diff --git a/doc/pub/week42/html/._week42-bs000.html b/doc/pub/week42/html/._week42-bs000.html index 006fb35be..5012da2cf 100644 --- a/doc/pub/week42/html/._week42-bs000.html +++ b/doc/pub/week42/html/._week42-bs000.html @@ -8,8 +8,8 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 42 Solving differential equations and Convolutional (CNN) + +Week 42 Constructing a Neural Network code with introduction to Tensor flow @@ -37,235 +37,147 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d @@ -293,7 +205,7 @@ MathJax.Hub.Config({ - Week 42 Solving differential equations and Convolutional (CNN) + Week 42 Constructing a Neural Network code with introduction to Tensor flow
@@ -461,7 +331,7 @@ MathJax.Hub.Config({ -->
- © 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
diff --git a/doc/pub/week42/html/._week42-bs001.html b/doc/pub/week42/html/._week42-bs001.html index 9179b8c41..9066f6c1d 100644 --- a/doc/pub/week42/html/._week42-bs001.html +++ b/doc/pub/week42/html/._week42-bs001.html @@ -8,8 +8,8 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 42 Solving differential equations and Convolutional (CNN) + +Week 42 Constructing a Neural Network code with introduction to Tensor flow @@ -37,235 +37,147 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d @@ -293,7 +205,7 @@ MathJax.Hub.Config({ - Week 42 Solving differential equations and Convolutional (CNN) + Week 42 Constructing a Neural Network code with introduction to Tensor flow
- +
-
-
- -

m

- -
-
- +
  • Building our own Feed-forward Neural Network and discussion of project 2
    • +
  • Readings and Videos:
    • + +

    I also recommend Michael Nielsen's intuitive approach to the neural networks and the universal approximation theorem, see the slides at http://neuralnetworksanddeeplearning.com/chap4.html.

    @@ -471,7 +325,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs002.html b/doc/pub/week42/html/._week42-bs002.html index ac1947aad..a072a86cb 100644 --- a/doc/pub/week42/html/._week42-bs002.html +++ b/doc/pub/week42/html/._week42-bs002.html @@ -8,8 +8,8 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 42 Solving differential equations and Convolutional (CNN) + +Week 42 Constructing a Neural Network code with introduction to Tensor flow @@ -37,235 +37,147 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d @@ -293,7 +205,7 @@ MathJax.Hub.Config({ - Week 42 Solving differential equations and Convolutional (CNN) + Week 42 Constructing a Neural Network code with introduction to Tensor flow
    diff --git a/doc/pub/week42/html/._week42-bs003.html b/doc/pub/week42/html/._week42-bs003.html index c4b7d9426..76c3fdda2 100644 --- a/doc/pub/week42/html/._week42-bs003.html +++ b/doc/pub/week42/html/._week42-bs003.html @@ -8,8 +8,8 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 42 Solving differential equations and Convolutional (CNN) + +Week 42 Constructing a Neural Network code with introduction to Tensor flow @@ -37,235 +37,147 @@ doconce format html week42.do.txt --html_style=bootstrap --pygments_html_style=d @@ -293,7 +205,7 @@ MathJax.Hub.Config({ - Week 42 Solving differential equations and Convolutional (CNN) + Week 42 Constructing a Neural Network code with introduction to Tensor flow
    + +
    +
    +
    +
    +
    +
    +
    +
    + + + +
    +

    Choose cost function and optimizer

    + +

    To measure how well our neural network is doing we need to introduce a cost function. +We will call the function that gives the error of a single sample output the loss function, and the function +that gives the total error of our network across all samples the cost function. +A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood. +

    + +

    In multiclass classification it is common to treat each integer label as a so called one-hot vector:

    + +

     
    +$$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ +

     

    + +

     
    +$$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ +

     

    + +

    i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.

    + +

    Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector. +We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \boldsymbol{x}_i \) in the dataset. +

    + +

    In the one-hot representation only one of the terms in the loss function is non-zero, namely the +probability of the correct category \( c' \) +(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong +you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases. +

    +
    + +
    +

    Optimizing the cost function

    + +

    The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent +is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function. +Each parameter \( \theta \) is iteratively adjusted according to the rule +

    + +

     
    +$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$ +

     

    + +

    where \( \eta \) is known as the learning rate, which controls how big a step we take towards the minimum. +This update can be repeated for any number of iterations, or until we are satisfied with the result. +

    + +

    A simple and effective improvement is a variant called Batch Gradient Descent. +Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient +on a subset of the data called a minibatch. +If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches +is \( N/M \). +We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes: +

    + +

     
    +$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad +\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$ +

     
    +

    + +

    i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.

    + +

    This has two important benefits:

    +
      +

    1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.
      2. + +

    2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.
      3. +
    +

    +

    The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.

    +
    + +
    +

    Regularization

    + +

    It is common to add an extra term to the cost function, proportional +to the size of the weights. This is equivalent to constraining the +size of the weights, so that they do not grow out of control. +Constraining the size of the weights means that the weights cannot +grow arbitrarily large to fit the training data, and in this way +reduces overfitting. +

    + +

    We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:

    + +

     
    +$$ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad +\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2 += \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$ +

     
    +

    + +

    i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.

    + +

    In order to train the model, we need to calculate the derivative of +the cost function with respect to every bias and weight in the +network. In total our network has \( (64 + 1)\times 50=3250 \) weights in +the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output +layer (\( +1 \) for the bias), and the gradient must be calculated for +every parameter. We use the backpropagation algorithm discussed +above. This is a clever use of the chain rule that allows us to +calculate the gradient efficently. +

    +
    + +
    +

    Matrix multiplication

    + +

    To more efficently train our network these equations are implemented using matrix operations. +The error in the output layer is calculated simply as, with \( \boldsymbol{t} \) being our targets, +

    + +

     
    +$$ \delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .$$ +

     

    + +

    The gradient for the output weights is calculated as

    + +

     
    +$$ \nabla W_{L} = \boldsymbol{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$ +

     

    + +

    where \( \boldsymbol{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input. +Since we are going backwards we have to transpose the activation matrix. +

    + +

    The gradient with respect to the output bias is then

    + +

     
    +$$ \nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$ +

     

    + +

    The error in the hidden layer is

    + +

     
    +$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$ +

     

    + +

    where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean +that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes +the Hadamard product, meaning element-wise multiplication. +

    + +

    This again gives us the gradients in the hidden layer:

    + +

     
    +$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$ +

     

    + +

     
    +$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$ +

     

    + + + +
    +
    +
    +
    +
    + 
    # to categorical turns our integer vector into a onehot representation
+from sklearn.metrics import accuracy_score
+
+# one-hot in numpy
+def to_categorical_numpy(integer_vector):
+    n_inputs = len(integer_vector)
+    n_categories = np.max(integer_vector) + 1
+    onehot_vector = np.zeros((n_inputs, n_categories))
+    onehot_vector[range(n_inputs), integer_vector] = 1
+    
+    return onehot_vector
+
+#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
+Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
+
+def feed_forward_train(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    exp_term = np.exp(z_o)
+    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+    
+    # for backpropagation need activations in hidden and output layers
+    return a_h, probabilities
+
+def backpropagation(X, Y):
+    a_h, probabilities = feed_forward_train(X)
+    
+    # error in the output layer
+    error_output = probabilities - Y
+    # error in the hidden layer
+    error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
+    
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_h.T, error_output)
+    output_bias_gradient = np.sum(error_output, axis=0)
+    
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(X.T, error_hidden)
+    hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
+eta = 0.01
+lmbd = 0.01
+for i in range(1000):
+    # calculate gradients
+    dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
+    
+    # regularization term gradients
+    dWo += lmbd * output_weights
+    dWh += lmbd * hidden_weights
+    
+    # update weights and biases
+    output_weights -= eta * dWo
+    output_bias -= eta * dBo
+    hidden_weights -= eta * dWh
+    hidden_bias -= eta * dBh
+
+print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Improving performance

    + +

    As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image. +In order to obtain a network that does something useful, we will have to do a bit more work. +

    + +

    The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).

    + +

    Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period +going through the entire dataset (\( n/M \) batches) an epoch. +

    + +

    If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. +Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here. +

    +
    + +
    +

    Full object-oriented implementation

    + +

    It is very natural to think of the network as an object, with specific instances of the network +being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below. +

    + + + +
    +
    +
    +
    +
    + 
    class NeuralNetwork:
+    def __init__(
+            self,
+            X_data,
+            Y_data,
+            n_hidden_neurons=50,
+            n_categories=10,
+            epochs=10,
+            batch_size=100,
+            eta=0.1,
+            lmbd=0.0):
+
+        self.X_data_full = X_data
+        self.Y_data_full = Y_data
+
+        self.n_inputs = X_data.shape[0]
+        self.n_features = X_data.shape[1]
+        self.n_hidden_neurons = n_hidden_neurons
+        self.n_categories = n_categories
+
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
+
+        self.create_biases_and_weights()
+
+    def create_biases_and_weights(self):
+        self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
+        self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
+
+        self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
+        self.output_bias = np.zeros(self.n_categories) + 0.01
+
+    def feed_forward(self):
+        # feed-forward for training
+        self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
+        self.a_h = sigmoid(self.z_h)
+
+        self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
+
+        exp_term = np.exp(self.z_o)
+        self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
+    def feed_forward_out(self, X):
+        # feed-forward for output
+        z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
+        a_h = sigmoid(z_h)
+
+        z_o = np.matmul(a_h, self.output_weights) + self.output_bias
         
+        exp_term = np.exp(z_o)
+        probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+        return probabilities
+
+    def backpropagation(self):
+        error_output = self.probabilities - self.Y_data
+        error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
+
+        self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
+        self.output_bias_gradient = np.sum(error_output, axis=0)
+
+        self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
+        self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+        if self.lmbd > 0.0:
+            self.output_weights_gradient += self.lmbd * self.output_weights
+            self.hidden_weights_gradient += self.lmbd * self.hidden_weights
+
+        self.output_weights -= self.eta * self.output_weights_gradient
+        self.output_bias -= self.eta * self.output_bias_gradient
+        self.hidden_weights -= self.eta * self.hidden_weights_gradient
+        self.hidden_bias -= self.eta * self.hidden_bias_gradient
+
+    def predict(self, X):
+        probabilities = self.feed_forward_out(X)
+        return np.argmax(probabilities, axis=1)
+
+    def predict_probabilities(self, X):
+        probabilities = self.feed_forward_out(X)
+        return probabilities
+
+    def train(self):
+        data_indices = np.arange(self.n_inputs)
+
+        for i in range(self.epochs):
+            for j in range(self.iterations):
+                # pick datapoints with replacement
+                chosen_datapoints = np.random.choice(
+                    data_indices, size=self.batch_size, replace=False
+                )
+
+                # minibatch training data
+                self.X_data = self.X_data_full[chosen_datapoints]
+                self.Y_data = self.Y_data_full[chosen_datapoints]
+
+                self.feed_forward()
+                self.backpropagation()
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Evaluate model performance on test data

    + +

    To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data. +We measure the performance of the network using the accuracy score. +The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \). +

    + +

     
    +$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,$$ +

     

    + +

    where \( I \) is the indicator function, \( 1 \) if \( \tilde{y}_i = y_i \) and \( 0 \) otherwise.

    + + + +
    +
    +
    +
    +
    + 
    epochs = 100
+batch_size = 100
+
+dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+                    n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+dnn.train()
+test_predict = dnn.predict(X_test)
+
+# accuracy score from scikit library
+print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+
+# equivalent in numpy
+def accuracy_score_numpy(Y_test, Y_pred):
+    return np.sum(Y_test == Y_pred) / len(Y_test)
+
+#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Adjust hyperparameters

    + +

    We now perform a grid search to find the optimal hyperparameters for the network. +Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate). +

    + + + +
    +
    +
    +
    +
    + 
    eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
+# grid search
 for i, eta in enumerate(eta_vals):
     for j, lmbd in enumerate(lmbd_vals):
-        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd)
-        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = CNN.evaluate(X_test, Y_test)
+        dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+                            n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+        dnn.train()
         
-        CNN_keras[i][j] = CNN
+        DNN_numpy[i][j] = dnn
         
-        print("Learning rate = ", eta)
+        test_predict = dnn.predict(X_test)
+        
+        print("Learning rate  = ", eta)
         print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
+        print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
         print()
    @@ -4154,7 +1501,7 @@ lmbd_vals = np.logspace(-5, Final visualization +

    Visualization

    @@ -4164,7 +1511,7 @@ lmbd_vals = np.logspace(-5, 
    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
+# uses seaborn heatmap, you can also do this with matplotlib imshow
 import seaborn as sns
 
 sns.set()
@@ -4174,10 +1521,13 @@ test_accuracy = np.zeros((len(eta_vals), for i in range(len(eta_vals)):
     for j in range(len(lmbd_vals)):
-        CNN = CNN_keras[i][j]
+        dnn = DNN_numpy[i][j]
+        
+        train_pred = dnn.predict(X_train) 
+        test_pred = dnn.predict(X_test)
 
-        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
 
         
 fig, ax = plt.subplots(figsize = (10, 10))
@@ -4210,12 +1560,20 @@ plt.show()
    -

    The CIFAR01 data set

    +

    scikit-learn implementation

    -

    The CIFAR10 dataset contains 60,000 color images in 10 classes, with -6,000 images in each class. The dataset is divided into 50,000 -training images and 10,000 testing images. The classes are mutually -exclusive and there is no overlap between them. +

    scikit-learn focuses more +on traditional machine learning methods, such as regression, +clustering, decision trees, etc. As such, it has only two types of +neural networks: Multi Layer Perceptron outputting continuous values, +MPLRegressor, and Multi Layer Perceptron outputting labels, +MLPClassifier. We will see how simple it is to use these classes. +

    + +

    scikit-learn implements a few improvements from our neural network, +such as early stopping, a varying learning rate, different +optimization methods, etc. We would therefore expect a better +performance overall.

    @@ -4225,16 +1583,22 @@ exclusive and there is no overlap between them.
    - 
    import tensorflow as tf
+  
    from sklearn.neural_network import MLPClassifier
+# store models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
 
-from tensorflow.keras import datasets, layers, models
-import matplotlib.pyplot as plt
-
-# We import the data set
-(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
-
-# Normalize pixel values to be between 0 and 1 by dividing by 255. 
-train_images, test_images = train_images / 255.0, test_images / 255.0
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X_train, Y_train)
+        
+        DNN_scikit[i][j] = dnn
+        
+        print("Learning rate  = ", eta)
+        print("Lambda = ", lmbd)
+        print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+        print()
    @@ -4252,10 +1616,7 @@ train_images, test_images = train_images / 255.0
    -

    Verifying the data set

    - -

    To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.

    - +

    Visualization

    @@ -4263,19 +1624,39 @@ train_images, test_images = train_images / 255.0
    - 
    class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
-               'dog', 'frog', 'horse', 'ship', 'truck']
-​
-plt.figure(figsize=(10,10))
-for i in range(25):
-    plt.subplot(5,5,i+1)
-    plt.xticks([])
-    plt.yticks([])
-    plt.grid(False)
-    plt.imshow(train_images[i], cmap=plt.cm.binary)
-    # The CIFAR labels happen to be arrays, 
-    # which is why you need the extra index
-    plt.xlabel(class_names[train_labels[i][0]])
+  
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        dnn = DNN_scikit[i][j]
+        
+        train_pred = dnn.predict(X_train) 
+        test_pred = dnn.predict(X_test)
+
+        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
 plt.show()
    @@ -4294,59 +1675,81 @@ plt.show()
    -

    Set up the model

    +

    Testing our code for the XOR, OR and AND gates

    -

    The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.

    - -

    As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.

    - - - -
    -
    -
    -
    -
    - 
    model = models.Sequential()
-model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-
-# Let's display the architecture of our model so far.
-
-model.summary()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -

    You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.

    -
    - -
    -

    Add Dense layers on top

    - -

    To complete our model, you will feed the last output tensor from the -convolutional base (of shape (4, 4, 64)) into one or more Dense layers -to perform classification. Dense layers take vectors as input (which -are 1D), while the current output is a 3D tensor. First, you will -flatten (or unroll) the 3D output to 1D, then add one or more Dense -layers on top. CIFAR has 10 output classes, so you use a final Dense -layer with 10 outputs and a softmax activation. +

    Last week we discussed three different types of gates, the so-called +XOR, the OR and the AND gates. Their inputs and outputs can be +summarized using the following tables, first for the OR gate with +inputs \( x_1 \) and \( x_2 \) and outputs \( y \):

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 1
    +
    + +
    +

    The AND and XOR Gates

    + +

    The AND gate is defined as

    + + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 0
    1 0 0
    1 1 1
    + +

    And finally we have the XOR gate

    + + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 0
    +
    + +
    +

    Representing the Data Sets

    + +

    Our design matrix is defined by the input values \( x_1 \) and \( x_2 \). Since we have four possible outputs, our design matrix reads

    + +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} 0 & 0 \\ + 0 & 1 \\ + 1 & 0 \\ + 1 & 1 \end{bmatrix}, +$$ +

     
    + +

    while the vector of outputs is \( \boldsymbol{y}^T=[0,1,1,0] \) for the XOR gate, \( \boldsymbol{y}^T=[0,0,0,1] \) for the AND gate and \( \boldsymbol{y}^T=[0,1,1,1] \) for the OR gate.

    +
    + +
    +

    Setting up the Neural Network

    + +

    We define first our design matrix and the various output vectors for the different gates.

    +
    @@ -4354,12 +1757,71 @@ layer with 10 outputs and a softmax activation.
    - 
    model.add(layers.Flatten())
-model.add(layers.Dense(64, activation='relu'))
-model.add(layers.Dense(10))
-Here's the complete architecture of our model.
+  
    """
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
 
-model.summary()
+# import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
+
+def sigmoid(x):
+    return 1/(1 + np.exp(-x))
+
+def feed_forward(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    probabilities = sigmoid(z_o)
+    return probabilities
+
+# we obtain a prediction by taking the class with the highest likelihood
+def predict(X):
+    probabilities = feed_forward(X)
+    return np.argmax(probabilities, axis=1)
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# Design matrix
+X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+# The AND gate
+yAND = np.array( [ 0, 0 ,0, 1])
+
+# Defining the neural network
+n_inputs, n_features = X.shape
+n_hidden_neurons = 2
+n_categories = 2
+n_features = 2
+
+# we make the weights normally distributed using numpy.random.randn
+
+# weights and bias in the hidden layer
+hidden_weights = np.random.randn(n_features, n_hidden_neurons)
+hidden_bias = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+output_weights = np.random.randn(n_hidden_neurons, n_categories)
+output_bias = np.zeros(n_categories) + 0.01
+
+probabilities = feed_forward(X)
+print(probabilities)
+
+
+predictions = predict(X)
+print(predictions)
    @@ -4375,11 +1837,11 @@ model.summary()
    -

    As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.

    +

    Not an impressive result, but this was our first forward pass with randomly assigned weights. Let us now add the full network with the back-propagation algorithm discussed above.

    -

    Compile and train the model

    +

    The Code using Scikit-Learn

    @@ -4388,12 +1850,63 @@ model.summary()
    - 
    model.compile(optimizer='adam',
-              loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
-              metrics=['accuracy'])
-​
-history = model.fit(train_images, train_labels, epochs=10, 
-                    validation_data=(test_images, test_labels))
+  
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.neural_network import MLPClassifier
+from sklearn.metrics import accuracy_score
+import seaborn as sns
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# Design matrix
+X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+# The AND gate
+yAND = np.array( [ 0, 0 ,0, 1])
+
+# Defining the neural network
+n_inputs, n_features = X.shape
+n_hidden_neurons = 2
+n_categories = 2
+n_features = 2
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+epochs = 100
+
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X, yXOR)
+        DNN_scikit[i][j] = dnn
+        print("Learning rate  = ", eta)
+        print("Lambda = ", lmbd)
+        print("Accuracy score on data set: ", dnn.score(X, yXOR))
+        print()
+
+sns.set()
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        dnn = DNN_scikit[i][j]
+        test_pred = dnn.predict(X)
+        test_accuracy[i][j] = accuracy_score(yXOR, test_pred)
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
    @@ -4411,8 +1924,49 @@ history = model.fit(train_images, train_labels, epochs=Finally, evaluate the model +

    Building neural networks in Tensorflow and Keras

    +

    Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn +and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy +and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer. +

    + +

    In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite +clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or +NumPy arrays. +

    +
    + +
    +

    Tensorflow

    + +

    Tensorflow is an open source library machine learning library +developed by the Google Brain team for internal use. It was released +under the Apache 2.0 open source license in November 9, 2015. +

    + +

    Tensorflow is a computational framework that allows you to construct +machine learning models at different levels of abstraction, from +high-level, object-oriented APIs like Keras, down to the C++ kernels +that Tensorflow is built upon. The higher levels of abstraction are +simpler to use, but less flexible, and our choice of implementation +should reflect the problems we are trying to solve. +

    + +

    Tensorflow uses so-called graphs to represent your computation +in terms of the dependencies between individual operations, such that you first build a Tensorflow graph +to represent your model, and then create a Tensorflow session to run the graph. +

    + +

    In this guide we will analyze the same data as we did in our NumPy and +scikit-learn tutorial, gathered from the MNIST database of images. We +will give an introduction to the lower level Python Application +Program Interfaces (APIs), and see how we use them to build our graph. +Then we will build (effectively) the same graph in Keras, to see just +how simple solving a machine learning problem can be. +

    + +

    To install tensorflow on Unix/Linux systems, use pip as

    @@ -4420,16 +1974,7 @@ history = model.fit(train_images, train_labels, epochs=
    - 
    plt.plot(history.history['accuracy'], label='accuracy')
-plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
-plt.xlabel('Epoch')
-plt.ylabel('Accuracy')
-plt.ylim([0.5, 1])
-plt.legend(loc='lower right')
-
-test_loss, test_acc = model.evaluate(test_images,  test_labels, verbose=2)
-
-print(test_acc)
+  
    pip3 install tensorflow
    @@ -4444,6 +1989,830 @@ test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=
    + +

    and/or if you use anaconda, just write (or install from the graphical user interface) +(current release of CPU-only TensorFlow) +

    + + +
    +
    +
    +
    +
    + 
    conda create -n tf tensorflow
+conda activate tf
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    To install the current release of GPU TensorFlow

    + + +
    +
    +
    +
    +
    + 
    conda create -n tf-gpu tensorflow-gpu
+conda activate tf-gpu
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Using Keras

    + +

    Keras is a high level neural network +that supports Tensorflow, CTNK and Theano as backends. +If you have Anaconda installed you may run the following command +

    + + +
    +
    +
    +
    +
    + 
    conda install keras
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    You can look up the instructions here for more information.

    + +

    We will to a large extent use keras in this course.

    +
    + +
    +

    Collect and pre-process data

    + +

    Let us look again at the MINST data set.

    + + + +
    +
    +
    +
    +
    + 
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+import tensorflow as tf
+from sklearn import datasets
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# flatten the image
+# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
+n_inputs = len(inputs)
+inputs = inputs.reshape(n_inputs, -1)
+print("X = (n_inputs, n_features) = " + str(inputs.shape))
+
+
+# choose some random images to display
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +
    +
    +
    +
    + 
    from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
+
+from sklearn.model_selection import train_test_split
+
+# one-hot representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +
    +
    +
    +
    + 
    epochs = 100
+batch_size = 100
+n_neurons_layer1 = 100
+n_neurons_layer2 = 50
+n_categories = 10
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
+    model = Sequential()
+    model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax'))
+    
+    sgd = optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +
    +
    +
    +
    + 
    DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
+                                         eta=eta, lmbd=lmbd)
+        DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = DNN.evaluate(X_test, Y_test)
+        
+        DNN_keras[i][j] = DNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +
    +
    +
    +
    + 
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        DNN = DNN_keras[i][j]
+
+        train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    The Breast Cancer Data, now with Keras

    + + + +
    +
    +
    +
    +
    + 
    import tensorflow as tf
+from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
+import numpy as np
+import matplotlib.pyplot as plt
+import seaborn as sns
+from sklearn.model_selection import train_test_split as splitter
+from sklearn.datasets import load_breast_cancer
+import pickle
+import os 
+
+
+"""Load breast cancer dataset"""
+
+np.random.seed(0)        #create same seed for random number every time
+
+cancer=load_breast_cancer()      #Download breast cancer dataset
+
+inputs=cancer.data                     #Feature matrix of 569 rows (samples) and 30 columns (parameters)
+outputs=cancer.target                  #Label array of 569 rows (0 for benign and 1 for malignant)
+labels=cancer.feature_names[0:30]
+
+print('The content of the breast cancer dataset is:')      #Print information about the datasets
+print(labels)
+print('-------------------------')
+print("inputs =  " + str(inputs.shape))
+print("outputs =  " + str(outputs.shape))
+print("labels =  "+ str(labels.shape))
+
+x=inputs      #Reassign the Feature and Label matrices to other variables
+y=outputs
+
+#%% 
+
+# Visualisation of dataset (for correlation analysis)
+
+plt.figure()
+plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean radius',fontweight='bold')
+plt.ylabel('Mean perimeter',fontweight='bold')
+plt.show()
+
+plt.figure()
+plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)
+plt.xlabel('Mean compactness',fontweight='bold')
+plt.ylabel('Mean concavity',fontweight='bold')
+plt.show()
+
+
+plt.figure()
+plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean radius',fontweight='bold')
+plt.ylabel('Mean texture',fontweight='bold')
+plt.show()
+
+plt.figure()
+plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean perimeter',fontweight='bold')
+plt.ylabel('Mean compactness',fontweight='bold')
+plt.show()
+
+
+# Generate training and testing datasets
+
+#Select features relevant to classification (texture,perimeter,compactness and symmetery) 
+#and add to input matrix
+
+temp1=np.reshape(x[:,1],(len(x[:,1]),1))
+temp2=np.reshape(x[:,2],(len(x[:,2]),1))
+X=np.hstack((temp1,temp2))      
+temp=np.reshape(x[:,5],(len(x[:,5]),1))
+X=np.hstack((X,temp))       
+temp=np.reshape(x[:,8],(len(x[:,8]),1))
+X=np.hstack((X,temp))       
+
+X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1)   #Split datasets into training and testing
+
+y_train=to_categorical(y_train)     #Convert labels to categorical when using categorical cross entropy
+y_test=to_categorical(y_test)
+
+del temp1,temp2,temp
+
+# %%
+
+# Define tunable parameters"
+
+eta=np.logspace(-3,-1,3)                    #Define vector of learning rates (parameter to SGD optimiser)
+lamda=0.01                                  #Define hyperparameter
+n_layers=2                                  #Define number of hidden layers in the model
+n_neuron=np.logspace(0,3,4,dtype=int)       #Define number of neurons per layer
+epochs=100                                   #Number of reiterations over the input data
+batch_size=100                              #Number of samples per gradient update
+
+# %%
+
+"""Define function to return Deep Neural Network model"""
+
+def NN_model(inputsize,n_layers,n_neuron,eta,lamda):
+    model=Sequential()      
+    for i in range(n_layers):       #Run loop to add hidden layers to the model
+        if (i==0):                  #First layer requires input dimensions
+            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))
+        else:                       #Subsequent layers are capable of automatic shape inferencing
+            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))
+    model.add(Dense(2,activation='softmax'))  #2 outputs - ordered and disordered (softmax for prob)
+    sgd=optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])
+    return model
+
+    
+Train_accuracy=np.zeros((len(n_neuron),len(eta)))      #Define matrices to store accuracy scores as a function
+Test_accuracy=np.zeros((len(n_neuron),len(eta)))       #of learning rate and number of hidden neurons for 
+
+for i in range(len(n_neuron)):     #run loops over hidden neurons and learning rates to calculate 
+    for j in range(len(eta)):      #accuracy scores 
+        DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)
+        DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)
+        Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]
+        Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]
+               
+
+def plot_data(x,y,data,title=None):
+
+    # plot results
+    fontsize=16
+
+
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    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()
+    
+plot_data(eta,n_neuron,Train_accuracy, 'training')
+plot_data(eta,n_neuron,Test_accuracy, 'testing')
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +
    +

    Fine-tuning neural network hyperparameters

    + +

    The flexibility of neural networks is also one of their main +drawbacks: there are many hyperparameters to tweak. Not only can you +use any imaginable network topology (how neurons/nodes are interconnected), +but even in a simple FFNN you can change the number of layers, the +number of neurons per layer, the type of activation function to use in +each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you +know what combination of hyperparameters is the best for your task? +

    + + +

    +

    However,since there are many hyperparameters to tune, and since +training a neural network on a large dataset takes a lot of time, you +will only be able to explore a tiny part of the hyperparameter space. +

    + + +
    + +
    +

    Hidden layers

    + +

    For many problems you can start with just one or two hidden layers and it will work just fine. +For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a +few hundred neurons. +You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of +neurons, in roughly the same amount of training time. +

    + +

    For more complex problems, you can gradually +ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such +as large image classification or speech recognition, typically require networks with dozens of layers +and they need a huge amount +of training data. However, you will rarely have to train such networks from scratch: it is much more +common to reuse parts of a pretrained state-of-the-art network that performs a similar task. +

    +
    + +
    +

    Which activation function should I use?

    + +

    The Back propagation algorithm we derived above works by going from +the output layer to the input layer, propagating the error gradient on +the way. Once the algorithm has computed the gradient of the cost +function with regards to each parameter in the network, it uses these +gradients to update each parameter with a Gradient Descent (GD) step. +

    + +

    Unfortunately for us, the gradients often get smaller and smaller as the +algorithm progresses down to the first hidden layers. As a result, the +GD update leaves the lower layer connection weights +virtually unchanged, and training never converges to a good +solution. This is known in the literature as +the vanishing gradients problem. +

    + +

    In other cases, the opposite can happen, namely the the gradients can grow bigger and +bigger. The result is that many of the layers get large updates of the +weights the +algorithm diverges. This is the exploding gradients problem, which is +mostly encountered in recurrent neural networks. More generally, deep +neural networks suffer from unstable gradients, different layers may +learn at widely different speeds +

    +
    + +
    +

    Is the Logistic activation function (Sigmoid) our choice?

    + +

    Although this unfortunate behavior has been empirically observed for +quite a while (it was one of the reasons why deep neural networks were +mostly abandoned for a long time), it is only around 2010 that +significant progress was made in understanding it. +

    + +

    A paper titled Understanding the Difficulty of Training Deep +Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that +the problems with the popular logistic +sigmoid activation function and the weight initialization technique +that was most popular at the time, namely random initialization using +a normal distribution with a mean of 0 and a standard deviation of +1. +

    + +

    They showed that with this activation function and this +initialization scheme, the variance of the outputs of each layer is +much greater than the variance of its inputs. Going forward in the +network, the variance keeps increasing after each layer until the +activation function saturates at the top layers. This is actually made +worse by the fact that the logistic function has a mean of 0.5, not 0 +(the hyperbolic tangent function has a mean of 0 and behaves slightly +better than the logistic function in deep networks). +

    +
    + +
    +

    The derivative of the Logistic funtion

    + +

    Looking at the logistic activation function, when inputs become large +(negative or positive), the function saturates at 0 or 1, with a +derivative extremely close to 0. Thus when backpropagation kicks in, +it has virtually no gradient to propagate back through the network, +and what little gradient exists keeps getting diluted as +backpropagation progresses down through the top layers, so there is +really nothing left for the lower layers. +

    + +

    In their paper, Glorot and Bengio propose a way to significantly +alleviate this problem. We need the signal to flow properly in both +directions: in the forward direction when making predictions, and in +the reverse direction when backpropagating gradients. We don’t want +the signal to die out, nor do we want it to explode and saturate. For +the signal to flow properly, the authors argue that we need the +variance of the outputs of each layer to be equal to the variance of +its inputs, and we also need the gradients to have equal variance +before and after flowing through a layer in the reverse direction. +

    + +

    One of the insights in the 2010 paper by Glorot and Bengio was that +the vanishing/exploding gradients problems were in part due to a poor +choice of activation function. Until then most people had assumed that +if Nature had chosen to use roughly sigmoid activation functions in +biological neurons, they must be an excellent choice. But it turns out +that other activation functions behave much better in deep neural +networks, in particular the ReLU activation function, mostly because +it does not saturate for positive values (and also because it is quite +fast to compute). +

    +
    + +
    +

    The RELU function family

    + +

    The ReLU activation function suffers from a problem known as the dying +ReLUs: during training, some neurons effectively die, meaning they +stop outputting anything other than 0. +

    + +

    In some cases, you may find that half of your network’s neurons are +dead, especially if you used a large learning rate. During training, +if a neuron’s weights get updated such that the weighted sum of the +neuron’s inputs is negative, it will start outputting 0. When this +happen, the neuron is unlikely to come back to life since the gradient +of the ReLU function is 0 when its input is negative. +

    + +

    To solve this problem, nowadays practitioners use a variant of the ReLU +function, such as the leaky ReLU discussed above or the so-called +exponential linear unit (ELU) function +

    + +

     
    +$$ +ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right. +$$ +

     
    +

    + +
    +

    Which activation function should we use?

    + +

    In general it seems that the ELU activation function is better than +the leaky ReLU function (and its variants), which is better than +ReLU. ReLU performs better than \( \tanh \) which in turn performs better +than the logistic function. +

    + +

    If runtime +performance is an issue, then you may opt for the leaky ReLU function over the +ELU function If you don’t +want to tweak yet another hyperparameter, you may just use the default +\( \alpha \) of \( 0.01 \) for the leaky ReLU, and \( 1 \) for ELU. If you have +spare time and computing power, you can use cross-validation or +bootstrap to evaluate other activation functions. +

    +
    + +
    +

    More on activation functions, output layers

    + +

    In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).

    + +

    It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.

    + +For the output layer: + + +
    + +
    +

    Batch Normalization

    + +

    Batch Normalization +aims to address the vanishing/exploding gradients problems, and more generally the problem that the +distribution of each layer’s inputs changes during training, as the parameters of the previous layers change. +

    + +

    The technique consists of adding an operation in the model just before the activation function of each +layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new +parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model +learn the optimal scale and mean of the inputs for each layer. +In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and +standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current +mini-batch, from this the name batch normalization. +

    +
    + +
    +

    Dropout

    + +

    It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but +excluding the output neurons) has a probability \( p \) of being temporarily dropped out, meaning it will be +entirely ignored during this training step, but it may be active during the next step. +

    + +

    The +hyperparameter \( p \) is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore. + It is viewed as one of the most popular regularization techniques. +

    +
    + +
    +

    Gradient Clipping

    + +

    A popular technique to lessen the exploding gradients problem is to simply clip the gradients during +backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural +networks). +

    + +

    This technique is called Gradient Clipping.

    + +

    In general however, Batch +Normalization is preferred. +

    +
    + +
    +

    A very nice website on Neural Networks

    + +

    You may find this website very useful.

    +
    + +
    +

    A top-down perspective on Neural networks

    + +

    The first thing we would like to do is divide the data into two or three +parts. A training set, a validation or dev (development) set, and a +test set. The test set is the data on which we want to make +predictions. The dev set is a subset of the training data we use to +check how well we are doing out-of-sample, after training the model on +the training dataset. We use the validation error as a proxy for the +test error in order to make tweaks to our model. It is crucial that we +do not use any of the test data to train the algorithm. This is a +cardinal sin in ML. Then: +

    + + +

    +

    If the validation and test sets are drawn from the same distributions, +then a good performance on the validation set should lead to similarly +good performance on the test set. +

    + +

    However, sometimes +the training data and test data differ in subtle ways because, for +example, they are collected using slightly different methods, or +because it is cheaper to collect data in one way versus another. In +this case, there can be a mismatch between the training and test +data. This can lead to the neural network overfitting these small +differences between the test and training sets, and a poor performance +on the test set despite having a good performance on the validation +set. To rectify this, Andrew Ng suggests making two validation or dev +sets, one constructed from the training data and one constructed from +the test data. The difference between the performance of the algorithm +on these two validation sets quantifies the train-test mismatch. This +can serve as another important diagnostic when using DNNs for +supervised learning. +

    +
    + +
    +

    Limitations of supervised learning with deep networks

    + +

    Like all statistical methods, supervised learning using neural +networks has important limitations. This is especially important when +one seeks to apply these methods, especially to physics problems. Like +all tools, DNNs are not a universal solution. Often, the same or +better performance on a task can be achieved by using a few +hand-engineered features (or even a collection of random +features). +

    + +

    Here we list some of the important limitations of supervised neural network based models.

    + + +

    +

    Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.

    diff --git a/doc/pub/week42/html/week42-solarized.html b/doc/pub/week42/html/week42-solarized.html index 4385f6122..50fa4de84 100644 --- a/doc/pub/week42/html/week42-solarized.html +++ b/doc/pub/week42/html/week42-solarized.html @@ -8,8 +8,8 @@ doconce format html week42.do.txt --pygments_html_style=perldoc --html_style=sol - -Week 42 Solving differential equations and Convolutional (CNN) + +Week 42 Constructing a Neural Network code with introduction to Tensor flow @@ -64,235 +64,147 @@ div.toc p,a { @@ -314,7 +226,7 @@ MathJax.Hub.Config({
    -

    Week 42 Solving differential equations and Convolutional (CNN)

    +

    Week 42 Constructing a Neural Network code with introduction to Tensor flow

    @@ -326,11 +238,11 @@ MathJax.Hub.Config({ [1] Department of Physics, University of Oslo
    -[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University +[2] Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University

    -

    Oct 21, 2022

    +

    Week 42


    @@ -338,3514 +250,328 @@ MathJax.Hub.Config({

    Plan for week 42

    - +Material for the active learning sessions on Tuesday and Wednesday

    - +
    -Excellent lectures on CNNs and Neural Networks +Material for the lecture on Thursday October 12, 2023

    -
    - -

    m

    - -
    -And Lecture material on CNNs -

    +

  • Building our own Feed-forward Neural Network and discussion of project 2
    • +
  • Readings and Videos:
    • + +

    I also recommend Michael Nielsen's intuitive approach to the neural networks and the universal approximation theorem, see the slides at http://neuralnetworksanddeeplearning.com/chap4.html.











    -

    Using Automatic differentiation

    -

    a -In our discussions of ordinary differential equations -we will also study the usage of Autograd in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from week 39 and the Autograd documentation. -t +

    Lecture Thursday October 12

    + +









    +

    Review of the back propagation algorithm

    + +

    During the last lecture we discussed in detail the back propagation +algorithm. This algorithm is based on a repeated application of the +chain rule. Let us bring back the basic equation and at the same time +link this with the basic mathematics of automatic differentiation.











    -

    Back propagation and automatic differentiation

    +

    Setting up the Back propagation algorithm

    -

    For more details on the back propagation algorithm and automatic differentiation see

    -
      -
    1. https://www.jmlr.org/papers/volume18/17-468/17-468.pdf
      2. -
    2. https://deepimaging.github.io/lectures/lecture_11_Backpropagation.pdf
      3. -
    3. Slides 12-44 at URL":http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf"
      4. -
    -









    -

    Solving ODEs with Deep Learning

    +

    The four equations derived last week provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    -

    The Universal Approximation Theorem states that a neural network can -approximate any function at a single hidden layer along with one input -and output layer to any given precision. +

    First, we set up the input data \( \boldsymbol{x} \) and the activations +\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and +the pertinent outputs \( \boldsymbol{a}^1 \).

    -Book on solving differential equations with ML methods +

    -

    An Introduction to Neural Network Methods for Differential Equations, by Yadav and Kumar.

    +

    Secondly, we perform then the feed forward till we reach the output +layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the +activation function and the pertinent outputs \( \boldsymbol{a}^l \) for +\( l=2,3,\dots,L \). +

    -Master thesis on applying deep learning to problems in mechanics +

    -

    Using Deep Reinforcement Learning for Active Flow Control, by Marius Holm

    +

    Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all

    +$$ +\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. +$$
    -Thanks to Kristine Baluka Hein +

    -

    The lectures on differential equations were developed by Kristine Baluka Hein, now PhD student at IFI. -A great thanks to Kristine. -

    +

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    +$$ +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +$$
    -









    -

    Ordinary Differential Equations

    - -

    An ordinary differential equation (ODE) is an equation involving functions having one variable.

    - -

    In general, an ordinary differential equation looks like

    - +
    + +

    +

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    $$ -\begin{equation} \label{ode} -f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right) = 0 -\end{equation} -$$ - -

    where \( g(x) \) is the function to find, and \( g^{(n)}(x) \) is the \( n \)-th derivative of \( g(x) \).

    - -

    The \( f\left(x, g(x), g'(x), g''(x), \, \dots \, , g^{(n)}(x)\right) \) is just a way to write that there is an expression involving \( x \) and \( g(x), \ g'(x), \ g''(x), \, \dots \, , \text{ and } g^{(n)}(x) \) on the left side of the equality sign in \eqref{ode}. -The highest order of derivative, that is the value of \( n \), determines to the order of the equation. -The equation is referred to as a \( n \)-th order ODE. -Along with \eqref{ode}, some additional conditions of the function \( g(x) \) are typically given -for the solution to be unique. -

    - -









    -

    The trial solution

    - -

    Let the trial solution \( g_t(x) \) be

    - -$$ -\begin{equation} - g_t(x) = h_1(x) + h_2(x,N(x,P)) -\label{_auto1} -\end{equation} -$$ - -

    where \( h_1(x) \) is a function that makes \( g_t(x) \) satisfy a given set -of conditions, \( N(x,P) \) a neural network with weights and biases -described by \( P \) and \( h_2(x, N(x,P)) \) some expression involving the -neural network. The role of the function \( h_2(x, N(x,P)) \), is to -ensure that the output from \( N(x,P) \) is zero when \( g_t(x) \) is -evaluated at the values of \( x \) where the given conditions must be -satisfied. The function \( h_1(x) \) should alone make \( g_t(x) \) satisfy -the conditions. -

    - -

    But what about the network \( N(x,P) \)?

    - -

    As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.

    - -









    -

    Minimization process

    - -

    For the minimization to be defined, we need to have a cost function at hand to minimize.

    - -

    It is given that \( f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right) \) should be equal to zero in \eqref{ode}. -We can choose to consider the mean squared error as the cost function for an input \( x \). -Since we are looking at one input, the cost function is just \( f \) squared. -The cost function \( c\left(x, P \right) \) can therefore be expressed as -

    - -$$ -C\left(x, P\right) = \big(f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)\big)^2 -$$ - -

    If \( N \) inputs are given as a vector \( \boldsymbol{x} \) with elements \( x_i \) for \( i = 1,\dots,N \), -the cost function becomes -

    - -$$ -\begin{equation} \label{cost} - C\left(\boldsymbol{x}, P\right) = \frac{1}{N} \sum_{i=1}^N \big(f\left(x_i, \, g(x_i), \, g'(x_i), \, g''(x_i), \, \dots \, , \, g^{(n)}(x_i)\right)\big)^2 -\end{equation} -$$ - -

    The neural net should then find the parameters \( P \) that minimizes the cost function in -\eqref{cost} for a set of \( N \) training samples \( x_i \). -

    - -









    -

    Minimizing the cost function using gradient descent and automatic differentiation

    - -

    To perform the minimization using gradient descent, the gradient of \( C\left(\boldsymbol{x}, P\right) \) is needed. -It might happen so that finding an analytical expression of the gradient of \( C(\boldsymbol{x}, P) \) from \eqref{cost} gets too messy, depending on which cost function one desires to use. -

    - -

    Luckily, there exists libraries that makes the job for us through automatic differentiation. -Automatic differentiation is a method of finding the derivatives numerically with very high precision. -

    - -









    -

    Example: Exponential decay

    - -

    An exponential decay of a quantity \( g(x) \) is described by the equation

    - -$$ -\begin{equation} \label{solve_expdec} - g'(x) = -\gamma g(x) -\end{equation} -$$ - -

    with \( g(0) = g_0 \) for some chosen initial value \( g_0 \).

    - -

    The analytical solution of \eqref{solve_expdec} is

    - -$$ -\begin{equation} - g(x) = g_0 \exp\left(-\gamma x\right) -\label{_auto2} -\end{equation} -$$ - -

    Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of \eqref{solve_expdec}.

    - -









    -

    The function to solve for

    - -

    The program will use a neural network to solve

    - -$$ -\begin{equation} \label{solveode} -g'(x) = -\gamma g(x) -\end{equation} -$$ - -

    where \( g(0) = g_0 \) with \( \gamma \) and \( g_0 \) being some chosen values.

    - -

    In this example, \( \gamma = 2 \) and \( g_0 = 10 \).

    - -









    -

    The trial solution

    -

    To begin with, a trial solution \( g_t(t) \) must be chosen. A general trial solution for ordinary differential equations could be

    - -$$ -g_t(x, P) = h_1(x) + h_2(x, N(x, P)) -$$ - -

    with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and \( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the neural network \( N(x,P) \) with \( P \) being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.

    - -









    -

    Setup of Network

    - -

    In this network, there are no weights and bias at the input layer, so \( P = \{ P_{\text{hidden}}, P_{\text{output}} \} \). -If there are \( N_{\text{hidden} } \) neurons in the hidden layer, then \( P_{\text{hidden}} \) is a \( N_{\text{hidden} } \times (1 + N_{\text{input}}) \) matrix, given that there are \( N_{\text{input}} \) neurons in the input layer. -

    - -

    The first column in \( P_{\text{hidden} } \) represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer. -If there are \( N_{\text{output} } \) neurons in the output layer, then \( P_{\text{output}} \) is a \( N_{\text{output} } \times (1 + N_{\text{hidden} }) \) matrix. -

    - -

    Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.

    - -

    It is given that \( g(0) = g_0 \). The trial solution must fulfill this condition to be a proper solution of \eqref{solveode}. A possible way to ensure that \( g_t(0, P) = g_0 \), is to let \( F(N(x,P)) = x \cdot N(x,P) \) and \( A(x) = g_0 \). This gives the following trial solution:

    - -$$ -\begin{equation} \label{trial} -g_t(x, P) = g_0 + x \cdot N(x, P) -\end{equation} +w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ -









    -

    Reformulating the problem

    - -

    We wish that our neural network manages to minimize a given cost function.

    - -

    A reformulation of out equation, \eqref{solveode}, must therefore be done, -such that it describes the problem a neural network can solve for. -

    - -

    The neural network must find the set of weights and biases \( P \) such that the trial solution in \eqref{trial} satisfies \eqref{solveode}.

    - -

    The trial solution

    - $$ -g_t(x, P) = g_0 + x \cdot N(x, P) +b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ - -

    has been chosen such that it already solves the condition \( g(0) = g_0 \). What remains, is to find \( P \) such that

    - -$$ -\begin{equation} \label{nnmin} -g_t'(x, P) = - \gamma g_t(x, P) -\end{equation} -$$ - -

    is fulfilled as best as possible.

    - -









    -

    More technicalities

    - -

    The left hand side and right hand side of \eqref{nnmin} must be computed separately, and then the neural network must choose weights and biases, contained in \( P \), such that the sides are equal as best as possible. -This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero. -In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to \( P \) of the neural network. -

    - -

    This gives the following cost function our neural network must solve for:

    - -$$ -\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} -$$ - -

    (the notation \( \min_{P}\{ f(x, P) \} \) means that we desire to find \( P \) that yields the minimum of \( f(x, P) \))

    - -

    or, in terms of weights and biases for the hidden and output layer in our network:

    - -$$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\} -$$ - -

    for an input value \( x \).

    - -









    -

    More details

    - -

    If the neural network evaluates \( g_t(x, P) \) at more values for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the total error to minimize becomes

    - -$$ -\begin{equation} \label{min} -\min_{P}\Big\{\frac{1}{N} \sum_{i=1}^N \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} -\end{equation} -$$ - -

    Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes

    - -$$ -\min_{P} C(\boldsymbol{x}, P) -$$ - -

    In terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \), this could also be expressed as

    - -

    $$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) -$$ -

    - -









    -

    A possible implementation of a neural network

    - -

    For simplicity, it is assumed that the input is an array \( \boldsymbol{x} = (x_1, \dots, x_N) \) with \( N \) elements. It is at these points the neural network should find \( P \) such that it fulfills \eqref{min}.

    - -

    First, the neural network must feed forward the inputs. -This means that \( \boldsymbol{x}s \) must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further. -The input layer will consist of \( N_{\text{input} } \) neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be \( N_{\text{hidden} } \). -

    - -









    -

    Technicalities

    - -

    For the \( i \)-th in the hidden layer with weight \( w_i^{\text{hidden} } \) and bias \( b_i^{\text{hidden} } \), the weighting from the \( j \)-th neuron at the input layer is:

    - -$$ -\begin{aligned} -z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\ -&= -\begin{pmatrix} -b_i^{\text{hidden}} & w_i^{\text{hidden}} -\end{pmatrix} -\begin{pmatrix} -1 \\ -x_j -\end{pmatrix} -\end{aligned} -$$ - - -









    -

    Final technicalities I

    - -

    The result after weighting the inputs at the \( i \)-th hidden neuron can be written as a vector:

    - -$$ -\begin{aligned} -\boldsymbol{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\ -&= -\begin{pmatrix} - b_i^{\text{hidden}} & w_i^{\text{hidden}} -\end{pmatrix} -\begin{pmatrix} -1 & 1 & \dots & 1 \\ -x_1 & x_2 & \dots & x_N -\end{pmatrix} \\ -&= \boldsymbol{p}_{i, \text{hidden}}^T X -\end{aligned} -$$ - - -









    -

    Final technicalities II

    - -

    The vector \( \boldsymbol{p}_{i, \text{hidden}}^T \) constitutes each row in \( P_{\text{hidden} } \), which contains the weights for the neural network to minimize according to \eqref{min}.

    - -

    After having found \( \boldsymbol{z}_{i}^{\text{hidden}} \) for every \( i \)-th neuron within the hidden layer, the vector will be sent to an activation function \( a_i(\boldsymbol{z}) \).

    - -

    In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:

    - -$$ -f(z) = \frac{1}{1 + \exp{(-z)}} -$$ - -

    It is possible to use other activations functions for the hidden layer also.

    - -

    The output \( \boldsymbol{x}_i^{\text{hidden}} \) from each \( i \)-th hidden neuron is:

    - -

    $$ -\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big) -$$ -

    - -

    The outputs \( \boldsymbol{x}_i^{\text{hidden} } \) are then sent to the output layer.

    - -

    The output layer consists of one neuron in this case, and combines the -output from each of the neurons in the hidden layers. The output layer -combines the results from the hidden layer using some weights \( w_i^{\text{output}} \) -and biases \( b_i^{\text{output}} \). In this case, -it is assumes that the number of neurons in the output layer is one. -

    - -









    -

    Final technicalities III

    - -

    The procedure of weighting the output neuron \( j \) in the hidden layer to the \( i \)-th neuron in the output layer is similar as for the hidden layer described previously.

    - -$$ -\begin{aligned} -z_{1,j}^{\text{output}} & = -\begin{pmatrix} -b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}} -\end{pmatrix} -\begin{pmatrix} -1 \\ -\boldsymbol{x}_j^{\text{hidden}} -\end{pmatrix} -\end{aligned} -$$ - - -









    -

    Final technicalities IV

    - -

    Expressing \( z_{1,j}^{\text{output}} \) as a vector gives the following way of weighting the inputs from the hidden layer:

    - -$$ -\boldsymbol{z}_{1}^{\text{output}} = -\begin{pmatrix} -b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}} -\end{pmatrix} -\begin{pmatrix} -1 & 1 & \dots & 1 \\ -\boldsymbol{x}_1^{\text{hidden}} & \boldsymbol{x}_2^{\text{hidden}} & \dots & \boldsymbol{x}_N^{\text{hidden}} -\end{pmatrix} -$$ - -

    In this case we seek a continuous range of values since we are approximating a function. This means that after computing \( \boldsymbol{z}_{1}^{\text{output}} \) the neural network has finished its feed forward step, and \( \boldsymbol{z}_{1}^{\text{output}} \) is the final output of the network.

    - -









    -

    Back propagation

    - -

    The next step is to decide how the parameters should be changed such that they minimize the cost function.

    - -

    The chosen cost function for this problem is

    - -$$ -C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 -$$ - -

    In order to minimize the cost function, an optimization method must be chosen.

    - -

    Here, gradient descent with a constant step size has been chosen.

    - -









    -

    Gradient descent

    - -

    The idea of the gradient descent algorithm is to update parameters in -a direction where the cost function decreases goes to a minimum. -

    - -

    In general, the update of some parameters \( \boldsymbol{\omega} \) given a cost -function defined by some weights \( \boldsymbol{\omega} \), \( C(\boldsymbol{x}, -\boldsymbol{\omega}) \), goes as follows: -

    - -$$ -\boldsymbol{\omega}_{\text{new} } = \boldsymbol{\omega} - \lambda \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega}) -$$ - -

    for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|$ becomes smaller than some given tolerance.

    - -

    The value of \( \lambda \) decides how large steps the algorithm must take -in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$. -The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect -to the elements in \( \boldsymbol{\omega} \). -

    - -

    In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with -respect to the two sets of weights and biases, that is for the hidden -layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} -} \) . -

    - -

    This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is updated by

    - -$$ -\begin{aligned} -P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\ -P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P) -\end{aligned} -$$ - - -









    -

    The code for solving the ODE

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-# Assuming one input, hidden, and output layer
-def neural_network(params, x):
-
-    # Find the weights (including and biases) for the hidden and output layer.
-    # Assume that params is a list of parameters for each layer.
-    # The biases are the first element for each array in params,
-    # and the weights are the remaning elements in each array in params.
-
-    w_hidden = params[0]
-    w_output = params[1]
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    ## Hidden layer:
-
-    # Add a row of ones to include bias
-    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
-
-    z_hidden = np.matmul(w_hidden, x_input)
-    x_hidden = sigmoid(z_hidden)
-
-    ## Output layer:
-
-    # Include bias:
-    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
-
-    z_output = np.matmul(w_output, x_hidden)
-    x_output = z_output
-
-    return x_output
-
-# The trial solution using the deep neural network:
-def g_trial(x,params, g0 = 10):
-    return g0 + x*neural_network(params,x)
-
-# The right side of the ODE:
-def g(x, g_trial, gamma = 2):
-    return -gamma*g_trial
-
-# The cost function:
-def cost_function(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial(x,P)
-
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(neural_network,1)(P,x)
-
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial,0)(x,P)
-
-    # The right side of the ODE
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum / np.size(err_sqr)
-
-# Solve the exponential decay ODE using neural network with one input, hidden, and output layer
-def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
-    ## Set up initial weights and biases
-
-    # For the hidden layer
-    p0 = npr.randn(num_neurons_hidden, 2 )
-
-    # For the output layer
-    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
-
-    P = [p0, p1]
-
-    print('Initial cost: %g'%cost_function(P, x))
-
-    ## Start finding the optimal weights using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_grad = grad(cost_function,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of two arrays;
-        # one for the gradient w.r.t P_hidden and
-        # one for the gradient w.r.t P_output
-        cost_grad =  cost_function_grad(P, x)
-
-        P[0] = P[0] - lmb * cost_grad[0]
-        P[1] = P[1] - lmb * cost_grad[1]
-
-    print('Final cost: %g'%cost_function(P, x))
-
-    return P
-
-def g_analytic(x, gamma = 2, g0 = 10):
-    return g0*np.exp(-gamma*x)
-
-# Solve the given problem
-if __name__ == '__main__':
-    # Set seed such that the weight are initialized
-    # with same weights and biases for every run.
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    N = 10
-    x = np.linspace(0, 1, N)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = 10
-    num_iter = 10000
-    lmb = 0.001
-
-    # Use the network
-    P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    # Print the deviation from the trial solution and true solution
-    res = g_trial(x,P)
-    res_analytical = g_analytic(x)
-
-    print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
-
-    # Plot the results
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, res_analytical)
-    plt.plot(x, res[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('x')
-    plt.ylabel('g(x)')
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -









    -

    The network with one input layer, specified number of hidden layers, and one output layer

    - -

    It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.

    - -

    The number of neurons within each hidden layer are given as a list of integers in the program below.

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-# The neural network with one input layer and one output layer,
-# but with number of hidden layers specified by the user.
-def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers
-
-    N_hidden = np.size(deep_params) - 1 # -1 since params consists of
-                                        # parameters to all the hidden
-                                        # layers AND the output layer.
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-# The trial solution using the deep neural network:
-def g_trial_deep(x,params, g0 = 10):
-    return g0 + x*deep_neural_network(params, x)
-
-# The right side of the ODE:
-def g(x, g_trial, gamma = 2):
-    return -gamma*g_trial
-
-# The same cost function as before, but calls deep_neural_network instead.
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(deep_neural_network,1)(P,x)
-
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
-
-    # The right side of the ODE
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum / np.size(err_sqr)
-
-# Solve the exponential decay ODE using neural network with one input and one output layer,
-# but with specified number of hidden layers from the user.
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # The number of elements in the list num_hidden_neurons thus represents
-    # the number of hidden layers.
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weights and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weights using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-def g_analytic(x, gamma = 2, g0 = 10):
-    return g0*np.exp(-gamma*x)
-
-# Solve the given problem
-if __name__ == '__main__':
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    N = 10
-    x = np.linspace(0, 1, N)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = np.array([10,10])
-    num_iter = 10000
-    lmb = 0.001
-
-    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    res = g_trial_deep(x,P)
-    res_analytical = g_analytic(x)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, res_analytical)
-    plt.plot(x, res[0,:])
-    plt.legend(['analytical','dnn'])
-    plt.ylabel('g(x)')
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Example: Population growth

    - -

    A logistic model of population growth assumes that a population converges toward an equilibrium. -The population growth can be modeled by +

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. +Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.

    -$$ -\begin{equation} \label{log} - g'(t) = \alpha g(t)(A - g(t)) -\end{equation} -$$ + +

    Setting up a Multi-layer perceptron model for classification

    -

    where \( g(t) \) is the population density at time \( t \), \( \alpha > 0 \) the growth rate and \( A > 0 \) is the maximum population number in the environment. -Also, at \( t = 0 \) the population has the size \( g(0) = g_0 \), where \( g_0 \) is some chosen constant. +

    We are now gong to develop an example based on the MNIST data +base. This is a classification problem and we need to use our +cross-entropy function we discussed in connection with logistic +regression. The cross-entropy defines our cost function for the +classificaton problems with neural networks.

    -

    In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability -and high execution time (this might be more apparent in the examples solving PDEs), -using a library like TensorFlow is recommended. -Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method. +

    In binary classification with two classes \( (0, 1) \) we define the +logistic/sigmoid function as the probability that a particular input +is in class \( 0 \) or \( 1 \). This is possible because the logistic +function takes any input from the real numbers and inputs a number +between 0 and 1, and can therefore be interpreted as a probability. It +also has other nice properties, such as a derivative that is simple to +calculate.

    -









    -

    Setting up the problem

    - -

    Here, we will model a population \( g(t) \) in an environment having carrying capacity \( A \). -The population follows the model +

    For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \) +is in class 0 or 1 is just. We let \( \theta \) represent the unknown weights and biases to be adjusted by our equations). The variable \( x \) +represents our activation values \( z \). We have

    - $$ -\begin{equation} \label{solveode_population} -g'(t) = \alpha g(t)(A - g(t)) -\end{equation} -$$ - -

    where \( g(0) = g_0 \).

    - -

    In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).

    - -









    -

    The trial solution

    - -

    We will get a slightly different trial solution, as the boundary conditions are different -compared to the case for exponential decay. -

    - -

    A possible trial solution satisfying the condition \( g(0) = g_0 \) could be

    - -

    $$ -h_1(t) = g_0 + t \cdot N(t,P) -$$ -

    - -

    with \( N(t,P) \) being the output from the neural network with weights and biases for each layer collected in the set \( P \).

    - -

    The analytical solution is

    - -

    $$ -g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)} -$$ -

    - -









    -

    The program using Autograd

    - -

    The network will be the similar as for the exponential decay example, but with some small modifications for our problem.

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-# Function to get the parameters.
-# Done such that one can easily change the paramaters after one's liking.
-def get_parameters():
-    alpha = 2
-    A = 1
-    g0 = 1.2
-    return alpha, A, g0
-
-def deep_neural_network(P, x):
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = P[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = P[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
-
-    # The right side of the ODE
-    func = f(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum / np.size(err_sqr)
-
-# The right side of the ODE:
-def f(x, g_trial):
-    alpha,A, g0 = get_parameters()
-    return alpha*g_trial*(A - g_trial)
-
-# The trial solution using the deep neural network:
-def g_trial_deep(x, params):
-    alpha,A, g0 = get_parameters()
-    return g0 + x*deep_neural_network(params,x)
-
-# The analytical solution:
-def g_analytic(t):
-    alpha,A, g0 = get_parameters()
-    return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weigths using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nt = 10
-    T = 1
-    t = np.linspace(0,T, Nt)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [100, 50, 25]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(t,P)
-    g_analytical = g_analytic(t)
-
-    # Find the maximum absolute difference between the solutons:
-    diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))
-    print("The max absolute difference between the solutions is: %g"%diff_ag)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(t, g_analytical)
-    plt.plot(t, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('t')
-    plt.ylabel('g(t)')
-
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Using forward Euler to solve the ODE

    - -

    A straightforward way of solving an ODE numerically, is to use Euler's method.

    - -

    Euler's method uses Taylor series to approximate the value at a function \( f \) at a step \( \Delta x \) from \( x \):

    - -

    $$ -f(x + \Delta x) \approx f(x) + \Delta x f'(x) -$$ -

    - -

    In our case, using Euler's method to approximate the value of \( g \) at a step \( \Delta t \) from \( t \) yields

    - -$$ -\begin{aligned} - g(t + \Delta t) &\approx g(t) + \Delta t g'(t) \\ - &= g(t) + \Delta t \big(\alpha g(t)(A - g(t))\big) -\end{aligned} -$$ - -

    along with the condition that \( g(0) = g_0 \).

    - -

    Let \( t_i = i \cdot \Delta t \) where \( \Delta t = \frac{T}{N_t-1} \) where \( T \) is the final time our solver must solve for and \( N_t \) the number of values for \( t \in [0, T] \) for \( i = 0, \dots, N_t-1 \).

    - -

    For \( i \geq 1 \), we have that

    -$$ -\begin{aligned} -t_i &= i\Delta t \\ -&= (i - 1)\Delta t + \Delta t \\ -&= t_{i-1} + \Delta t -\end{aligned} -$$ - -

    Now, if \( g_i = g(t_i) \) then

    - -$$ -\begin{equation} - \begin{aligned} - g_i &= g(t_i) \\ - &= g(t_{i-1} + \Delta t) \\ - &\approx g(t_{i-1}) + \Delta t \big(\alpha g(t_{i-1})(A - g(t_{i-1}))\big) \\ - &= g_{i-1} + \Delta t \big(\alpha g_{i-1}(A - g_{i-1})\big) - \end{aligned} -\end{equation} \label{odenum} -$$ - -

    for \( i \geq 1 \) and \( g_0 = g(t_0) = g(0) = g_0 \).

    - -

    Equation \eqref{odenum} could be implemented in the following way, -extending the program that uses the network using Autograd: -

    - - - -
    -
    -
    -
    -
    - 
    # Assume that all function definitions from the example program using Autograd
-# are located here.
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nt = 10
-    T = 1
-    t = np.linspace(0,T, Nt)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [100,50,25]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(t,P)
-    g_analytical = g_analytic(t)
-
-    # Find the maximum absolute difference between the solutons:
-    diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))
-    print("The max absolute difference between the solutions is: %g"%diff_ag)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(t, g_analytical)
-    plt.plot(t, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('t')
-    plt.ylabel('g(t)')
-
-    ## Find an approximation to the funtion using forward Euler
-
-    alpha, A, g0 = get_parameters()
-    dt = T/(Nt - 1)
-
-    # Perform forward Euler to solve the ODE
-    g_euler = np.zeros(Nt)
-    g_euler[0] = g0
-
-    for i in range(1,Nt):
-        g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))
-
-    # Print the errors done by each method
-    diff1 = np.max(np.abs(g_euler - g_analytical))
-    diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))
-
-    print('Max absolute difference between Euler method and analytical: %g'%diff1)
-    print('Max absolute difference between deep neural network and analytical: %g'%diff2)
-
-    # Plot results
-    plt.figure(figsize=(10,10))
-
-    plt.plot(t,g_euler)
-    plt.plot(t,g_analytical)
-    plt.plot(t,g_dnn_ag[0,:])
-
-    plt.legend(['euler','analytical','dnn'])
-    plt.xlabel('Time t')
-    plt.ylabel('g(t)')
-
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Example: Solving the one dimensional Poisson equation

    - -

    The Poisson equation for \( g(x) \) in one dimension is

    - -$$ -\begin{equation} \label{poisson} - -g''(x) = f(x) -\end{equation} -$$ - -

    where \( f(x) \) is a given function for \( x \in (0,1) \).

    - -

    The conditions that \( g(x) \) is chosen to fulfill, are

    -$$ -\begin{align*} - g(0) &= 0 \\ - g(1) &= 0 -\end{align*} -$$ - -

    This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used. -The results from the networks can then be compared to the analytical solution. -In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks. -

    - -









    -

    The specific equation to solve for

    - -

    Here, the function \( g(x) \) to solve for follows the equation

    - -$$ --g''(x) = f(x),\qquad x \in (0,1) -$$ - -

    where \( f(x) \) is a given function, along with the chosen conditions

    - -$$ -\begin{aligned} -g(0) = g(1) = 0 -\end{aligned}\label{cond} -$$ - -

    In this example, we consider the case when \( f(x) = (3x + x^2)\exp(x) \).

    - -

    For this case, a possible trial solution satisfying the conditions could be

    - -$$ -g_t(x) = x \cdot (1-x) \cdot N(P,x) -$$ - -

    The analytical solution for this problem is

    - -$$ -g(x) = x(1 - x)\exp(x) -$$ - - -









    -

    Solving the equation using Autograd

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weigths using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-## Set up the cost function specified for this Poisson equation:
-
-# The right side of the ODE
-def f(x):
-    return (3*x + x**2)*np.exp(x)
-
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the trial function
-    d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)
-
-    right_side = f(x)
-
-    err_sqr = (-d2_g_t - right_side)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum/np.size(err_sqr)
-
-# The trial solution:
-def g_trial_deep(x,P):
-    return x*(1-x)*deep_neural_network(P,x)
-
-# The analytic solution;
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10
-    x = np.linspace(0,1, Nx)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [200,100]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(x,P)
-    g_analytical = g_analytic(x)
-
-    # Find the maximum absolute difference between the solutons:
-    max_diff = np.max(np.abs(g_dnn_ag - g_analytical))
-    print("The max absolute difference between the solutions is: %g"%max_diff)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, g_analytical)
-    plt.plot(x, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('x')
-    plt.ylabel('g(x)')
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Comparing with a numerical scheme

    - -

    The Poisson equation is possible to solve using Taylor series to approximate the second derivative.

    - -

    Using Taylor series, the second derivative can be expressed as

    - -

    $$ -g''(x) = \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} + E_{\Delta x}(x) -$$ -

    - -

    where \( \Delta x \) is a small step size and \( E_{\Delta x}(x) \) being the error term.

    - -

    Looking away from the error terms gives an approximation to the second derivative:

    - -$$ -\begin{equation} \label{approx} -g''(x) \approx \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} -\end{equation} -$$ - -

    If \( x_i = i \Delta x = x_{i-1} + \Delta x \) and \( g_i = g(x_i) \) for \( i = 1,\dots N_x - 2 \) with \( N_x \) being the number of values for \( x \), \eqref{approx} becomes

    - -$$ -\begin{aligned} -g''(x_i) &\approx \frac{g(x_i + \Delta x) - 2g(x_i) + g(x_i -\Delta x)}{\Delta x^2} \\ -&= \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} -\end{aligned} -$$ - -

    Since we know from our problem that

    - -$$ -\begin{aligned} --g''(x) &= f(x) \\ -&= (3x + x^2)\exp(x) -\end{aligned} -$$ - -

    along with the conditions \( g(0) = g(1) = 0 \), -the following scheme can be used to find an approximate solution for \( g(x) \) numerically: -

    - -$$ -\begin{equation} - \begin{aligned} - -\Big( \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} \Big) &= f(x_i) \\ - -g_{i+1} + 2g_i - g_{i-1} &= \Delta x^2 f(x_i) - \end{aligned} -\end{equation} \label{odesys} -$$ - -

    for \( i = 1, \dots, N_x - 2 \) where \( g_0 = g_{N_x - 1} = 0 \) and \( f(x_i) = (3x_i + x_i^2)\exp(x_i) \), which is given for our specific problem.

    - -

    The equation can be rewritten into a matrix equation:

    - -$$ -\begin{aligned} -\begin{pmatrix} -2 & -1 & 0 & \dots & 0 \\ --1 & 2 & -1 & \dots & 0 \\ -\vdots & & \ddots & & \vdots \\ -0 & \dots & -1 & 2 & -1 \\ -0 & \dots & 0 & -1 & 2\\ -\end{pmatrix} -\begin{pmatrix} -g_1 \\ -g_2 \\ -\vdots \\ -g_{N_x - 3} \\ -g_{N_x - 2} -\end{pmatrix} -&= -\Delta x^2 -\begin{pmatrix} -f(x_1) \\ -f(x_2) \\ -\vdots \\ -f(x_{N_x - 3}) \\ -f(x_{N_x - 2}) -\end{pmatrix} \\ -\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f}, -\end{aligned} -$$ - -

    which makes it possible to solve for the vector \( \boldsymbol{g} \).

    - -









    -

    Setting up the code

    - -

    We can then compare the result from this numerical scheme with the output from our network using Autograd:

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weigths using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-## Set up the cost function specified for this Poisson equation:
-
-# The right side of the ODE
-def f(x):
-    return (3*x + x**2)*np.exp(x)
-
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the trial function
-    d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)
-
-    right_side = f(x)
-
-    err_sqr = (-d2_g_t - right_side)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum/np.size(err_sqr)
-
-# The trial solution:
-def g_trial_deep(x,P):
-    return x*(1-x)*deep_neural_network(P,x)
-
-# The analytic solution;
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10
-    x = np.linspace(0,1, Nx)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [200,100]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(x,P)
-    g_analytical = g_analytic(x)
-
-    # Find the maximum absolute difference between the solutons:
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, g_analytical)
-    plt.plot(x, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('x')
-    plt.ylabel('g(x)')
-
-    ## Perform the computation using the numerical scheme
-
-    dx = 1/(Nx - 1)
-
-    # Set up the matrix A
-    A = np.zeros((Nx-2,Nx-2))
-
-    A[0,0] = 2
-    A[0,1] = -1
-
-    for i in range(1,Nx-3):
-        A[i,i-1] = -1
-        A[i,i] = 2
-        A[i,i+1] = -1
-
-    A[Nx - 3, Nx - 4] = -1
-    A[Nx - 3, Nx - 3] = 2
-
-    # Set up the vector f
-    f_vec = dx**2 * f(x[1:-1])
-
-    # Solve the equation
-    g_res = np.linalg.solve(A,f_vec)
-
-    g_vec = np.zeros(Nx)
-    g_vec[1:-1] = g_res
-
-    # Print the differences between each method
-    max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))
-    max_diff2 = np.max(np.abs(g_vec - g_analytical))
-    print("The max absolute difference between the analytical solution and DNN Autograd: %g"%max_diff1)
-    print("The max absolute difference between the analytical solution and numerical scheme: %g"%max_diff2)
-
-    # Plot the results
-    plt.figure(figsize=(10,10))
-
-    plt.plot(x,g_vec)
-    plt.plot(x,g_analytical)
-    plt.plot(x,g_dnn_ag[0,:])
-
-    plt.legend(['numerical scheme','analytical','dnn'])
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Partial Differential Equations

    - -

    A partial differential equation (PDE) has a solution here the function -is defined by multiple variables. The equation may involve all kinds -of combinations of which variables the function is differentiated with -respect to. -

    - -

    In general, a partial differential equation for a function \( g(x_1,\dots,x_N) \) with \( N \) variables may be expressed as

    - -$$ -\begin{equation} \label{PDE} - f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) = 0 -\end{equation} -$$ - -

    where \( f \) is an expression involving all kinds of possible mixed derivatives of \( g(x_1,\dots,x_N) \) up to an order \( n \). In order for the solution to be unique, some additional conditions must also be given.

    - -









    -

    Type of problem

    - -

    The problem our network must solve for, is similar to the ODE case. -We must have a trial solution \( g_t \) at hand. -

    - -

    For instance, the trial solution could be expressed as

    -$$ -\begin{align*} - g_t(x_1,\dots,x_N) = h_1(x_1,\dots,x_N) + h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) -\end{align*} -$$ - -

    where \( h_1(x_1,\dots,x_N) \) is a function that ensures \( g_t(x_1,\dots,x_N) \) satisfies some given conditions. -The neural network \( N(x_1,\dots,x_N,P) \) has weights and biases described by \( P \) and \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \) is an expression using the output from the neural network in some way. -

    - -

    The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensure that the output of \( N(x_1,\dots,x_N,P) \) is zero when \( g_t(x_1,\dots,x_N) \) is evaluated at the values of \( x_1,\dots,x_N \) where the given conditions must be satisfied. The function \( h_1(x_1,\dots,x_N) \) should alone make \( g_t(x_1,\dots,x_N) \) satisfy the conditions.

    - -









    -

    Network requirements

    - -

    The network tries then the minimize the cost function following the -same ideas as described for the ODE case, but now with more than one -variables to consider. The concept still remains the same; find a set -of parameters \( P \) such that the expression \( f \) in \eqref{PDE} is as -close to zero as possible. -

    - -

    As for the ODE case, the cost function is the mean squared error that -the network must try to minimize. The cost function for the network to -minimize is -

    - -$$ -\begin{equation*} -C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2 -\end{equation*} -$$ - - -









    -

    More details

    - -

    If we let \( \boldsymbol{x} = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:

    -$$ - C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2 -$$ - -

    If we also have \( M \) different sets of values for \( x_1, \dots, x_N \), that is \( \boldsymbol{x}_i = \big(x_1^{(i)}, \dots, x_N^{(i)}\big) \) for \( i = 1,\dots,M \) being the rows in matrix \( X \), the cost function can be generalized into

    -$$ -\begin{equation*} -C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2. -\end{equation*} -$$ - - -









    -

    Example: The diffusion equation

    - -

    In one spatial dimension, the equation reads

    -$$ -\begin{equation*} - \frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation*} -$$ - -

    where a possible choice of conditions are

    -$$ -\begin{align*} -g(0,t) &= 0 ,\qquad t \geq 0 \\ -g(1,t) &= 0, \qquad t \geq 0 \\ -g(x,0) &= u(x),\qquad x\in [0,1] -\end{align*} -$$ - -

    with \( u(x) \) being some given function.

    - -









    -

    Defining the problem

    - -

    For this case, we want to find \( g(x,t) \) such that

    - -$$ -\begin{equation} - \frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation} \label{diffonedim} +P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp{(- \boldsymbol{x}})} , $$

    and

    - $$ -\begin{align*} -g(0,t) &= 0 ,\qquad t \geq 0 \\ -g(1,t) &= 0, \qquad t \geq 0 \\ -g(x,0) &= u(x),\qquad x\in [0,1] -\end{align*} +P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) , $$ -

    with \( u(x) = \sin(\pi x) \).

    - -

    First, let us set up the deep neural network. -The deep neural network will follow the same structure as discussed in the examples solving the ODEs. -First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions. +

    where \( y \in \{0, 1\} \) and \( \boldsymbol{\theta} \) represents the weights and biases +of our network.

    -









    -

    Setting up the network using Autograd

    - -

    The only change to do here, is to extend our network such that -functions of multiple parameters are correctly handled. In this case -we have two variables in our function to solve for, that is time \( t \) -and position \( x \). The variables will be represented by a -one-dimensional array in the program. The program will evaluate the -network at each possible pair \( (x,t) \), given an array for the desired -\( x \)-values and \( t \)-values to approximate the solution at. -

    - - - -
    -
    -
    -
    -
    - 
    def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # x is now a point and a 1D numpy array; make it a column vector
-    num_coordinates = np.size(x,0)
-    x = x.reshape(num_coordinates,-1)
-
-    num_points = np.size(x,1)
-
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output[0][0]
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -









    -

    Setting up the network using Autograd; The trial solution

    +

    Defining the cost function

    -

    The cost function must then iterate through the given arrays -containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep -neural network and the trial solution is evaluated at, and then finds -the Jacobian of the trial solution. -

    - -

    A possible trial solution for this PDE is

    - -

    $$ -g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P) +

    Our cost function is given as (see the Logistic regression lectures)

    $$ -

    - -

    with \( A(x,t) \) being a function ensuring that \( g_t(x,t) \) satisfies our given conditions, and \( N(x,t,P) \) being the output from the deep neural network using weights and biases for each layer from \( P \).

    - -

    To fulfill the conditions, \( A(x,t) \) could be:

    - -

    $$ -h_1(x,t) = (1-t)\Big(u(x) - \big((1-x)u(0) + x u(1)\big)\Big) = (1-t)u(x) = (1-t)\sin(\pi x) +\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n +y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) . $$ -since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \). + +

    This last equality means that we can interpret our cost function as a sum over the loss function +for each point in the dataset \( \mathcal{L}_i(\boldsymbol{\theta}) \). +The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather +than maximizing a negative number.

    +

    In multiclass classification it is common to treat each integer label as a so called one-hot vector:

    + +

    \( y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and

    + +\( y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \) + +

    i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \))..

    + +

    If \( \boldsymbol{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th +output vector \( \boldsymbol{y}_i \). +The probability of \( \boldsymbol{x}_i \) being in class \( c \) will be given by the softmax function: +

    + +$$ +P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}} +{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} , +$$ + +

    which reduces to the logistic function in the binary case. +The likelihood of this \( C \)-class classifier +is now given as: +

    + +$$ +P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} . +$$ + +

    Again we take the negative log-likelihood to define our cost function:

    + +$$ +\mathcal{C}(\boldsymbol{\theta}) = - \log{P(\mathcal{D} \mid \boldsymbol{\theta})}. +$$ + +

    See the logistic regression lectures for a full definition of the cost function.

    + +

    The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!

    +









    -

    Why the jacobian?

    +

    Example: binary classification problem

    -

    The Jacobian is used because the program must find the derivative of -the trial solution with respect to \( x \) and \( t \). +

    As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \( \beta \) as

    +$$ +\mathcal{C}(\boldsymbol{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\boldsymbol{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\boldsymbol{\beta})}\right), +$$ + +

    where we had defined the logistic (sigmoid) function

    +$$ +p(y_i =1\vert x_i,\boldsymbol{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, +$$ + +

    and

    +$$ +p(y_i =0\vert x_i,\boldsymbol{\beta})=1-p(y_i =1\vert x_i,\boldsymbol{\beta}). +$$ + +

    The parameters \( \boldsymbol{\beta} \) were defined using a minimization method like gradient descent or Newton-Raphson's method.

    + +

    Now we replace \( x_i \) with the activation \( z_i^l \) for a given layer \( l \) and the outputs as \( y_i=a_i^l=f(z_i^l) \), with \( z_i^l \) now being a function of the weights \( w_{ij}^l \) and biases \( b_i^l \). +We have then

    +$$ +a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, +$$ -

    This gives the necessity of computing the Jacobian matrix, as we want -to evaluate the gradient with respect to \( x \) and \( t \) (note that the -Jacobian of a scalar-valued multivariate function is simply its -gradient). +

    with

    +$$ +z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, +$$ + +

    where the superscript \( l-1 \) indicates that these are the outputs from layer \( l-1 \). +Our cost function at the final layer \( l=L \) is now

    +$$ +\mathcal{C}(\boldsymbol{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), +$$ -

    In Autograd, the differentiation is by default done with respect to -the first input argument of your Python function. Since the points is -an array representing \( x \) and \( t \), the Jacobian is calculated using -the values of \( x \) and \( t \). -

    - -

    To find the second derivative with respect to \( x \) and \( t \), the -Jacobian can be found for the second time. The result is a Hessian -matrix, which is the matrix containing all the possible second order -mixed derivatives of \( g(x,t) \). -

    - - - -
    -
    -
    -
    -
    - 
    # Set up the trial function:
-def u(x):
-    return np.sin(np.pi*x)
-
-def g_trial(point,P):
-    x,t = point
-    return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)
-
-# The right side of the ODE:
-def f(point):
-    return 0.
-
-# The cost function:
-def cost_function(P, x, t):
-    cost_sum = 0
-
-    g_t_jacobian_func = jacobian(g_trial)
-    g_t_hessian_func = hessian(g_trial)
-
-    for x_ in x:
-        for t_ in t:
-            point = np.array([x_,t_])
-
-            g_t = g_trial(point,P)
-            g_t_jacobian = g_t_jacobian_func(point,P)
-            g_t_hessian = g_t_hessian_func(point,P)
-
-            g_t_dt = g_t_jacobian[1]
-            g_t_d2x = g_t_hessian[0][0]
-
-            func = f(point)
-
-            err_sqr = ( (g_t_dt - g_t_d2x) - func)**2
-            cost_sum += err_sqr
-
-    return cost_sum
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    +

    where we have defined the targets \( t_i \). The derivatives of the cost function with respect to the output \( a_i^L \) are then easily calculated and we get

    +$$ +\frac{\partial \mathcal{C}(\boldsymbol{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. +$$ +

    In case we use another activation function than the logistic one, we need to evaluate other derivatives.











    -

    Setting up the network using Autograd; The full program

    - -

    Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.

    - -

    The analytical solution of our problem is

    - -

    $$ -g(x,t) = \exp(-\pi^2 t)\sin(\pi x) +

    The Softmax function

    +

    In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need

    $$ -

    - -

    A possible way to implement a neural network solving the PDE, is given below. -Be aware, though, that it is fairly slow for the parameters used. -A better result is possible, but requires more iterations, and thus longer time to complete. -

    - -

    Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE. -Using TensorFlow results in a much better execution time. Try it! -

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import jacobian,hessian,grad
-import autograd.numpy.random as npr
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-## Set up the network
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # x is now a point and a 1D numpy array; make it a column vector
-    num_coordinates = np.size(x,0)
-    x = x.reshape(num_coordinates,-1)
-
-    num_points = np.size(x,1)
-
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output[0][0]
-
-## Define the trial solution and cost function
-def u(x):
-    return np.sin(np.pi*x)
-
-def g_trial(point,P):
-    x,t = point
-    return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)
-
-# The right side of the ODE:
-def f(point):
-    return 0.
-
-# The cost function:
-def cost_function(P, x, t):
-    cost_sum = 0
-
-    g_t_jacobian_func = jacobian(g_trial)
-    g_t_hessian_func = hessian(g_trial)
-
-    for x_ in x:
-        for t_ in t:
-            point = np.array([x_,t_])
-
-            g_t = g_trial(point,P)
-            g_t_jacobian = g_t_jacobian_func(point,P)
-            g_t_hessian = g_t_hessian_func(point,P)
-
-            g_t_dt = g_t_jacobian[1]
-            g_t_d2x = g_t_hessian[0][0]
-
-            func = f(point)
-
-            err_sqr = ( (g_t_dt - g_t_d2x) - func)**2
-            cost_sum += err_sqr
-
-    return cost_sum /( np.size(x)*np.size(t) )
-
-## For comparison, define the analytical solution
-def g_analytic(point):
-    x,t = point
-    return np.exp(-np.pi**2*t)*np.sin(np.pi*x)
-
-## Set up a function for training the network to solve for the equation
-def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):
-    ## Set up initial weigths and biases
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: ',cost_function(P, x, t))
-
-    cost_function_grad = grad(cost_function,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        cost_grad =  cost_function_grad(P, x , t)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_grad[l]
-
-    print('Final cost: ',cost_function(P, x, t))
-
-    return P
-
-if __name__ == '__main__':
-    ### Use the neural network:
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10; Nt = 10
-    x = np.linspace(0, 1, Nx)
-    t = np.linspace(0,1,Nt)
-
-    ## Set up the parameters for the network
-    num_hidden_neurons = [100, 25]
-    num_iter = 250
-    lmb = 0.01
-
-    P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
-
-    ## Store the results
-    g_dnn_ag = np.zeros((Nx, Nt))
-    G_analytical = np.zeros((Nx, Nt))
-    for i,x_ in enumerate(x):
-        for j, t_ in enumerate(t):
-            point = np.array([x_, t_])
-            g_dnn_ag[i,j] = g_trial(point,P)
-
-            G_analytical[i,j] = g_analytic(point)
-
-    # Find the map difference between the analytical and the computed solution
-    diff_ag = np.abs(g_dnn_ag - G_analytical)
-    print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag))
-
-    ## Plot the solutions in two dimensions, that being in position and time
-
-    T,X = np.meshgrid(t,x)
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-    s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Analytical solution')
-    s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Difference')
-    s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-    ## Take some slices of the 3D plots just to see the solutions at particular times
-    indx1 = 0
-    indx2 = int(Nt/2)
-    indx3 = Nt-1
-
-    t1 = t[indx1]
-    t2 = t[indx2]
-    t3 = t[indx3]
-
-    # Slice the results from the DNN
-    res1 = g_dnn_ag[:,indx1]
-    res2 = g_dnn_ag[:,indx2]
-    res3 = g_dnn_ag[:,indx3]
-
-    # Slice the analytical results
-    res_analytical1 = G_analytical[:,indx1]
-    res_analytical2 = G_analytical[:,indx2]
-    res_analytical3 = G_analytical[:,indx3]
-
-    # Plot the slices
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t1)
-    plt.plot(x, res1)
-    plt.plot(x,res_analytical1)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t2)
-    plt.plot(x, res2)
-    plt.plot(x,res_analytical2)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t3)
-    plt.plot(x, res3)
-    plt.plot(x,res_analytical3)
-    plt.legend(['dnn','analytical'])
-
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Example: Solving the wave equation with Neural Networks

    - -

    The wave equation is

    -$$ -\begin{equation*} - \frac{\partial^2 g(x,t)}{\partial t^2} = c^2\frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation*} +\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = +\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}. $$ -

    with \( c \) being the specified wave speed.

    - -

    Here, the chosen conditions are

    +

    For the Softmax function we have

    $$ -\begin{align*} - g(0,t) &= 0 \\ - g(1,t) &= 0 \\ - g(x,0) &= u(x) \\ - \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} &= v(x) -\end{align*} +f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}. $$ -

    where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivative of \( g(x,t) \) with respect to \( t \) is evaluated at \( t = 0 \), and \( u(x) \) and \( v(x) \) being given functions.

    - -









    -

    The problem to solve for

    - -

    The wave equation to solve for, is

    - +

    Its derivative with respect to \( z_j^l \) gives

    $$ -\begin{equation} \label{wave} -\frac{\partial^2 g(x,t)}{\partial t^2} = c^2 \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation} +\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), $$ -

    where \( c \) is the given wave speed. -The chosen conditions for this equation are -

    +

    which in case of the simply binary model reduces to having \( i=j \).

    -$$ -\begin{aligned} -g(0,t) &= 0, &t \geq 0 \\ -g(1,t) &= 0, &t \geq 0 \\ -g(x,0) &= u(x), &x\in[0,1] \\ -\frac{\partial g(x,t)}{\partial t}\Big |_{t = 0} &= v(x), &x \in [0,1] -\end{aligned} \label{condwave} -$$ + +

    Developing a code for doing neural networks with back propagation

    -

    In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi\sin(\pi x) \).

    - -









    -

    The trial solution

    -

    Setting up the network is done in similar matter as for the example of solving the diffusion equation. -The only things we have to change, is the trial solution such that it satisfies the conditions from \eqref{condwave} and the cost function. -

    - -

    The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution \( g_t(x,t) \) is

    - -

    $$ -g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P) -$$ -

    - -

    where

    - -

    $$ -h_1(x,t) = (1-t^2)u(x) + tv(x) -$$ -

    - -

    Note that this trial solution satisfies the conditions only if \( u(0) = v(0) = u(1) = v(1) = 0 \), which is the case in this example.

    - -









    -

    The analytical solution

    - -

    The analytical solution for our specific problem, is

    - -

    $$ -g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t) -$$ -

    - -









    -

    Solving the wave equation - the full program using Autograd

    - - - -
    -
    -
    -
    -
    - 
    import autograd.numpy as np
-from autograd import hessian,grad
-import autograd.numpy.random as npr
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-## Set up the trial function:
-def u(x):
-    return np.sin(np.pi*x)
-
-def v(x):
-    return -np.pi*np.sin(np.pi*x)
-
-def h1(point):
-    x,t = point
-    return (1 - t**2)*u(x) + t*v(x)
-
-def g_trial(point,P):
-    x,t = point
-    return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)
-
-## Define the cost function
-def cost_function(P, x, t):
-    cost_sum = 0
-
-    g_t_hessian_func = hessian(g_trial)
-
-    for x_ in x:
-        for t_ in t:
-            point = np.array([x_,t_])
-
-            g_t_hessian = g_t_hessian_func(point,P)
-
-            g_t_d2x = g_t_hessian[0][0]
-            g_t_d2t = g_t_hessian[1][1]
-
-            err_sqr = ( (g_t_d2t - g_t_d2x) )**2
-            cost_sum += err_sqr
-
-    return cost_sum / (np.size(t) * np.size(x))
-
-## The neural network
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # x is now a point and a 1D numpy array; make it a column vector
-    num_coordinates = np.size(x,0)
-    x = x.reshape(num_coordinates,-1)
-
-    num_points = np.size(x,1)
-
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output[0][0]
-
-## The analytical solution
-def g_analytic(point):
-    x,t = point
-    return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)
-
-def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):
-    ## Set up initial weigths and biases
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: ',cost_function(P, x, t))
-
-    cost_function_grad = grad(cost_function,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        cost_grad =  cost_function_grad(P, x , t)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_grad[l]
-
-
-    print('Final cost: ',cost_function(P, x, t))
-
-    return P
-
-if __name__ == '__main__':
-    ### Use the neural network:
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10; Nt = 10
-    x = np.linspace(0, 1, Nx)
-    t = np.linspace(0,1,Nt)
-
-    ## Set up the parameters for the network
-    num_hidden_neurons = [50,20]
-    num_iter = 1000
-    lmb = 0.01
-
-    P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
-
-    ## Store the results
-    res = np.zeros((Nx, Nt))
-    res_analytical = np.zeros((Nx, Nt))
-    for i,x_ in enumerate(x):
-        for j, t_ in enumerate(t):
-            point = np.array([x_, t_])
-            res[i,j] = g_trial(point,P)
-
-            res_analytical[i,j] = g_analytic(point)
-
-    diff = np.abs(res - res_analytical)
-    print("Max difference between analytical and solution from nn: %g"%np.max(diff))
-
-    ## Plot the solutions in two dimensions, that being in position and time
-
-    T,X = np.meshgrid(t,x)
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-    s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Analytical solution')
-    s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Difference')
-    s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-    ## Take some slices of the 3D plots just to see the solutions at particular times
-    indx1 = 0
-    indx2 = int(Nt/2)
-    indx3 = Nt-1
-
-    t1 = t[indx1]
-    t2 = t[indx2]
-    t3 = t[indx3]
-
-    # Slice the results from the DNN
-    res1 = res[:,indx1]
-    res2 = res[:,indx2]
-    res3 = res[:,indx3]
-
-    # Slice the analytical results
-    res_analytical1 = res_analytical[:,indx1]
-    res_analytical2 = res_analytical[:,indx2]
-    res_analytical3 = res_analytical[:,indx3]
-
-    # Plot the slices
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t1)
-    plt.plot(x, res1)
-    plt.plot(x,res_analytical1)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t2)
-    plt.plot(x, res2)
-    plt.plot(x,res_analytical2)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t3)
-    plt.plot(x, res3)
-    plt.plot(x,res_analytical3)
-    plt.legend(['dnn','analytical'])
-
-    plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Resources on differential equations and deep learning

    +

    One can identify a set of key steps when using neural networks to solve supervised learning problems:

      -
    1. Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al
      2. -
    2. Neural networks for solving differential equations by A. Honchar
      3. -
    3. Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener
      4. -
    4. Introduction to Partial Differential Equations by A. Tveito, R. Winther
      5. +
    5. Collect and pre-process data
      6. +
    6. Define model and architecture
      7. +
    7. Choose cost function and optimizer
      8. +
    8. Train the model
      9. +
    9. Evaluate model performance on test data
      10. +
    10. Adjust hyperparameters (if necessary, network architecture)










    -

    Convolutional Neural Networks (recognizing images)

    +

    Collect and pre-process data

    -

    Convolutional neural networks (CNNs) were developed during the last -decade of the previous century, with a focus on character recognition -tasks. Nowadays, CNNs are a central element in the spectacular success -of deep learning methods. The success in for example image -classifications have made them a central tool for most machine -learning practitioners. +

    Here we will be using the MNIST dataset, which is readily available through the scikit-learn +package. You may also find it for example here. +The MNIST (Modified National Institute of Standards and Technology) database is a large database +of handwritten digits that is commonly used for training various image processing systems. +The MNIST dataset consists of 70 000 images of size \( 28\times 28 \) pixels, each labeled from 0 to 9. +The scikit-learn dataset we will use consists of a selection of 1797 images of size \( 8\times 8 \) collected and processed from this database.

    -

    CNNs are very similar to ordinary Neural Networks. -They are made up of neurons that have learnable weights and -biases. Each neuron receives some inputs, performs a dot product and -optionally follows it with a non-linearity. The whole network still -expresses a single differentiable score function: from the raw image -pixels on one end to class scores at the other. And they still have a -loss function (for example Softmax) on the last (fully-connected) layer -and all the tips/tricks we developed for learning regular Neural -Networks still apply (back propagation, gradient descent etc etc). +

    To feed data into a feed-forward neural network we need to represent +the inputs as a design/feature matrix \( X = (n_{inputs}, n_{features}) \). Each +row represents an input, in this case a handwritten digit, and +each column represents a feature, in this case a pixel. The +correct answers, also known as labels or targets are +represented as a 1D array of integers +\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \).

    -









    -

    What is the Difference

    - -

    CNN architectures make the explicit assumption that -the inputs are images, which allows us to encode certain properties -into the architecture. These then make the forward function more -efficient to implement and vastly reduce the amount of parameters in -the network. +

    As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from +measurements of height (in m) +and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:

    -

    Here we provide only a superficial overview, for the more interested, we recommend highly the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231. +

    $$ X = \begin{bmatrix} +1.85 & 81\\ +1.71 & 65\\ +1.95 & 103\\ +1.55 & 42\\ +1.63 & 56 +\end{bmatrix} ,$$

    -

    Another good read is the article here https://arxiv.org/pdf/1603.07285.pdf.

    +

    and the targets would be:

    -









    -

    Neural Networks vs CNNs

    +

    $$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$

    -

    Neural networks are defined as affine transformations, that is -a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an -output (to which a bias vector is usually added before passing the result -through a nonlinear activation function). This is applicable to any type of input, be it an -image, a sound clip or an unordered collection of features: whatever their -dimensionality, their representation can always be flattened into a vector -before the transformation. +

    Since each input image is a 2D matrix, we need to flatten the image +(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a +design/feature matrix. This means we lose all spatial information in the +image, such as locality and translational invariance. More complicated +architectures such as Convolutional Neural Networks can take advantage +of such information, and are most commonly applied when analyzing +images.

    -









    -

    Why CNNS for images, sound files, medical images from CT scans etc?

    - -

    However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic -structure. More formally, they share these important properties: -

    - -

    These properties are not exploited when an affine transformation is applied; in -fact, all the axes are treated in the same way and the topological information -is not taken into account. Still, taking advantage of the implicit structure of -the data may prove very handy in solving some tasks, like computer vision and -speech recognition, and in these cases it would be best to preserve it. This is -where discrete convolutions come into play. -

    - -

    A discrete convolution is a linear transformation that preserves this notion of -ordering. It is sparse (only a few input units contribute to a given output -unit) and reuses parameters (the same weights are applied to multiple locations -in the input). -

    - -









    -

    Regular NNs don’t scale well to full images

    - -

    As an example, consider -an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a -single fully-connected neuron in a first hidden layer of a regular -Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still -seems manageable, but clearly this fully-connected structure does not -scale to larger images. For example, an image of more respectable -size, say \( 200\times 200\times 3 \), would lead to neurons that have -\( 200\times 200\times 3 = 120,000 \) weights. -

    - -

    We could have -several such neurons, and the parameters would add up quickly! Clearly, -this full connectivity is wasteful and the huge number of parameters -would quickly lead to possible overfitting. -

    - -
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    Figure 1: A regular 3-layer Neural Network.

    -
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    -

    3D volumes of neurons

    - -

    Convolutional Neural Networks take advantage of the fact that the -input consists of images and they constrain the architecture in a more -sensible way. -

    - -

    In particular, unlike a regular Neural Network, the -layers of a CNN have neurons arranged in 3 dimensions: width, -height, depth. (Note that the word depth here refers to the third -dimension of an activation volume, not to the depth of a full Neural -Network, which can refer to the total number of layers in a network.) -

    - -

    To understand it better, the above example of an image -with an input volume of -activations has dimensions \( 32\times 32\times 3 \) (width, height, -depth respectively). -

    - -

    The neurons in a layer will -only be connected to a small region of the layer before it, instead of -all of the neurons in a fully-connected manner. Moreover, the final -output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), -because by the -end of the CNN architecture we will reduce the full image into a -single vector of class scores, arranged along the depth -dimension. -

    - -
    -
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    Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    -
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    -
    - - -

    Layers used to build CNNs

    - -

    A simple CNN is a sequence of layers, and every layer of a CNN -transforms one volume of activations to another through a -differentiable function. We use three main types of layers to build -CNN architectures: Convolutional Layer, Pooling Layer, and -Fully-Connected Layer (exactly as seen in regular Neural Networks). We -will stack these layers to form a full CNN architecture. -

    - -

    A simple CNN for image classification could have the architecture:

    - - -









    -

    Transforming images

    - -

    CNNs transform the original image layer by layer from the original -pixel values to the final class scores. -

    - -

    Observe that some layers contain -parameters and other don’t. In particular, the CNN layers perform -transformations that are a function of not only the activations in the -input volume, but also of the parameters (the weights and biases of -the neurons). On the other hand, the RELU/POOL layers will implement a -fixed function. The parameters in the CONV/FC layers will be trained -with gradient descent so that the class scores that the CNN computes -are consistent with the labels in the training set for each image. -

    - -









    -

    CNNs in brief

    - -

    In summary:

    - - -

    For more material on convolutional networks, we strongly recommend -the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. -

    - -

    The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well.

    - -









    -

    Key Idea

    - -

    A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.

    - -

    The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect -only neighboring neurons in the input instead of connecting all with the first hidden layer. -

    - -

    We say we perform a filtering (convolution is the mathematical operation).

    - -









    -

    Mathematics of CNNs

    - -

    The mathematics of CNNs is based on the mathematical operation of -convolution. In mathematics (in particular in functional analysis), -convolution is represented by mathematical operation (integration, -summation etc) on two function in order to produce a third function -that expresses how the shape of one gets modified by the other. -Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning. -

    - -

    Mathematically, convolution is defined as follows (one-dimensional example): -Let us define a continuous function \( y(t) \) given by -

    -$$ -y(t) = \int x(a) w(t-a) da, -$$ - -

    where \( x(a) \) represents a so-called input and \( w(t-a) \) is normally called the weight function or kernel.

    - -

    The above integral is written in a more compact form as

    -$$ -y(t) = \left(x * w\right)(t). -$$ - -

    The discretized version reads

    -$$ -y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). -$$ - -

    Computing the inverse of the above convolution operations is known as deconvolution.

    - -

    How can we use this? And what does it mean? Let us study some familiar examples first.

    - -









    -

    Convolution Examples: Polynomial multiplication

    - -

    We have already met such an example in project 1 when we tried to set -up the design matrix for a two-dimensional function. This was an -example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. -Let us look a the following polynomials to second and third order, respectively: -

    -$$ -p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, -$$ - -

    and

    -$$ -s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. -$$ - -

    The polynomial multiplication gives us a new polynomial of degree \( 5 \)

    -$$ -z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. -$$ - - -









    -

    Efficient Polynomial Multiplication

    - -

    Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. -We note first that the new coefficients are given as -

    - -$$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} -$$ - -

    We note that \( \alpha_i=0 \) except for \( i\in \left\{0,1,2\right\} \) and \( \beta_i=0 \) except for \( i\in\left\{0,1,2,3\right\} \).

    - -

    We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as

    -$$ -\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, -$$ - -

    or as a double sum with restriction \( l=i+j \)

    -$$ -\delta_l = \sum_{ij}\alpha_i\beta_{j}. -$$ - -

    Do you see a potential drawback with these equations?

    - -









    -

    A more efficient way of coding the above Convolution

    - -

    Since we only have a finite number of \( \alpha \) and \( \beta \) values -which are non-zero, we can rewrite the above convolution expressions -as a matrix-vector multiplication -

    - -$$ -\boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ - \alpha_1 & \alpha_0 & 0 & 0 \\ - \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ - 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ - 0 & 0 & \alpha_2 & \alpha_1 \\ - 0 & 0 & 0 & \alpha_2 - \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. -$$ - -

    The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \( \beta \) and a vector holding \( \alpha \). -In this case we have -

    -$$ -\boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ - \beta_1 & \beta_0 & 0 \\ - \beta_2 & \beta_1 & \beta_0 \\ - \beta_3 & \beta_2 & \beta_1 \\ - 0 & \beta_3 & \beta_2 \\ - 0 & 0 & \beta_3 - \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. -$$ - -

    Note that the use of these matrices is for mathematical purposes only and not implementation purposes. -When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. -We rather code the convolutions in the minimal memory footprint that they require. -

    - -

    Does the number of floating point operations change here when we use the commutative property?

    - -









    -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    - -

    For problems with so-called harmonic oscillations, given by for example the following differential equation

    -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ - -

    where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.

    - -

    If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like -

    - -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\label{_auto3} -\end{equation} -$$ - - -









    -

    Principle of Superposition

    - -

    This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. -

    - -

    Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) -

    - -$$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} -$$ - -

    One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. -

    - -









    -

    Simple Code Example

    - -

    The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \).

    - - - -
    -
    -
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    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
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    - -

    For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, -

    - -$$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\label{_auto4} -\end{equation} -$$ - - -









    -

    Wrapping up Fourier transforms

    - -

    We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), -

    - -$$ -\begin{equation} -\label{eq:fourierdef1} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). -\end{equation} -$$ - -

    The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, -

    - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ - - -









    -

    Finding the Coefficients

    - -

    Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). -

    - -

    The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by -

    - -$$ -\begin{eqnarray} -\label{eq:fourierdef2} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -

    To check the consistency of these expressions and to verify -Eq. \eqref{eq:fourierdef2}, one can insert the expansion of \( F(t) \) in -Eq. \eqref{eq:fourierdef1} into the expression for the coefficients in -Eq. \eqref{eq:fourierdef2} and see whether -

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -

    Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives -

    - -$$ -\begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\label{_auto5} -\end{equation} -$$ - -

    and

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ -\nonumber -&=&f_n~\checkmark. -\end{eqnarray} -$$ - -

    The same method can be used to check for the consistency of \( g_n \).

    - -









    -

    Final words on Fourier Transforms

    - -

    The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. -

    - - - -
    -
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    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-T =0.2
-# Max value of square signal                                                                             
-Fmax= 2.0
-# Width of signal   
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
-    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
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    -

    Two-dimensional Objects

    - -

    We often use convolutions over more than one dimension at a time. If -we have a two-dimensional image \( I \) as input, we can have a filter -defined by a two-dimensional kernel \( K \). This leads to an output \( S \) -

    - -$$ -S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n). -$$ - -

    Convolution is a commutatitave process, which means we can rewrite this equation as

    -$$ -S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n). -$$ - -

    Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \).

    - -









    -

    Cross-Correlation

    - -

    Many deep learning libraries implement cross-correlation instead of convolution

    -$$ -S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n). -$$ - - -









    -

    More on Dimensionalities

    - -

    In feilds like signal processing (and imaging as well), one designs -so-called filters. These filters are defined by the convolutions and -are often hand-crafted. One may specify filters for smoothing, edge -detection, frequency reshaping, and similar operations. However with -neural networks the idea is to automatically learn the filters and use -many of them in conjunction with non-linear operations (activation -functions). -

    - -

    As an example consider a neural network operating on sound sequence -data. Assume that we an input vector \( \boldsymbol{x} \) of length \( d=10^6 \). We -construct then a neural network with onle hidden layer only with -\( 10^4 \) nodes. This means that we will have a weight matrix with -\( 10^4\times 10^6=10^{10} \) weights to be determined, together with \( 10^4 \) biases. -

    - -

    Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). -It means that we have only one output node. But since this output node connects to \( 10^4 \) nodes in the hidden layer, there are in total \( 10^4 \) weights to be determined for the output layer, plus one bias. In total we have -

    - -$$ -\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, -$$ - -

    that is ten billion parameters to determine.

    - -









    -

    Further Dimensionality Remarks

    - -

    In today’s architecture one can train such neural networks, however -this is a huge number of parameters for the task at hand. In general, -it is a very wasteful and inefficient use of dense matrices as -parameters. Just as importantly, such trained network parameters are -very specific for the type of input data on which they were trained -and the network is not likely to generalize easily to variations in -the input. -

    - -

    The main principles that justify convolutions is locality of -information and repetion of patterns within the signal. Sound samples -of the input in adjacent spots are much more likely to affect each -other than those that are very far away. Similarly, sounds are -repeated in multiple times in the signal. While slightly simplistic, -reasoning about such a sound example demonstrates this. The same -principles then apply to images and other similar data. -

    - -









    -

    CNNs in more detail, Lecture from IN5400

    - - -









    -

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    - -

    As discussed above, CNNs are neural networks built from the assumption that the inputs -to the network are 2D images. This is important because the number of features or pixels in images -grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. -

    - -

    As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks -are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. -In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D -matrices, typically 1 for each color dimension (Red, Green, Blue). -

    - -









    -

    Setting it up

    - -

    It means that to represent the entire -dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: -

    -$$ -(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . -$$ - - -









    -

    The MNIST dataset again

    - -

    The MNIST dataset consists of grayscale images with a pixel size of -\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each -neuron in the first hidden layer. -

    - -

    If we were to analyze images of size \( 128\times 128 \) we would require -\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were -dealing with color images, as most images are, we have an image matrix -of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), -meaning 3 times the number of weights \( = 49152 \) are required for every -single neuron in the first hidden layer. -

    - - -









    -

    Strong correlations

    - -

    Images typically have strong local correlations, meaning that a small -part of the image varies little from its neighboring regions. If for -example we have an image of a blue car, we can roughly assume that a -small blue part of the image is surrounded by other blue regions. -

    - -

    Therefore, instead of connecting every single pixel to a neuron in the -first hidden layer, as we have previously done with deep neural -networks, we can instead connect each neuron to a small part of the -image (in all 3 RGB depth dimensions). The size of each small area is -fixed, and known as a receptive. -

    - - - -

    Layers of a CNN

    -

    The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. -The input image is typically a square matrix of depth 3. -

    - -

    A convolution is performed on the image which outputs -a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. -

    - -

    Each filter slides along the input image, taking the dot product -between each small part of the image and the filter, in all depth -dimensions. This is then passed through a non-linear function, -typically the Rectified Linear (ReLu) function, which serves as the -activation of the neurons in the first convolutional layer. This is -further passed through a pooling layer, which reduces the size of the -convolutional layer, e.g. by taking the maximum or average across some -small regions, and this serves as input to the next convolutional -layer. -

    - -









    -

    Systematic reduction

    - -

    By systematically reducing the size of the input volume, through -convolution and pooling, the network should create representations of -small parts of the input, and then from them assemble representations -of larger areas. The final pooling layer is flattened to serve as -input to a hidden layer, such that each neuron in the final pooling -layer is connected to every single neuron in the hidden layer. This -then serves as input to the output layer, e.g. a softmax output for -classification. -

    - - -









    -

    Prerequisites: Collect and pre-process data

    @@ -3874,16 +600,18 @@ digits = datasets.load_digits() inputs = digits.images labels = digits.target -# RGB images have a depth of 3 -# our images are grayscale so they should have a depth of 1 -inputs = inputs[:,:,:,np.newaxis] - -print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape)) print("labels = (n_inputs) = " + str(labels.shape)) -# choose some random images to display +# flatten the image +# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64 n_inputs = len(inputs) +inputs = inputs.reshape(n_inputs, -1) +print("X = (n_inputs, n_features) = " + str(inputs.shape)) + + +# choose some random images to display indices = np.arange(n_inputs) random_indices = np.random.choice(indices, size=5) @@ -3910,7 +638,19 @@ plt.show()









    -

    Importing Keras and Tensorflow

    +

    Train and test datasets

    + +

    Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.

    + +

    We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.

    + +

    It is important that the train and test datasets are drawn randomly from our dataset, to ensure +no bias in the sampling. +Say you are taking measurements of weather data to predict the weather in the coming 5 days. +You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data +collected from 12.00 to 24.00. +

    +
    @@ -3918,80 +658,33 @@ plt.show()
    - 
    from tensorflow.keras import datasets, layers, models
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-#from tensorflow.keras import Conv2D
-#from tensorflow.keras import MaxPooling2D
-#from tensorflow.keras import Flatten
+  
    from sklearn.model_selection import train_test_split
 
-from sklearn.model_selection import train_test_split
-
-# representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
 # one-liner from scikit-learn library
 train_size = 0.8
 test_size = 1 - train_size
 X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
                                                     test_size=test_size)
-
    
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -

    Running with Keras

    - - - -
    -
    -
    -
    -
    - 
    def create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd):
-    model = Sequential()
-    model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
-              activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(layers.MaxPooling2D(pool_size=(2, 2)))
-    model.add(layers.Flatten())
-    model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
+# equivalently in numpy
+def train_test_split_numpy(inputs, labels, train_size, test_size):
+    n_inputs = len(inputs)
+    inputs_shuffled = inputs.copy()
+    labels_shuffled = labels.copy()
     
-    sgd = optimizers.SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    np.random.shuffle(inputs_shuffled)
+    np.random.shuffle(labels_shuffled)
     
-    return model
+    train_end = int(n_inputs*train_size)
+    X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
+    Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
+    
+    return X_train, X_test, Y_train, Y_test
 
-epochs = 100
-batch_size = 100
-input_shape = X_train.shape[1:4]
-receptive_field = 3
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
+#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
 
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
+print("Number of training images: " + str(len(X_train)))
+print("Number of test images: " + str(len(X_test)))
    @@ -4009,7 +702,194 @@ lmbd_vals = np.logspace(-5, Final part +

    Define model and architecture

    + +

    Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have

    + +

    $$ z = \sum_{i=1}^n w_i a_i ,$$

    + +

    $$ y = f(z) ,$$

    + +

    where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer +and \( w_i \) is the weight to input \( i \). +The activation of the neurons in the input layer is just the features (e.g. a pixel value). +

    + +

    The simplest activation function for a neuron is the Heaviside function:

    + +

    $$ f(z) = +\begin{cases} +1, & z > 0\\ +0, & \text{otherwise} +\end{cases} +$$ +

    + +

    A feed-forward neural network with this activation is known as a perceptron. +For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer. +This activation can be generalized to \( k \) classes (using e.g. the one-against-all strategy), +and we call these architectures multiclass perceptrons. +

    + +

    However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and +Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function. +

    + +

    Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU). +We will be using the sigmoid function \( \sigma(x) \): +

    + +

    $$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$

    + +

    which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.

    + + +

    Layers

    + +
      +
    • Input
      • +
    +

    Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.

    + +
      +
    • Hidden layer
      • +
    +

    We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer. +Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer. +

    + +
      +
    • Output
      • +
    +

    If we were building a binary classifier, it would be sufficient with a single neuron in the output layer, +which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1. +

    + +

    For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.

    + +

    Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the softmax function:

    + +

    $$ P(\text{class \( j \)} \mid \text{input \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}} +{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$ +

    + +

    i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) the weights of neuron \( j \) to the inputs. +The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1. +The exponent is just the weighted sum of inputs as before: +

    + +

    $$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$

    + +

    Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 +weights to the output layer. +

    + + +

    Weights and biases

    + +

    Typically weights are initialized with small values distributed around zero, drawn from a uniform +or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. +

    + +

    Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range +of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \( j \), \( b_j \): +

    + +

    $$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$

    + +

    The bias weights \( \boldsymbol{b} \) are often initialized to zero, but a small value like \( 0.01 \) ensures all neurons have some output which can be backpropagated in the first training cycle.

    + + +
    +
    +
    +
    +
    + 
    # building our neural network
+
+n_inputs, n_features = X_train.shape
+n_hidden_neurons = 50
+n_categories = 10
+
+# we make the weights normally distributed using numpy.random.randn
+
+# weights and bias in the hidden layer
+hidden_weights = np.random.randn(n_features, n_hidden_neurons)
+hidden_bias = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+output_weights = np.random.randn(n_hidden_neurons, n_categories)
+output_bias = np.zeros(n_categories) + 0.01
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + + +









    +

    Feed-forward pass

    + +

    Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories. +For each input image we calculate a weighted sum of input features (pixel values) to each neuron \( j \) in the hidden layer \( l \): +

    + +

    $$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$

    + +

    this is then passed through our activation function

    + +

    $$ a_{j}^{l} = f(z_{j}^{l}) .$$

    + +

    We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:

    + +

    $$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$

    + +

    Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:

    + +

    $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}} +{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$ +

    + + +

    Matrix multiplications

    + +

    Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden +layer have the dimensions +\( W_{hidden} = (n_{features}, n_{hidden}) \), +we can easily feed the network all our training data in one go by taking the matrix product +

    + +

    $$ X W^{h} = (n_{inputs}, n_{hidden}),$$

    + +

    and obtain a matrix that holds the weighted sum of inputs to the hidden layer +for each input image and each hidden neuron. +We also add the bias to obtain a matrix of weighted sums to the hidden layer \( Z^{h} \): +

    + +

    $$ \boldsymbol{z}^{l} = \boldsymbol{X} \boldsymbol{W}^{l} + \boldsymbol{b}^{l} ,$$

    + +

    meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image. +This is then passed through the activation: +

    + +

    $$ \boldsymbol{a}^{l} = f(\boldsymbol{z}^l) .$$

    + +

    This is fed to the output layer:

    + +

    $$ \boldsymbol{z}^{L} = \boldsymbol{a}^{L} \boldsymbol{W}^{L} + \boldsymbol{b}^{L} .$$

    + +

    Finally we receive our output values for each image and each category by passing it through the softmax function:

    + +

    $$ output = softmax (\boldsymbol{z}^{L}) = (n_{inputs}, n_{categories}) .$$

    @@ -4018,21 +898,507 @@ lmbd_vals = np.logspace(-5,
    - 
    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+  
    # setup the feed-forward pass, subscript h = hidden layer
+
+def sigmoid(x):
+    return 1/(1 + np.exp(-x))
+
+def feed_forward(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    exp_term = np.exp(z_o)
+    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+    
+    return probabilities
+
+probabilities = feed_forward(X_train)
+print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
+print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
+print("probabilities sum up to: " + str(probabilities[0].sum()))
+print()
+
+# we obtain a prediction by taking the class with the highest likelihood
+def predict(X):
+    probabilities = feed_forward(X)
+    return np.argmax(probabilities, axis=1)
+
+predictions = predict(X_train)
+print("predictions = (n_inputs) = " + str(predictions.shape))
+print("prediction for image 0: " + str(predictions[0]))
+print("correct label for image 0: " + str(Y_train[0]))
+
    
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + + +









    +

    Choose cost function and optimizer

    + +

    To measure how well our neural network is doing we need to introduce a cost function. +We will call the function that gives the error of a single sample output the loss function, and the function +that gives the total error of our network across all samples the cost function. +A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood. +

    + +

    In multiclass classification it is common to treat each integer label as a so called one-hot vector:

    + +

    $$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$

    + +

    $$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$

    + +

    i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.

    + +

    Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector. +We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \boldsymbol{x}_i \) in the dataset. +

    + +

    In the one-hot representation only one of the terms in the loss function is non-zero, namely the +probability of the correct category \( c' \) +(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong +you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases. +

    + + +









    +

    Optimizing the cost function

    + +

    The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent +is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function. +Each parameter \( \theta \) is iteratively adjusted according to the rule +

    + +

    $$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$

    + +

    where \( \eta \) is known as the learning rate, which controls how big a step we take towards the minimum. +This update can be repeated for any number of iterations, or until we are satisfied with the result. +

    + +

    A simple and effective improvement is a variant called Batch Gradient Descent. +Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient +on a subset of the data called a minibatch. +If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches +is \( N/M \). +We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes: +

    + +

    $$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad +\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$ +

    + +

    i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.

    + +

    This has two important benefits:

    +
      +
    1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.
      2. +
    2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.
      3. +
    +

    The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.

    + + +

    Regularization

    + +

    It is common to add an extra term to the cost function, proportional +to the size of the weights. This is equivalent to constraining the +size of the weights, so that they do not grow out of control. +Constraining the size of the weights means that the weights cannot +grow arbitrarily large to fit the training data, and in this way +reduces overfitting. +

    + +

    We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:

    + +

    $$ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad +\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2 += \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$ +

    + +

    i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.

    + +

    In order to train the model, we need to calculate the derivative of +the cost function with respect to every bias and weight in the +network. In total our network has \( (64 + 1)\times 50=3250 \) weights in +the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output +layer (\( +1 \) for the bias), and the gradient must be calculated for +every parameter. We use the backpropagation algorithm discussed +above. This is a clever use of the chain rule that allows us to +calculate the gradient efficently. +

    + +









    +

    Matrix multiplication

    + +

    To more efficently train our network these equations are implemented using matrix operations. +The error in the output layer is calculated simply as, with \( \boldsymbol{t} \) being our targets, +

    + +

    $$ \delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .$$

    + +

    The gradient for the output weights is calculated as

    + +

    $$ \nabla W_{L} = \boldsymbol{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$

    + +

    where \( \boldsymbol{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input. +Since we are going backwards we have to transpose the activation matrix. +

    + +

    The gradient with respect to the output bias is then

    + +

    $$ \nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$

    + +

    The error in the hidden layer is

    + +

    $$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$

    + +

    where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean +that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes +the Hadamard product, meaning element-wise multiplication. +

    + +

    This again gives us the gradients in the hidden layer:

    + +

    $$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$

    + +

    $$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$

    + + + +
    +
    +
    +
    +
    + 
    # to categorical turns our integer vector into a onehot representation
+from sklearn.metrics import accuracy_score
+
+# one-hot in numpy
+def to_categorical_numpy(integer_vector):
+    n_inputs = len(integer_vector)
+    n_categories = np.max(integer_vector) + 1
+    onehot_vector = np.zeros((n_inputs, n_categories))
+    onehot_vector[range(n_inputs), integer_vector] = 1
+    
+    return onehot_vector
+
+#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
+Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
+
+def feed_forward_train(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    exp_term = np.exp(z_o)
+    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+    
+    # for backpropagation need activations in hidden and output layers
+    return a_h, probabilities
+
+def backpropagation(X, Y):
+    a_h, probabilities = feed_forward_train(X)
+    
+    # error in the output layer
+    error_output = probabilities - Y
+    # error in the hidden layer
+    error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
+    
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_h.T, error_output)
+    output_bias_gradient = np.sum(error_output, axis=0)
+    
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(X.T, error_hidden)
+    hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
+eta = 0.01
+lmbd = 0.01
+for i in range(1000):
+    # calculate gradients
+    dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
+    
+    # regularization term gradients
+    dWo += lmbd * output_weights
+    dWh += lmbd * hidden_weights
+    
+    # update weights and biases
+    output_weights -= eta * dWo
+    output_bias -= eta * dBo
+    hidden_weights -= eta * dWh
+    hidden_bias -= eta * dBh
+
+print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + + +









    +

    Improving performance

    + +

    As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image. +In order to obtain a network that does something useful, we will have to do a bit more work. +

    + +

    The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).

    + +

    Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period +going through the entire dataset (\( n/M \) batches) an epoch. +

    + +

    If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. +Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here. +

    + +









    +

    Full object-oriented implementation

    + +

    It is very natural to think of the network as an object, with specific instances of the network +being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below. +

    + + + +
    +
    +
    +
    +
    + 
    class NeuralNetwork:
+    def __init__(
+            self,
+            X_data,
+            Y_data,
+            n_hidden_neurons=50,
+            n_categories=10,
+            epochs=10,
+            batch_size=100,
+            eta=0.1,
+            lmbd=0.0):
+
+        self.X_data_full = X_data
+        self.Y_data_full = Y_data
+
+        self.n_inputs = X_data.shape[0]
+        self.n_features = X_data.shape[1]
+        self.n_hidden_neurons = n_hidden_neurons
+        self.n_categories = n_categories
+
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
+
+        self.create_biases_and_weights()
+
+    def create_biases_and_weights(self):
+        self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
+        self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
+
+        self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
+        self.output_bias = np.zeros(self.n_categories) + 0.01
+
+    def feed_forward(self):
+        # feed-forward for training
+        self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
+        self.a_h = sigmoid(self.z_h)
+
+        self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
+
+        exp_term = np.exp(self.z_o)
+        self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
+    def feed_forward_out(self, X):
+        # feed-forward for output
+        z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
+        a_h = sigmoid(z_h)
+
+        z_o = np.matmul(a_h, self.output_weights) + self.output_bias
         
+        exp_term = np.exp(z_o)
+        probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+        return probabilities
+
+    def backpropagation(self):
+        error_output = self.probabilities - self.Y_data
+        error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
+
+        self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
+        self.output_bias_gradient = np.sum(error_output, axis=0)
+
+        self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
+        self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+        if self.lmbd > 0.0:
+            self.output_weights_gradient += self.lmbd * self.output_weights
+            self.hidden_weights_gradient += self.lmbd * self.hidden_weights
+
+        self.output_weights -= self.eta * self.output_weights_gradient
+        self.output_bias -= self.eta * self.output_bias_gradient
+        self.hidden_weights -= self.eta * self.hidden_weights_gradient
+        self.hidden_bias -= self.eta * self.hidden_bias_gradient
+
+    def predict(self, X):
+        probabilities = self.feed_forward_out(X)
+        return np.argmax(probabilities, axis=1)
+
+    def predict_probabilities(self, X):
+        probabilities = self.feed_forward_out(X)
+        return probabilities
+
+    def train(self):
+        data_indices = np.arange(self.n_inputs)
+
+        for i in range(self.epochs):
+            for j in range(self.iterations):
+                # pick datapoints with replacement
+                chosen_datapoints = np.random.choice(
+                    data_indices, size=self.batch_size, replace=False
+                )
+
+                # minibatch training data
+                self.X_data = self.X_data_full[chosen_datapoints]
+                self.Y_data = self.Y_data_full[chosen_datapoints]
+
+                self.feed_forward()
+                self.backpropagation()
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + + +









    +

    Evaluate model performance on test data

    + +

    To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data. +We measure the performance of the network using the accuracy score. +The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \). +

    + +

    $$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,$$

    + +

    where \( I \) is the indicator function, \( 1 \) if \( \tilde{y}_i = y_i \) and \( 0 \) otherwise.

    + + + +
    +
    +
    +
    +
    + 
    epochs = 100
+batch_size = 100
+
+dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+                    n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+dnn.train()
+test_predict = dnn.predict(X_test)
+
+# accuracy score from scikit library
+print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+
+# equivalent in numpy
+def accuracy_score_numpy(Y_test, Y_pred):
+    return np.sum(Y_test == Y_pred) / len(Y_test)
+
+#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + + +









    +

    Adjust hyperparameters

    + +

    We now perform a grid search to find the optimal hyperparameters for the network. +Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate). +

    + + + +
    +
    +
    +
    +
    + 
    eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
+# grid search
 for i, eta in enumerate(eta_vals):
     for j, lmbd in enumerate(lmbd_vals):
-        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd)
-        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = CNN.evaluate(X_test, Y_test)
+        dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+                            n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+        dnn.train()
         
-        CNN_keras[i][j] = CNN
+        DNN_numpy[i][j] = dnn
         
-        print("Learning rate = ", eta)
+        test_predict = dnn.predict(X_test)
+        
+        print("Learning rate  = ", eta)
         print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
+        print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
         print()
    @@ -4051,7 +1417,7 @@ lmbd_vals = np.logspace(-5, Final visualization +

    Visualization

    @@ -4061,7 +1427,7 @@ lmbd_vals = np.logspace(-5, 
    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
+# uses seaborn heatmap, you can also do this with matplotlib imshow
 import seaborn as sns
 
 sns.set()
@@ -4071,10 +1437,13 @@ test_accuracy = np.zeros((len(eta_vals), for i in range(len(eta_vals)):
     for j in range(len(lmbd_vals)):
-        CNN = CNN_keras[i][j]
+        dnn = DNN_numpy[i][j]
+        
+        train_pred = dnn.predict(X_train) 
+        test_pred = dnn.predict(X_test)
 
-        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
 
         
 fig, ax = plt.subplots(figsize = (10, 10))
@@ -4107,12 +1476,20 @@ plt.show()
 
 
 









    
-
    The CIFAR01 data set 
    
+
    scikit-learn implementation 
    
 
-
    The CIFAR10 dataset contains 60,000 color images in 10 classes, with
-6,000 images in each class. The dataset is divided into 50,000
-training images and 10,000 testing images. The classes are mutually
-exclusive and there is no overlap between them.
+
    scikit-learn focuses more
+on traditional machine learning methods, such as regression,
+clustering, decision trees, etc. As such, it has only two types of
+neural networks: Multi Layer Perceptron outputting continuous values,
+MPLRegressor, and Multi Layer Perceptron outputting labels,
+MLPClassifier. We will see how simple it is to use these classes.
+
    
+
+
    scikit-learn implements a few improvements from our neural network,
+such as early stopping, a varying learning rate, different
+optimization methods, etc. We would therefore expect a better
+performance overall.
 
    
 
 
@@ -4122,16 +1499,22 @@ exclusive and there is no overlap between them.
     
    
       
    
         
    
-  
    import tensorflow as tf
+  
    from sklearn.neural_network import MLPClassifier
+# store models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
 
-from tensorflow.keras import datasets, layers, models
-import matplotlib.pyplot as plt
-
-# We import the data set
-(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
-
-# Normalize pixel values to be between 0 and 1 by dividing by 255. 
-train_images, test_images = train_images / 255.0, test_images / 255.0
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X_train, Y_train)
+        
+        DNN_scikit[i][j] = dnn
+        
+        print("Learning rate  = ", eta)
+        print("Lambda = ", lmbd)
+        print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+        print()
 
    
 
    
       
    
@@ -4149,10 +1532,7 @@ train_images, test_images = train_images / 255.0









    
-
    Verifying the data set 
    
-
-
    To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
    
-
+
    Visualization 
    
 
 
 
    
@@ -4160,19 +1540,39 @@ train_images, test_images = train_images / 255.0
       
    
         
    
-  
    class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
-               'dog', 'frog', 'horse', 'ship', 'truck']
-​
-plt.figure(figsize=(10,10))
-for i in range(25):
-    plt.subplot(5,5,i+1)
-    plt.xticks([])
-    plt.yticks([])
-    plt.grid(False)
-    plt.imshow(train_images[i], cmap=plt.cm.binary)
-    # The CIFAR labels happen to be arrays, 
-    # which is why you need the extra index
-    plt.xlabel(class_names[train_labels[i][0]])
+  
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        dnn = DNN_scikit[i][j]
+        
+        train_pred = dnn.predict(X_train) 
+        test_pred = dnn.predict(X_test)
+
+        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
 plt.show()
 
    
 
    
@@ -4191,58 +1591,76 @@ plt.show()
 
 
 









    
-
    Set up  the model 
    
+
    Testing our code for the XOR, OR and AND gates 
    
 
-
    The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
    
-
-
    As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
    
-
-
-
-
    
-  
    
-    
    
-      
    
-        
    
-  
    model = models.Sequential()
-model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-
-# Let's display the architecture of our model so far.
-
-model.summary()
-
    
-
    
-      
    
-    
    
-  
    
-  
    
-    
    
-      
    
-        
              
-        
    
-      
    
-    
    
-  
    
-
    
-
-
    You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.
    
-
-









    
-
    Add Dense layers on top 
    
-
-
    To complete our model, you will feed the last output tensor from the
-convolutional base (of shape (4, 4, 64)) into one or more Dense layers
-to perform classification. Dense layers take vectors as input (which
-are 1D), while the current output is a 3D tensor. First, you will
-flatten (or unroll) the 3D output to 1D, then add one or more Dense
-layers on top. CIFAR has 10 output classes, so you use a final Dense
-layer with 10 outputs and a softmax activation.
+
    Last week we discussed three different types of gates, the so-called
+XOR, the OR and the AND gates.  Their inputs and outputs can be
+summarized using the following tables, first for the OR gate with
+inputs \( x_1 \) and \( x_2 \) and outputs \( y \):
 
    
 
+
+
+
+
+
+
+
+
+
+
+
    \( x_1 \) \( x_2 \) \( y \) 
       0                0                0           
       0                1                1           
       1                0                1           
       1                1                1           
    
+
+









    
+
    The AND and XOR Gates 
    
+
+
    The AND gate is defined as
    
+
+
+
+
+
+
+
+
+
+
+
+
    \( x_1 \) \( x_2 \) \( y \) 
       0                0                0           
       0                1                0           
       1                0                0           
       1                1                1           
    
+
+
    And finally we have the XOR gate
    
+
+
+
+
+
+
+
+
+
+
+
+
    \( x_1 \) \( x_2 \) \( y \) 
       0                0                0           
       0                1                1           
       1                0                1           
       1                1                0           
    
+
+









    
+
    Representing the Data Sets 
    
+
+
    Our design matrix is defined by the input values \( x_1 \) and \( x_2 \). Since we have four possible outputs, our design matrix reads
    
+
+$$
+\boldsymbol{X}=\begin{bmatrix} 0 & 0 \\
+                       0 & 1 \\
+		       1 & 0 \\
+		       1 & 1 \end{bmatrix},
+$$
+
+
    while the vector of outputs is \( \boldsymbol{y}^T=[0,1,1,0] \) for the XOR gate, \( \boldsymbol{y}^T=[0,0,0,1] \) for the AND gate and \( \boldsymbol{y}^T=[0,1,1,1] \) for the OR gate.
    
+
+









    
+
    Setting up the Neural Network 
    
+
+
    We define first our design matrix and the various output vectors for the different gates.
    
+
 
 
 
    
@@ -4250,12 +1668,71 @@ layer with 10 outputs and a softmax activation.
     
    
       
    
         
    
-  
    model.add(layers.Flatten())
-model.add(layers.Dense(64, activation='relu'))
-model.add(layers.Dense(10))
-Here's the complete architecture of our model.
+  
    """
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
 
-model.summary()
+# import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
+
+def sigmoid(x):
+    return 1/(1 + np.exp(-x))
+
+def feed_forward(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    probabilities = sigmoid(z_o)
+    return probabilities
+
+# we obtain a prediction by taking the class with the highest likelihood
+def predict(X):
+    probabilities = feed_forward(X)
+    return np.argmax(probabilities, axis=1)
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# Design matrix
+X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+# The AND gate
+yAND = np.array( [ 0, 0 ,0, 1])
+
+# Defining the neural network
+n_inputs, n_features = X.shape
+n_hidden_neurons = 2
+n_categories = 2
+n_features = 2
+
+# we make the weights normally distributed using numpy.random.randn
+
+# weights and bias in the hidden layer
+hidden_weights = np.random.randn(n_features, n_hidden_neurons)
+hidden_bias = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+output_weights = np.random.randn(n_hidden_neurons, n_categories)
+output_bias = np.zeros(n_categories) + 0.01
+
+probabilities = feed_forward(X)
+print(probabilities)
+
+
+predictions = predict(X)
+print(predictions)
 
    
 
    
       
    
@@ -4271,10 +1748,10 @@ model.summary()
   
    
 
    
 
-
    As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
    
+
    Not an impressive result, but this was our first forward pass with randomly assigned weights. Let us now add the full network with the back-propagation algorithm discussed above.
    
 
 









    
-
    Compile and train the model 
    
+
    The Code using Scikit-Learn 
    
 
 
 
@@ -4283,12 +1760,63 @@ model.summary()
     
    
       
    
         
    
-  
    model.compile(optimizer='adam',
-              loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
-              metrics=['accuracy'])
-​
-history = model.fit(train_images, train_labels, epochs=10, 
-                    validation_data=(test_images, test_labels))
+  
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.neural_network import MLPClassifier
+from sklearn.metrics import accuracy_score
+import seaborn as sns
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# Design matrix
+X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+# The AND gate
+yAND = np.array( [ 0, 0 ,0, 1])
+
+# Defining the neural network
+n_inputs, n_features = X.shape
+n_hidden_neurons = 2
+n_categories = 2
+n_features = 2
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+epochs = 100
+
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X, yXOR)
+        DNN_scikit[i][j] = dnn
+        print("Learning rate  = ", eta)
+        print("Lambda = ", lmbd)
+        print("Accuracy score on data set: ", dnn.score(X, yXOR))
+        print()
+
+sns.set()
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        dnn = DNN_scikit[i][j]
+        test_pred = dnn.predict(X)
+        test_accuracy[i][j] = accuracy_score(yXOR, test_pred)
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
 
    
 
    
       
    
@@ -4306,8 +1834,48 @@ history = model.fit(train_images, train_labels, epochs=Finally, evaluate the model 
+
    Building neural networks in Tensorflow and Keras 
    
 
+
    Now we want  to build on the experience gained from our neural network implementation in NumPy and scikit-learn
+and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
+and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.  
+
    
+
+
    In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
+clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
+NumPy arrays.
+
    
+
+









    
+
    Tensorflow 
    
+
+
    Tensorflow is an open source library machine learning library
+developed by the Google Brain team for internal use. It was released
+under the Apache 2.0 open source license in November 9, 2015.
+
    
+
+
    Tensorflow is a computational framework that allows you to construct
+machine learning models at different levels of abstraction, from
+high-level, object-oriented APIs like Keras, down to the C++ kernels
+that Tensorflow is built upon. The higher levels of abstraction are
+simpler to use, but less flexible, and our choice of implementation
+should reflect the problems we are trying to solve.
+
    
+
+
    Tensorflow uses so-called graphs to represent your computation
+in terms of the dependencies between individual operations, such that you first build a Tensorflow graph
+to represent your model, and then create a Tensorflow session to run the graph.
+
    
+
+
    In this guide we will analyze the same data as we did in our NumPy and
+scikit-learn tutorial, gathered from the MNIST database of images. We
+will give an introduction to the lower level Python Application
+Program Interfaces (APIs), and see how we use them to build our graph.
+Then we will build (effectively) the same graph in Keras, to see just
+how simple solving a machine learning problem can be.
+
    
+
+
    To install tensorflow on Unix/Linux systems, use pip as
    
 
 
 
    
@@ -4315,16 +1883,7 @@ history = model.fit(train_images, train_labels, epochs=
       
    
         
    
-  
    plt.plot(history.history['accuracy'], label='accuracy')
-plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
-plt.xlabel('Epoch')
-plt.ylabel('Accuracy')
-plt.ylim([0.5, 1])
-plt.legend(loc='lower right')
-
-test_loss, test_acc = model.evaluate(test_images,  test_labels, verbose=2)
-
-print(test_acc)
+  
    pip3 install tensorflow
 
    
 
    
       
    
@@ -4340,10 +1899,813 @@ test_loss, test_acc = model.evaluate(test_images,  test_labels, verbose=
 
    
 
+
    and/or if you use anaconda, just write (or install from the graphical user interface)
+(current release of CPU-only TensorFlow)
+
    
+
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    conda create -n tf tensorflow
+conda activate tf
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
    
+
+
    To install the current release of GPU TensorFlow
    
+
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    conda create -n tf-gpu tensorflow-gpu
+conda activate tf-gpu
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
    
+
+
+









    
+
    Using Keras 
    
+
+
    Keras is a high level neural network
+that supports Tensorflow, CTNK and Theano as backends.  
+If you have Anaconda installed you may run the following command
+
    
+
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    conda install keras
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
    
+
+
    You can look up the instructions here for more information.
    
+
+
    We will to a large extent use keras in this course. 
    
+
+









    
+
    Collect and pre-process data 
    
+
+
    Let us look again at the MINST data set.
    
+
+
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+import tensorflow as tf
+from sklearn import datasets
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# flatten the image
+# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
+n_inputs = len(inputs)
+inputs = inputs.reshape(n_inputs, -1)
+print("X = (n_inputs, n_features) = " + str(inputs.shape))
+
+
+# choose some random images to display
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
+
+from sklearn.model_selection import train_test_split
+
+# one-hot representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    epochs = 100
+batch_size = 100
+n_neurons_layer1 = 100
+n_neurons_layer2 = 50
+n_categories = 10
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
+    model = Sequential()
+    model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax'))
+    
+    sgd = optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
+                                         eta=eta, lmbd=lmbd)
+        DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = DNN.evaluate(X_test, Y_test)
+        
+        DNN_keras[i][j] = DNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        DNN = DNN_keras[i][j]
+
+        train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
    
+
+
+









    
+
    The Breast Cancer Data, now with Keras 
    
+
+
+
+
    
+  
    
+    
    
+      
    
+        
    
+  
    import tensorflow as tf
+from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
+import numpy as np
+import matplotlib.pyplot as plt
+import seaborn as sns
+from sklearn.model_selection import train_test_split as splitter
+from sklearn.datasets import load_breast_cancer
+import pickle
+import os 
+
+
+"""Load breast cancer dataset"""
+
+np.random.seed(0)        #create same seed for random number every time
+
+cancer=load_breast_cancer()      #Download breast cancer dataset
+
+inputs=cancer.data                     #Feature matrix of 569 rows (samples) and 30 columns (parameters)
+outputs=cancer.target                  #Label array of 569 rows (0 for benign and 1 for malignant)
+labels=cancer.feature_names[0:30]
+
+print('The content of the breast cancer dataset is:')      #Print information about the datasets
+print(labels)
+print('-------------------------')
+print("inputs =  " + str(inputs.shape))
+print("outputs =  " + str(outputs.shape))
+print("labels =  "+ str(labels.shape))
+
+x=inputs      #Reassign the Feature and Label matrices to other variables
+y=outputs
+
+#%% 
+
+# Visualisation of dataset (for correlation analysis)
+
+plt.figure()
+plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean radius',fontweight='bold')
+plt.ylabel('Mean perimeter',fontweight='bold')
+plt.show()
+
+plt.figure()
+plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)
+plt.xlabel('Mean compactness',fontweight='bold')
+plt.ylabel('Mean concavity',fontweight='bold')
+plt.show()
+
+
+plt.figure()
+plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean radius',fontweight='bold')
+plt.ylabel('Mean texture',fontweight='bold')
+plt.show()
+
+plt.figure()
+plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean perimeter',fontweight='bold')
+plt.ylabel('Mean compactness',fontweight='bold')
+plt.show()
+
+
+# Generate training and testing datasets
+
+#Select features relevant to classification (texture,perimeter,compactness and symmetery) 
+#and add to input matrix
+
+temp1=np.reshape(x[:,1],(len(x[:,1]),1))
+temp2=np.reshape(x[:,2],(len(x[:,2]),1))
+X=np.hstack((temp1,temp2))      
+temp=np.reshape(x[:,5],(len(x[:,5]),1))
+X=np.hstack((X,temp))       
+temp=np.reshape(x[:,8],(len(x[:,8]),1))
+X=np.hstack((X,temp))       
+
+X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1)   #Split datasets into training and testing
+
+y_train=to_categorical(y_train)     #Convert labels to categorical when using categorical cross entropy
+y_test=to_categorical(y_test)
+
+del temp1,temp2,temp
+
+# %%
+
+# Define tunable parameters"
+
+eta=np.logspace(-3,-1,3)                    #Define vector of learning rates (parameter to SGD optimiser)
+lamda=0.01                                  #Define hyperparameter
+n_layers=2                                  #Define number of hidden layers in the model
+n_neuron=np.logspace(0,3,4,dtype=int)       #Define number of neurons per layer
+epochs=100                                   #Number of reiterations over the input data
+batch_size=100                              #Number of samples per gradient update
+
+# %%
+
+"""Define function to return Deep Neural Network model"""
+
+def NN_model(inputsize,n_layers,n_neuron,eta,lamda):
+    model=Sequential()      
+    for i in range(n_layers):       #Run loop to add hidden layers to the model
+        if (i==0):                  #First layer requires input dimensions
+            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))
+        else:                       #Subsequent layers are capable of automatic shape inferencing
+            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))
+    model.add(Dense(2,activation='softmax'))  #2 outputs - ordered and disordered (softmax for prob)
+    sgd=optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])
+    return model
+
+    
+Train_accuracy=np.zeros((len(n_neuron),len(eta)))      #Define matrices to store accuracy scores as a function
+Test_accuracy=np.zeros((len(n_neuron),len(eta)))       #of learning rate and number of hidden neurons for 
+
+for i in range(len(n_neuron)):     #run loops over hidden neurons and learning rates to calculate 
+    for j in range(len(eta)):      #accuracy scores 
+        DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)
+        DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)
+        Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]
+        Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]
+               
+
+def plot_data(x,y,data,title=None):
+
+    # plot results
+    fontsize=16
+
+
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    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()
+    
+plot_data(eta,n_neuron,Train_accuracy, 'training')
+plot_data(eta,n_neuron,Test_accuracy, 'testing')
+
    
+
    
+      
    
+    
    
+  
    
+  
    
+    
    
+      
    
+        
              
+        
    
+      
    
+    
    
+  
    
+
    
+
+
+









    
+
    Fine-tuning neural network hyperparameters 
    
+
+
    The flexibility of neural networks is also one of their main
+drawbacks: there are many hyperparameters to tweak. Not only can you
+use any imaginable network topology (how neurons/nodes are interconnected),
+but even in a simple FFNN you can change the number of layers, the
+number of neurons per layer, the type of activation function to use in
+each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you
+know what combination of hyperparameters is the best for your task?
+
    
+
+
      
+
    •  You can use grid search with cross-validation to find the right hyperparameters.
      • 
+
    
+
    However,since there are many hyperparameters to tune, and since
+training a neural network on a large dataset takes a lot of time, you
+will only be able to explore a tiny part of the hyperparameter space.
+
    
+
+
      
+
    •  You can use randomized search.
      • 
+
    •  Or use tools like Oscar, which implements more complex algorithms to help you find a good set of hyperparameters quickly.
      •   
+
    
+









    
+
    Hidden layers 
    
+
+
    For many problems you can start with just one or two hidden layers and it will work just fine.
+For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a
+few hundred neurons.
+You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of
+neurons, in roughly the same amount of training time. 
+
    
+
+
    For more complex problems, you can gradually
+ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such
+as large image classification or speech recognition, typically require networks with dozens of layers
+and they need a huge amount
+of training data. However, you will rarely have to train such networks from scratch: it is much more
+common to reuse parts of a pretrained state-of-the-art network that performs a similar task.
+
    
+
+
+
    Which activation function should I use? 
    
+
+
    The Back propagation algorithm we derived above works by going from
+the output layer to the input layer, propagating the error gradient on
+the way. Once the algorithm has computed the gradient of the cost
+function with regards to each parameter in the network, it uses these
+gradients to update each parameter with a Gradient Descent (GD) step.
+
    
+
+
    Unfortunately for us, the gradients often get smaller and smaller as the
+algorithm progresses down to the first hidden layers. As a result, the
+GD update leaves the lower layer connection weights
+virtually unchanged, and training never converges to a good
+solution. This is known in the literature as 
+the vanishing gradients problem. 
+
    
+
+
    In other cases, the opposite can happen, namely the the gradients can grow bigger and
+bigger. The result is that many of the layers get large updates of the 
+weights the
+algorithm diverges. This is the exploding gradients problem, which is
+mostly encountered in recurrent neural networks. More generally, deep
+neural networks suffer from unstable gradients, different layers may
+learn at widely different speeds
+
    
+
+
+
    Is the Logistic activation function (Sigmoid)  our choice? 
    
+
+
    Although this unfortunate behavior has been empirically observed for
+quite a while (it was one of the reasons why deep neural networks were
+mostly abandoned for a long time), it is only around 2010 that
+significant progress was made in understanding it.
+
    
+
+
    A paper titled Understanding the Difficulty of Training Deep
+Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that
+the problems with the popular logistic
+sigmoid activation function and the weight initialization technique
+that was most popular at the time, namely random initialization using
+a normal distribution with a mean of 0 and a standard deviation of
+1. 
+
    
+
+
    They showed that with this activation function and this
+initialization scheme, the variance of the outputs of each layer is
+much greater than the variance of its inputs. Going forward in the
+network, the variance keeps increasing after each layer until the
+activation function saturates at the top layers. This is actually made
+worse by the fact that the logistic function has a mean of 0.5, not 0
+(the hyperbolic tangent function has a mean of 0 and behaves slightly
+better than the logistic function in deep networks).
+
    
+
+









    
+
    The derivative of the Logistic funtion 
    
+
+
    Looking at the logistic activation function, when inputs become large
+(negative or positive), the function saturates at 0 or 1, with a
+derivative extremely close to 0. Thus when backpropagation kicks in,
+it has virtually no gradient to propagate back through the network,
+and what little gradient exists keeps getting diluted as
+backpropagation progresses down through the top layers, so there is
+really nothing left for the lower layers.
+
    
+
+
    In their paper, Glorot and Bengio propose a way to significantly
+alleviate this problem. We need the signal to flow properly in both
+directions: in the forward direction when making predictions, and in
+the reverse direction when backpropagating gradients. We don’t want
+the signal to die out, nor do we want it to explode and saturate. For
+the signal to flow properly, the authors argue that we need the
+variance of the outputs of each layer to be equal to the variance of
+its inputs, and we also need the gradients to have equal variance
+before and after flowing through a layer in the reverse direction.
+
    
+
+
    One of the insights in the 2010 paper by Glorot and Bengio was that
+the vanishing/exploding gradients problems were in part due to a poor
+choice of activation function. Until then most people had assumed that
+if Nature had chosen to use roughly sigmoid activation functions in
+biological neurons, they must be an excellent choice. But it turns out
+that other activation functions behave much better in deep neural
+networks, in particular the ReLU activation function, mostly because
+it does not saturate for positive values (and also because it is quite
+fast to compute).
+
    
+
+









    
+
    The RELU function family 
    
+
+
    The ReLU activation function suffers from a problem known as the dying
+ReLUs: during training, some neurons effectively die, meaning they
+stop outputting anything other than 0.
+
    
+
+
    In some cases, you may find that half of your network’s neurons are
+dead, especially if you used a large learning rate. During training,
+if a neuron’s weights get updated such that the weighted sum of the
+neuron’s inputs is negative, it will start outputting 0. When this
+happen, the neuron is unlikely to come back to life since the gradient
+of the ReLU function is 0 when its input is negative.
+
    
+
+
    To solve this problem, nowadays practitioners use a  variant of the ReLU
+function, such as the leaky ReLU discussed above or the so-called
+exponential linear unit (ELU) function
+
    
+
+$$
+ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\  z & z \ge 0.\end{array}\right. 
+$$
+
+
+









    
+
    Which activation function should we use? 
    
+
+
    In general it seems that the ELU activation function is better than
+the leaky ReLU function (and its variants), which is better than
+ReLU. ReLU performs better than \( \tanh \) which in turn performs better
+than the logistic function. 
+
    
+
+
    If runtime
+performance is an issue, then you may opt for the  leaky ReLU function  over the 
+ELU function If you don’t
+want to tweak yet another hyperparameter, you may just use the default
+\( \alpha \) of \( 0.01 \) for the leaky ReLU, and \( 1 \) for ELU. If you have
+spare time and computing power, you can use cross-validation or
+bootstrap to evaluate other activation functions.
+
    
+
+









    
+
    More on activation functions, output layers 
    
+
+
    In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).
    
+
+
    It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.
    
+
+For the output layer:
+
+
      
+
    •  For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).
      • 
+
    •  For regression tasks, you can simply use no activation function at all.
      • 
+
    
+









    
+
    Batch Normalization 
    
+
+
    Batch Normalization
+aims to address the vanishing/exploding gradients problems, and more generally the problem that the
+distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.
+
    
+
+
    The technique consists of adding an operation in the model just before the activation function of each
+layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new
+parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model
+learn the optimal scale and mean of the inputs for each layer.
+In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and
+standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current
+mini-batch, from this the name batch normalization.
+
    
+
+









    
+
    Dropout 
    
+
+
    It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but
+excluding the output neurons) has a probability \( p \) of being temporarily dropped out, meaning it will be
+entirely ignored during this training step, but it may be active during the next step.
+
    
+
+
    The
+hyperparameter \( p \) is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.
+ It is viewed as one of the most popular regularization techniques.
+
    
+
+









    
+
    Gradient Clipping 
    
+
+
    A popular technique to lessen the exploding gradients problem is to simply clip the gradients during
+backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural
+networks).
+
    
+
+
    This technique is called Gradient Clipping.
    
+
+
    In general however, Batch
+Normalization is preferred.
+
    
+
+









    
+
    A very nice website on Neural Networks 
    
+
+
    You may find this website very useful. 
    
+
+
+
    A top-down perspective on Neural networks 
    
+
+
    The first thing we would like to do is divide the data into two or three
+parts. A training set, a validation or dev (development) set, and a
+test set. The test set is the data on which we want to make
+predictions. The dev set is a subset of the training data we use to
+check how well we are doing out-of-sample, after training the model on
+the training dataset. We use the validation error as a proxy for the
+test error in order to make tweaks to our model. It is crucial that we
+do not use any of the test data to train the algorithm. This is a
+cardinal sin in ML. Then:
+
    
+
+
      
+
    •  Estimate optimal error rate
      • 
+
    •  Minimize underfitting (bias) on training data set.
      • 
+
    •  Make sure you are not overfitting.
      • 
+
    
+
    If the validation and test sets are drawn from the same distributions,
+then a good performance on the validation set should lead to similarly
+good performance on the test set. 
+
    
+
+
    However, sometimes
+the training data and test data differ in subtle ways because, for
+example, they are collected using slightly different methods, or
+because it is cheaper to collect data in one way versus another. In
+this case, there can be a mismatch between the training and test
+data. This can lead to the neural network overfitting these small
+differences between the test and training sets, and a poor performance
+on the test set despite having a good performance on the validation
+set. To rectify this, Andrew Ng suggests making two validation or dev
+sets, one constructed from the training data and one constructed from
+the test data. The difference between the performance of the algorithm
+on these two validation sets quantifies the train-test mismatch. This
+can serve as another important diagnostic when using DNNs for
+supervised learning.
+
    
+
+









    
+
    Limitations of supervised learning with deep networks 
    
+
+
    Like all statistical methods, supervised learning using neural
+networks has important limitations. This is especially important when
+one seeks to apply these methods, especially to physics problems. Like
+all tools, DNNs are not a universal solution. Often, the same or
+better performance on a task can be achieved by using a few
+hand-engineered features (or even a collection of random
+features). 
+
    
+
+
    Here we list some of the important limitations of supervised neural network based models. 
    
+
+
      
+
    •  Need labeled data. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).
      • 
+
    •  Supervised neural networks are extremely data intensive. DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.
      • 
+
    •  Homogeneous data. Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.
      • 
+
    •  Many problems are not about prediction. In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a wrong model. The model might or might not be useful for understanding the underlying science.
      • 
+
    
+
    Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.
    
 
 
 
    
- © 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+ © 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 
    
 
 
diff --git a/doc/pub/week42/html/week42.html b/doc/pub/week42/html/week42.html
index 5d58d611d..3bbba9c6d 100644
--- a/doc/pub/week42/html/week42.html
+++ b/doc/pub/week42/html/week42.html
@@ -8,8 +8,8 @@ doconce format html week42.do.txt --pygments_html_style=default --html_style=blo
 
 
 
-
-Week 42 Solving differential equations and Convolutional (CNN)
+
+Week 42 Constructing a Neural Network code with introduction to Tensor flow