Implementation of Artificial Neural Network for XOR Logic Gate with 2-bit Binary Input

Publicado Por Alfonso Moraleja Juárez el 15/03/2023

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. 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.

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. We are going to propagate backwards in order to the determine the weights and biases. In order to do so we need to represent the error in the layer before the final one \( L-1 \) in terms of the errors in the final output layer. The assertions in the book ‘Perceptrons’ by Minsky was inspite of his thorough knowledge that the powerful perceptrons have multiple layers and that Rosenblatt’s basic feed-forward perceptrons have three layers. In the book, to deceive unsuspecting readers, Minsky defined a perceptron as a two-layer machine that can handle only linearly separable problems and, for example, cannot solve the exclusive-OR problem.

The main principle behind it is that each parameter changes in proportion to how much it affects the network’s output. A weight that has barely any effect on the output of the model will show a very small change, while one that has a large negative impact will change drastically to improve the model’s prediction power. Though there are many kinds of activation functions, we’ll be using a simple linear activation function for our perceptron. The linear activation function has no effect on its input and outputs it as is. These parameters are what we update when we talk about “training” a model.

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. The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. Here it is convenient to use stochastic gradient descent with mini-batches with an outer loop that steps through multiple epochs of training. Let us first try to fit various gates using standard linear regression.

Visualize model architecture

This motivates the use of unsupervised methods which in part circumvent these problems. A popular technique to lessen the exploding gradients problem is to simply clip the gradients during backpropagation so that they never exceed some threshold . Or use tools like Oscar, which implements more complex algorithms to help you find a good set of hyperparameters quickly. The factor \( \lambda \) is known as a regularization parameter. # This is expensive because it uses the whole dataset, so we don’t want to do it too often. The classification problem can be summarized as creating a boundary between the red and the blue dots.

• We have some instance variables like the training data, the target, the number of input nodes and the learning rate.
• For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
• The findings of this study revealed that attitude is the most significant factor that affects the consumers’ behavioral intention.
• They found that MLP-based prediction tool produces better predictions than other examined approaches.
• This allows for the plotting of the errors over the training process.

So far, various schemes have been proposed and utilized for all-optical logic operations. Artificial neural networks , a popular nonlinear mapping technique, can overcome some of these challenges. This technique can be used as an alternative way to solve complex engineering problems such as a function approximator (Mesaritakis et al., 2016, Olyaee et al., 2009). Mathematical procedure in ANN algorithms that attain the formula of physical relationships of the concerned problem is not complicated and their processing time is fast. Excellent mapping can be obtained by this technique if the technician is trained correctly. Accurate and fast neural network models can be developed from measured or simulated data over a range of geometrical parameter values.

Designing an ANN to Solve the XOR Logic Problem

Dense is used to define layers of neural networks with parameters like the number of neurons, input_shape, and activation function. 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. The number of input nodes does not need to equal the number of output nodes. Each layer may have its own number of nodes and activation functions.

The loss function we used in our MLP model is the Mean Squared loss function. Though this is a very popular loss function, it makes some assumptions on the data and isn’t always convex when it comes to a classification problem. It was used here to make it easier to understand how a perceptron works, but for classification tasks, there are better alternatives, like binary cross-entropy loss. Adding more layers or nodes gives increasingly complex decision boundaries. But this could also lead to something called overfitting — where a model achieves very high accuracies on the training data, but fails to generalize.

Neural networks and back propagation algorithm

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.

To train our perceptron, we must ensure that we correctly classify all of our train data. Note that this is different from how you would train a neural network, where you wouldn’t try and correctly classify your entire training data. That would lead to something called overfitting in most cases.

In this tutorial I will not discuss exactly how these xor neural networks work, but instead I will show how flexible these models can be by training an ANN that will act as a XOR logic gate. We will choose one extra hidden layer apart from the input and output layers. For that, we also need to define the activation and loss function for them and update the parameters using the gradient descent optimization algorithm. An artificial neural network is made of layers, and a layer is made of many perceptrons . Perceptron is the basic computational unit of the neural network, which multiplies input with weight, adds bias, and passes the result from the activation function to deliver the output to the next layer.

Artificial intelligence-aided nanoplasmonic biosensor modeling

To bring everything together, we create a simple Perceptron class with the functions we just discussed. We have some instance variables like the training data, the target, the number of input nodes and the learning rate. It is also important to note that ANNs must undergo a ‘learning process’ with training data before they are ready to be implemented. Looking at the logistic activation function, when inputs become large , the function saturates at 0 or 1, with a derivative extremely close to 0. To measure how well our neural network is doing we need to introduce a cost function.

We use this value to update weights and we can multiply learning rate before we adjust the weight. However, usually the weights are much more important than the particular function chosen. These sigmoid functions are very similar, and the output differences are small. Note that all functions are normalized in such a way that their slope at the origin is 1. When the Perceptron is ‘upgraded’ to include one or more hidden layers, the network becomes a Multilayer Perceptron Network . We have defined the getORdata function for fetching inputs and outputs.

It fails to map the output for XOR because the data points are in a non-linear arrangement, and hence we need a model which can learn these complexities. Adding a hidden layer will help the Perceptron to learn that non-linearity. 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. Where \( f’ \) 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 unknowwn quantities are our weights \( w_ \) and we need to find an algorithm for changing them so that our errors are as small as possible. For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units. Hereafter we will call the various entities of a layer for nodes. A neural network with one or more layers of nodes between the input and the output nodes.

You can just use linear decision neurons for this with adjusting the biases for the tresholds. The inputs of the NOT AND gate should be negative for the 0/1 inputs. This picture should make it more clear, the values on the connections are the weights, the values in the neurons are the biases, the decision functions act as 0/1 decisions . The approach calculates quantum optical characteristics and the current flow in a semiconductor quantum dot device . In recent years, artificial neural networks algorithms have become successful -in a wide variety of tasks- for accurate modeling of systems [26–31]. Following this, two functions were created using the previously explained equations for the forward pass and backward pass.

To design a hidden layer, we need to define the key constituents again first. The basic principle of matrix multiplication says if the shape of X is and W is , then only they can be multiplied, and the shape of XW will be . As we can see the network does not seem to be learning at all.

https://forexhero.info/ networks in real-world are typically implemented using a deep-learning framework such as tensorflow. But, building a neural network with very minimal dependencies helps one gain an understanding of how neural networks work. This understanding is essential to designing effective neural network models.

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). For regression tasks, you can simply use no activation function at all. We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.

The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. 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. As stated earlier, an important theorem in studies of neural networks, restated without proof here, is the universal approximation theorem. For a more in depth discussion on neural networks we recommend Goodfellow et al chapters 6 and 7. Chapters 11 and 12 contain alot of material on practicalities and applications.

When the boolean argument is set as true, the sigmoid function calculates the derivative of x. Once we understood some basics and learn how to measure the performance of our network we can figure out a lot of exciting things through trial and error. If you made it this far we’ll have to say THANK YOU for bearing so long with us just for the sake of understanding a model to solve XOR. If there’s just one take away we hope it’s that we don’t have to be a mathematician to start with machine learning. The function simply returns it’s input without applying any math, so it’s essentially the same as using no activation function at all. Let’s see if we can hold our claim of solving XOR without any activation function at all.