 = self.bias_variable([self.n_categories],name='fc12',dtype=tf.float32)

self.Y_predicted = tf.matmul(h_fc1_drop, W_fc2) + b_fc2

def weight_variable(self, shape, name='', dtype=tf.float32):
"""weight_variable generates a weight variable of a given shape."""
initial = tf.truncated_normal(shape, stddev=0.1)
return tf.Variable(initial,dtype=dtype,name=name)

def bias_variable(self, shape, name='', dtype=tf.float32):
"""bias_variable generates a bias variable of a given shape."""
initial = tf.constant(0.1, shape=shape)
return tf.Variable(initial,dtype=dtype,name=name)

def conv2d(self, x, W, name=''):
"""conv2d returns a 2d convolution layer with full stride."""
return tf.nn.conv2d(x, W, strides=[1, 1, 1, 1], padding='VALID', name=name)

def max_pool_2x2(self, x,name=''):
"""max_pool_2x2 downsamples a feature map by 2X."""
return tf.nn.max_pool(x, ksize=[1, 2, 2, 1],
strides=[1, 2, 2, 1],
name=name
)

def create_loss(self):
with tf.name_scope('loss'):
self.loss = tf.reduce_mean(
tf.nn.softmax_cross_entropy_with_logits(labels=self.Y,logits=self.Y_predicted)
)

def create_optimiser(self,kwargs):
with tf.name_scope('optimiser'):

def create_accuracy(self):
with tf.name_scope('accuracy'):
correct_prediction = tf.equal(tf.argmax(self.Y, 1), tf.argmax(self.Y_predicted, 1))
correct_prediction = tf.cast(correct_prediction, tf.float64)
self.accuracy = tf.reduce_mean(correct_prediction)


## Exercises¶

• TensorFlow allows one to visualize the model as a graph. Read the TensorBoard documentation and implement this in the code above.
• Read about using TensorBoard to monitor the training and test errors.
• Use the CNN model defined above to run the Ising classification analysis. Choose the optimal network architecture to improve on the results of the non-convolutional layer. To test the performance of the CNN, use the critical data set.
• Does using the Adam optimizer improve on the performance on the critical data? Modify the CNN model class to do a grid search over the parameters of the Adam optimizer.