An artificial neural network is an interconnected group of nodes, inspired by a simplification of neurons in a brain. Here, each circular node represents an artificial neuron and an arrow represents a connection from the output of one artificial neuron to the input of another.

Artificial neural networks (ANNs, also shortened to neural networks (NNs) or neural nets) are a branch of machine learning models that are built using principles of neuronal organization discovered by connectionism in the biological neural networks constituting animal brains.^[1][2]

An ANN is based on a collection of connected units or nodes called artificial neurons, which loosely model the neurons in a biological brain. Each connection, like the synapses in a biological brain, can transmit a signal to other neurons. An artificial neuron receives signals then processes them and can signal neurons connected to it. The "signal" at a connection is a real number, and the output of each neuron is computed by some non-linear function of the sum of its inputs. The connections are called edges. Neurons and edges typically have a weight that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold.

Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.

A network is typically called a deep neural network if it has at least 2 hidden layers.^[3]

Training

Neural networks are typically trained through empirical risk minimization. This method is based on the idea of optimizing the network's parameters to minimize the difference, or empirical risk, between the predicted output and the actual target values in a given dataset.^[4] Gradient based methods such as backpropagation are usually used to estimate the parameters of the network.^[4] During the training phase, ANNs learn from labeled training data by iteratively updating their parameters to minimize a defined loss function.^[5] This method allows the network to generalize to unseen data.

Simplified example of training a neural network in object detection: The network is trained by multiple images that are known to depict starfish and sea urchins, which are correlated with "nodes" that represent visual features. The starfish match with a ringed texture and a star outline, whereas most sea urchins match with a striped texture and oval shape. However, the instance of a ring textured sea urchin creates a weakly weighted association between them.

Subsequent run of the network on an input image (left):^[6] The network correctly detects the starfish. However, the weakly weighted association between ringed texture and sea urchin also confers a weak signal to the latter from one of two intermediate nodes. In addition, a shell that was not included in the training gives a weak signal for the oval shape, also resulting in a weak signal for the sea urchin output. These weak signals may result in a false positive result for sea urchin.
In reality, textures and outlines would not be represented by single nodes, but rather by associated weight patterns of multiple nodes.

History

The simplest kind of feedforward neural network (FNN) is a linear network, which consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. The sum of the products of the weights and the inputs is calculated at each node. The mean squared errors between these calculated outputs and the given target values are minimized by creating an adjustment to the weights. This technique has been known for over two centuries as the method of least squares or linear regression. It was used as a means of finding a good rough linear fit to a set of points by Legendre (1805) and Gauss (1795) for the prediction of planetary movement.^{[7][8][9][10][11]}

Wilhelm Lenz and Ernst Ising created and analyzed the Ising model (1925)^[12] which is essentially a non-learning artificial recurrent neural network (RNN) consisting of neuron-like threshold elements.^[10] In 1972, Shun'ichi Amari made this architecture adaptive.^[13][10] His learning RNN was popularised by John Hopfield in 1982.^[14]

Warren McCulloch and Walter Pitts^[15] (1943) also considered a non-learning computational model for neural networks.^[16] In the late 1940s, D. O. Hebb^[17] created a learning hypothesis based on the mechanism of neural plasticity that became known as Hebbian learning. Farley and Wesley A. Clark^[18] (1954) first used computational machines, then called "calculators", to simulate a Hebbian network. In 1958, psychologist Frank Rosenblatt invented the perceptron, the first implemented artificial neural network,^{[19][20][21][22]} funded by the United States Office of Naval Research.^[23]

The invention of the perceptron raised public excitement for research in Artificial Neural Networks, causing the US government to drastically increase funding into deep learning research. This led to "the golden age of AI" fueled by the optimistic claims made by computer scientists regarding the ability of perceptrons to emulate human intelligence.^[24] For example, in 1957 Herbert Simon famously said:^[24]

It is not my aim to surprise or shock you—but the simplest way I can summarize is to say that there are now in the world machines that think, that learn and that create. Moreover, their ability to do these things is going to increase rapidly until—in a visible future—the range of problems they can handle will be coextensive with the range to which the human mind has been applied.

However, this wasn't the case as research stagnated following the work of Minsky and Papert (1969),^[25] who discovered that basic perceptrons were incapable of processing the exclusive-or circuit and that computers lacked sufficient power to train useful neural networks. This, along with other factors such as the 1973 Lighthill report by James Lighthill stating that research in Artificial Intelligence has not "produced the major impact that was then promised," shutting funding in research into the field of AI in all but two universities in the UK and in many major institutions across the world.^[26] This ushered an era called the AI Winter with reduced research into connectionism due to a decrease in government funding and an increased stress on symbolic artificial intelligence in the United States and other Western countries.^[27][26]

During the AI Winter era, however, research outside the United States continued, especially in Eastern Europe. By the time Minsky and Papert's book on Perceptrons came out, methods for training multilayer perceptrons (MLPs) were already known. The first deep learning Multi Layer Perceptron (MLP) was published by Alexey Grigorevich Ivakhnenko and Valentin Lapa in 1965, as the Group Method of Data Handling.^[28][29][30] The first deep learning MLP trained by stochastic gradient descent^[31] was published in 1967 by Shun'ichi Amari.^[32][33] In computer experiments conducted by Amari's student Saito, a five layer MLP with two modifiable layers learned useful internal representations to classify non-linearily separable pattern classes.^[33]

Self-organizing maps (SOMs) were described by Teuvo Kohonen in 1982.^[34][35] SOMs are neurophysiologically inspired^[36] neural networks that learn low-dimensional representations of high-dimensional data while preserving the topological structure of the data. They are trained using competitive learning.^[34]

The convolutional neural network (CNN) architecture with convolutional layers and downsampling layers was introduced by Kunihiko Fukushima in 1980.^[37] He called it the neocognitron. In 1969, he also introduced the ReLU (rectified linear unit) activation function.^[38][10] The rectifier has become the most popular activation function for CNNs and deep neural networks in general.^[39] CNNs have become an essential tool for computer vision.

The backpropagation algorithm is an efficient application of the Leibniz chain rule (1673)^[40] to networks of differentiable nodes.^[10] It is also known as the reverse mode of automatic differentiation or reverse accumulation, due to Seppo Linnainmaa (1970).^{[41][42][43][44][10]} The term "back-propagating errors" was introduced in 1962 by Frank Rosenblatt,^[45][10] but he did not have an implementation of this procedure, although Henry J. Kelley^[46] and Bryson^[47] had dynamic programming based continuous precursors of backpropagation^{[28][48][49][50]} already in 1960–61 in the context of control theory.^[10] In 1973, Dreyfus used backpropagation to adapt parameters of controllers in proportion to error gradients.^[51] In 1982, Paul Werbos applied backpropagation to MLPs in the way that has become standard.^[52][48] In 1986 Rumelhart, Hinton and Williams showed that backpropagation learned interesting internal representations of words as feature vectors when trained to predict the next word in a sequence.^[53]

The time delay neural network (TDNN) of Alex Waibel (1987) combined convolutions and weight sharing and backpropagation.^[54][55] In 1988, Wei Zhang et al. applied backpropagation to a CNN (a simplified Neocognitron with convolutional interconnections between the image feature layers and the last fully connected layer) for alphabet recognition.^[56][57] In 1989, Yann LeCun et al. trained a CNN to recognize handwritten ZIP codes on mail.^[58] In 1992, max-pooling for CNNs was introduced by Juan Weng et al. to help with least-shift invariance and tolerance to deformation to aid 3D object recognition.^[59][60][61] LeNet-5 (1998), a 7-level CNN by Yann LeCun et al.,^[62] that classifies digits, was applied by several banks to recognize hand-written numbers on checks digitized in 32x32 pixel images.

From 1988 onward,^[63][64] the use of neural networks transformed the field of protein structure prediction, in particular when the first cascading networks were trained on profiles (matrices) produced by multiple sequence alignments.^[65]

In the 1980s, backpropagation did not work well for deep FNNs and RNNs. To overcome this problem, Juergen Schmidhuber (1992) proposed a hierarchy of RNNs pre-trained one level at a time by self-supervised learning.^[66] It uses predictive coding to learn internal representations at multiple self-organizing time scales. This can substantially facilitate downstream deep learning. The RNN hierarchy can be collapsed into a single RNN, by distilling a higher level chunker network into a lower level automatizer network.^[66][10] In 1993, a chunker solved a deep learning task whose depth exceeded 1000.^[67]

In 1992, Juergen Schmidhuber also published an alternative to RNNs^[68] which is now called a linear Transformer or a Transformer with linearized self-attention^[69][70][10] (save for a normalization operator). It learns internal spotlights of attention:^[71] a slow feedforward neural network learns by gradient descent to control the fast weights of another neural network through outer products of self-generated activation patterns FROM and TO (which are now called key and value for self-attention).^[69] This fast weight attention mapping is applied to a query pattern.

The modern Transformer was introduced by Ashish Vaswani et al. in their 2017 paper "Attention Is All You Need."^[72] It combines this with a softmax operator and a projection matrix.^[10] Transformers have increasingly become the model of choice for natural language processing.^[73] Many modern large language models such as ChatGPT, GPT-4, and BERT use it. Transformers are also increasingly being used in computer vision.^[74]

In 1991, Juergen Schmidhuber also published adversarial neural networks that contest with each other in the form of a zero-sum game, where one network's gain is the other network's loss.^[75][76][77] The first network is a generative model that models a probability distribution over output patterns. The second network learns by gradient descent to predict the reactions of the environment to these patterns. This was called "artificial curiosity."

In 2014, this principle was used in a generative adversarial network (GAN) by Ian Goodfellow et al.^[78] Here the environmental reaction is 1 or 0 depending on whether the first network's output is in a given set. This can be used to create realistic deepfakes.^[79] Excellent image quality is achieved by Nvidia's StyleGAN (2018)^[80] based on the Progressive GAN by Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen.^[81] Here the GAN generator is grown from small to large scale in a pyramidal fashion.

Sepp Hochreiter's diploma thesis (1991)^[82] was called "one of the most important documents in the history of machine learning" by his supervisor Juergen Schmidhuber.^[10] Hochreiter identified and analyzed the vanishing gradient problem^[82][83] and proposed recurrent residual connections to solve it. This led to the deep learning method called long short-term memory (LSTM), published in Neural Computation (1997).^[84] LSTM recurrent neural networks can learn "very deep learning" tasks^[85] with long credit assignment paths that require memories of events that happened thousands of discrete time steps before. The "vanilla LSTM" with forget gate was introduced in 1999 by Felix Gers, Schmidhuber and Fred Cummins.^[86] LSTM has become the most cited neural network of the 20th century.^[10] In 2015, Rupesh Kumar Srivastava, Klaus Greff, and Schmidhuber used the LSTM principle to create the Highway network, a feedforward neural network with hundreds of layers, much deeper than previous networks.^[87][88] 7 months later, Kaiming He, Xiangyu Zhang; Shaoqing Ren, and Jian Sun won the ImageNet 2015 competition with an open-gated or gateless Highway network variant called Residual neural network.^[89] This has become the most cited neural network of the 21st century.^[10]

The development of metal–oxide–semiconductor (MOS) very-large-scale integration (VLSI), in the form of complementary MOS (CMOS) technology, enabled increasing MOS transistor counts in digital electronics. This provided more processing power for the development of practical artificial neural networks in the 1980s.^[90]

Neural networks' early successes included predicting the stock market and in 1995 a (mostly) self-driving car.^{[lower-alpha 1][91]}

Geoffrey Hinton et al. (2006) proposed learning a high-level representation using successive layers of binary or real-valued latent variables with a restricted Boltzmann machine^[92] to model each layer. In 2012, Ng and Dean created a network that learned to recognize higher-level concepts, such as cats, only from watching unlabeled images.^[93] Unsupervised pre-training and increased computing power from GPUs and distributed computing allowed the use of larger networks, particularly in image and visual recognition problems, which became known as "deep learning".^[5]

Ciresan and colleagues (2010)^[94] showed that despite the vanishing gradient problem, GPUs make backpropagation feasible for many-layered feedforward neural networks.^[95] Between 2009 and 2012, ANNs began winning prizes in image recognition contests, approaching human level performance on various tasks, initially in pattern recognition and handwriting recognition.^[96][97] For example, the bi-directional and multi-dimensional long short-term memory (LSTM)^[98][99] of Graves et al. won three competitions in connected handwriting recognition in 2009 without any prior knowledge about the three languages to be learned.^[98][99]

Ciresan and colleagues built the first pattern recognizers to achieve human-competitive/superhuman performance^[100] on benchmarks such as traffic sign recognition (IJCNN 2012).

Models

Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals

ANNs began as an attempt to exploit the architecture of the human brain to perform tasks that conventional algorithms had little success with. They soon reoriented towards improving empirical results, abandoning attempts to remain true to their biological precursors. ANNs have the ability to learn and model non-linearities and complex relationships. This is achieved by neurons being connected in various patterns, allowing the output of some neurons to become the input of others. The network forms a directed, weighted graph.^[101]

An artificial neural network consists of simulated neurons. Each neuron is connected to other nodes via links like a biological axon-synapse-dendrite connection. All the nodes connected by links take in some data and use it to perform specific operations and tasks on the data. Each link has a weight, determining the strength of one node's influence on another,^[102] allowing weights to choose the signal between neurons.

  In situation s perform action a;
  Receive consequence situation s';
  Compute emotion of being in consequence situation v(s');
  Update crossbar memory w'(a,s) = w(a,s) + v(s').

Types

ANNs have evolved into a broad family of techniques that have advanced the state of the art across multiple domains. The simplest types have one or more static components, including number of units, number of layers, unit weights and topology. Dynamic types allow one or more of these to evolve via learning. The latter is much more complicated but can shorten learning periods and produce better results. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers.

Some of the main breakthroughs include: convolutional neural networks that have proven particularly successful in processing visual and other two-dimensional data;^[141][142] long short-term memory avoid the vanishing gradient problem^[143] and can handle signals that have a mix of low and high frequency components aiding large-vocabulary speech recognition,^[144][145] text-to-speech synthesis,^{[146][48][147]} and photo-real talking heads;^[148] competitive networks such as generative adversarial networks in which multiple networks (of varying structure) compete with each other, on tasks such as winning a game^[149] or on deceiving the opponent about the authenticity of an input.^[78]

Network design

Neural architecture search (NAS) uses machine learning to automate ANN design. Various approaches to NAS have designed networks that compare well with hand-designed systems. The basic search algorithm is to propose a candidate model, evaluate it against a dataset, and use the results as feedback to teach the NAS network.^[150] Available systems include AutoML and AutoKeras.^[151] scikit-learn library provides functions to help with building a deep network from scratch. We can then implement a deep network with TensorFlow or Keras.

Design issues include deciding the number, type, and connectedness of network layers, as well as the size of each and the connection type (full, pooling, etc. ).

Hyperparameters must also be defined as part of the design (they are not learned), governing matters such as how many neurons are in each layer, learning rate, step, stride, depth, receptive field and padding (for CNNs), etc.^[152]

def train(X, y, n_hidden, learning_rate, n_iter):
    m, n_input = X.shape

    # 1. random initialize weights and biases
    w1 = np.random.randn(n_input, n_hidden)
    b1 = np.zeros((1, n_hidden))
    w2 = np.random.randn(n_hidden, 1)
    b2 = np.zeros((1, 1))

    # 2. in each iteration, feed all layers with the latest weights and biases
    for i in range(n_iter + 1):

        z2 = np.dot(X, w1) + b1

        a2 = sigmoid(z2)

        z3 = np.dot(a2, w2) + b2

        a3 = z3

        dz3 = a3 - y

        dw2 = np.dot(a2.T, dz3)

        db2 = np.sum(dz3, axis=0, keepdims=True)

        dz2 = np.dot(dz3, w2.T) * sigmoid_derivative(z2)

        dw1 = np.dot(X.T, dz2)

        db1 = np.sum(dz2, axis=0)

        # 3. update weights and biases with gradients
        w1 -= learning_rate * dw1 / m
        w2 -= learning_rate * dw2 / m
        b1 -= learning_rate * db1 / m
        b2 -= learning_rate * db2 / m

        if i % 1000 == 0:
            print("Epoch", i, "loss: ", np.mean(np.square(dz3)))

    model = {"w1": w1, "b1": b1, "w2": w2, "b2": b2}
    return model

Use

Using artificial neural networks requires an understanding of their characteristics.

• Choice of model: This depends on the data representation and the application. Overly complex models are slow learning.
• Learning algorithm: Numerous trade-offs exist between learning algorithms. Almost any algorithm will work well with the correct hyperparameters^[153] for training on a particular data set. However, selecting and tuning an algorithm for training on unseen data requires significant experimentation.
• Robustness: If the model, cost function and learning algorithm are selected appropriately, the resulting ANN can become robust.

ANN capabilities fall within the following broad categories:^[154]

• Function approximation,^[155] or regression analysis,^[156] including time series prediction, fitness approximation^[157] and modeling.
• Classification, including pattern and sequence recognition, novelty detection and sequential decision making.^[158]
• Data processing,^[159] including filtering, clustering, blind source separation^[160] and compression.
• Robotics, including directing manipulators and prostheses.

Applications

Because of their ability to reproduce and model nonlinear processes, artificial neural networks have found applications in many disciplines. Application areas include system identification and control (vehicle control, trajectory prediction,^[161] process control, natural resource management), quantum chemistry,^[162] general game playing,^[163] pattern recognition (radar systems, face identification, signal classification,^[164] 3D reconstruction,^[165] object recognition and more), sensor data analysis,^[166] sequence recognition (gesture, speech, handwritten and printed text recognition^[167]), medical diagnosis, finance^[168] (e.g. ex-ante models for specific financial long-run forecasts and artificial financial markets), data mining, visualization, machine translation, social network filtering^[169] and e-mail spam filtering. ANNs have been used to diagnose several types of cancers^[170][171] and to distinguish highly invasive cancer cell lines from less invasive lines using only cell shape information.^[172][173]

ANNs have been used to accelerate reliability analysis of infrastructures subject to natural disasters^[174][175] and to predict foundation settlements.^[176] It can also be useful to mitigate flood by the use of ANNs for modelling rainfall-runoff.^[177] ANNs have also been used for building black-box models in geoscience: hydrology,^[178][179] ocean modelling and coastal engineering,^[180][181] and geomorphology.^[182] ANNs have been employed in cybersecurity, with the objective to discriminate between legitimate activities and malicious ones. For example, machine learning has been used for classifying Android malware,^[183] for identifying domains belonging to threat actors and for detecting URLs posing a security risk.^[184] Research is underway on ANN systems designed for penetration testing, for detecting botnets,^[185] credit cards frauds^[186] and network intrusions.

ANNs have been proposed as a tool to solve partial differential equations in physics^{[187][188][189]} and simulate the properties of many-body open quantum systems.^{[190][191][192][193]} In brain research ANNs have studied short-term behavior of individual neurons,^[194] the dynamics of neural circuitry arise from interactions between individual neurons and how behavior can arise from abstract neural modules that represent complete subsystems. Studies considered long-and short-term plasticity of neural systems and their relation to learning and memory from the individual neuron to the system level.

Theoretical properties

Generalization and statistics

Applications whose goal is to create a system that generalizes well to unseen examples, face the possibility of over-training. This arises in convoluted or over-specified systems when the network capacity significantly exceeds the needed free parameters. Two approaches address over-training. The first is to use cross-validation and similar techniques to check for the presence of over-training and to select hyperparameters to minimize the generalization error.

The second is to use some form of regularization. This concept emerges in a probabilistic (Bayesian) framework, where regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting.

Confidence analysis of a neural network

Supervised neural networks that use a mean squared error (MSE) cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the confidence interval of network output, assuming a normal distribution. A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in a component-based network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is useful in classification as it gives a certainty measure on classifications.

The softmax activation function is:

${\displaystyle y_{i}={\frac {e^{x_{i}}}{\sum _{j=1}^{c}e^{x_{j}}}}}$

Criticism

Gallery

• 
  A single-layer feedforward artificial neural network. Arrows originating from ${\displaystyle \scriptstyle x_{2}}$ are omitted for clarity. There are p inputs to this network and q outputs. In this system, the value of the qth output, ${\displaystyle y_{q}}$, is calculated as ${\displaystyle \scriptstyle y_{q}=K*(\sum _{i}(x_{i}*w_{iq})-b_{q}).}$
• 
  A two-layer feedforward artificial neural network
• 
  An artificial neural network
• 
  An ANN dependency graph
• 
  A single-layer feedforward artificial neural network with 4 inputs, 6 hidden nodes and 2 outputs. Given position state and direction, it outputs wheel based control values.
• 
  A two-layer feedforward artificial neural network with 8 inputs, 2x8 hidden nodes and 2 outputs. Given position state, direction and other environment values, it outputs thruster based control values.
• 
  Parallel pipeline structure of CMAC neural network. This learning algorithm can converge in one step.

Recent Advancements and Future Directions

Artificial Neural Networks (ANNs) are a pivotal element in the realm of machine learning, resembling the structure and function of the human brain. ANNs have undergone significant advancements, particularly in their ability to model complex systems, handle large data sets, and adapt to various types of applications. Their evolution over the past few decades has been marked by notable methodological developments and a broad range of applications in fields such as image processing, speech recognition, natural language processing, finance, and medicine.

See also

Notes

References

Bibliography