Artificial neural network

An artificial neural network (ANN), usually called neural network (NN), is a mathematical model or computational model that is inspired by the structure and/or functional aspects of biological neural networks. A neural network consists of an interconnected group of artificial neurons, and it processes information using a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network during the learning phase. Modern neural networks are non-linear statistical data modeling tools. They are usually used to model complex relationships between inputs and outputs or to find patterns in data.

An artificial neural network is an interconnected group of nodes, akin to the vast network of neurons in the human brain.

Background

The original inspiration for the term Artificial Neural Network came from examination of central nervous systems and their neurons, axons, dendrites, and synapses, which constitute the processing elements of biological neural networks investigated by neuroscience. In an artificial neural network, simple artificial nodes, variously called "neurons", "neurodes", "processing elements" (PEs) or "units", are connected together to form a network of nodes mimicking the biological neural networks — hence the term "artificial neural network".

Because neuroscience is still full of unanswered questions, and since there are many levels of abstraction and therefore many ways to take inspiration from the brain, there is no single formal definition of what an artificial neural network is. Generally, it involves a network of simple processing elements that exhibit complex global behavior determined by connections between processing elements and element parameters. While an artificial neural network does not have to be adaptive per se, its practical use comes with algorithms designed to alter the strength (weights) of the connections in the network to produce a desired signal flow.

These networks are also similar to the biological neural networks in the sense that functions are performed collectively and in parallel by the units, rather than there being a clear delineation of subtasks to which various units are assigned (see also connectionism). Currently, the term Artificial Neural Network (ANN) tends to refer mostly to neural network models employed in statistics, cognitive psychology and artificial intelligence. Neural network models designed with emulation of the central nervous system (CNS) in mind are a subject of theoretical neuroscience and computational neuroscience.

In modern software implementations of artificial neural networks, the approach inspired by biology has been largely abandoned for a more practical approach based on statistics and signal processing. In some of these systems, neural networks or parts of neural networks (such as artificial neurons) are used as components in larger systems that combine both adaptive and non-adaptive elements. While the more general approach of such adaptive systems is more suitable for real-world problem solving, it has far less to do with the traditional artificial intelligence connectionist models. What they do have in common, however, is the principle of non-linear, distributed, parallel and local processing and adaptation.

Models

Neural network models in artificial intelligence are usually referred to as artificial neural networks (ANNs); these are essentially simple mathematical models defining a function $\scriptstyle f : X \rightarrow Y$ or a distribution over $\scriptstyle X$ or both $\scriptstyle X$ and $\scriptstyle Y$, but sometimes models are also intimately associated with a particular learning algorithm or learning rule. A common use of the phrase ANN model really means the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons or their connectivity).

Network function

See also: Graphical models

The word network in the term 'artificial neural network' refers to the inter–connections between the neurons in the different layers of each system. An example system has three layers. The first layer has input neurons, which send data via synapses to the second layer of neurons, and then via more synapses to the third layer of output neurons. More complex systems will have more layers of neurons with some having increased layers of input neurons and output neurons. The synapses store parameters called "weights" that manipulate the data in the calculations.

An ANN is typically defined by three types of parameters:

1. The interconnection pattern between different layers of neurons
2. The learning process for updating the weights of the interconnections
3. The activation function that converts a neuron's weighted input to its output activation.

Mathematically, a neuron's network function $\scriptstyle f(x)$ is defined as a composition of other functions $\scriptstyle g_i(x)$, which can further be defined as a composition of other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between variables. A widely used type of composition is the nonlinear weighted sum, where $\scriptstyle f (x) = K \left(\sum_i w_i g_i(x)\right)$, where $\scriptstyle K$ (commonly referred to as the activation function^[1]) is some predefined function, such as the hyperbolic tangent. It will be convenient for the following to refer to a collection of functions $\scriptstyle g_i$ as simply a vector $\scriptstyle g = (g_1, g_2, \ldots, g_n)$.

ANN dependency graph

This figure depicts such a decomposition of $\scriptstyle f$, with dependencies between variables indicated by arrows. These can be interpreted in two ways.

The first view is the functional view: the input $\scriptstyle x$ is transformed into a 3-dimensional vector $\scriptstyle h$, which is then transformed into a 2-dimensional vector $\scriptstyle g$, which is finally transformed into $\scriptstyle f$. This view is most commonly encountered in the context of optimization.

The second view is the probabilistic view: the random variable $\scriptstyle F = f(G)$ depends upon the random variable $\scriptstyle G = g(H)$, which depends upon $\scriptstyle H=h(X)$, which depends upon the random variable $\scriptstyle X$. This view is most commonly encountered in the context of graphical models.

The two views are largely equivalent. In either case, for this particular network architecture, the components of individual layers are independent of each other (e.g., the components of $\scriptstyle g$ are independent of each other given their input $\scriptstyle h$). This naturally enables a degree of parallelism in the implementation.

Recurrent ANN dependency graph

Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where $\scriptstyle f$ is shown as being dependent upon itself. However, an implied temporal dependence is not shown.

Learning

What has attracted the most interest in neural networks is the possibility of learning. Given a specific task to solve, and a class of functions $\scriptstyle F$, learning means using a set of observations to find $\scriptstyle f^* \in F$ which solves the task in some optimal sense.

This entails defining a cost function $\scriptstyle C : F \rightarrow \mathbb{R}$ such that, for the optimal solution $\scriptstyle f^*$, $\scriptstyle C(f^*) \leq C(f)$ $\scriptstyle \forall f \in F$ (i.e., no solution has a cost less than the cost of the optimal solution).

The cost function $\scriptstyle C$ is an important concept in learning, as it is a measure of how far away a particular solution is from an optimal solution to the problem to be solved. Learning algorithms search through the solution space to find a function that has the smallest possible cost.

For applications where the solution is dependent on some data, the cost must necessarily be a function of the observations, otherwise we would not be modelling anything related to the data. It is frequently defined as a statistic to which only approximations can be made. As a simple example, consider the problem of finding the model $\scriptstyle f$, which minimizes $\scriptstyle C=E\left[(f(x) - y)^2\right]$, for data pairs $\scriptstyle (x,y)$ drawn from some distribution $\scriptstyle \mathcal{D}$. In practical situations we would only have $\scriptstyle N$ samples from $\scriptstyle \mathcal{D}$ and thus, for the above example, we would only minimize $\scriptstyle \hat{C}=\frac{1}{N}\sum_{i=1}^N (f(x_i)-y_i)^2$. Thus, the cost is minimized over a sample of the data rather than the entire data set.

When $\scriptstyle N \rightarrow \infty$ some form of online machine learning must be used, where the cost is partially minimized as each new example is seen. While online machine learning is often used when $\scriptstyle \mathcal{D}$ is fixed, it is most useful in the case where the distribution changes slowly over time. In neural network methods, some form of online machine learning is frequently used for finite datasets.

See also: Mathematical optimization, Estimation theory, and Machine learning

Choosing a cost function

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (e.g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost). Ultimately, the cost function will depend on the desired task. An overview of the three main categories of learning tasks is provided below.

Learning paradigms

There are three major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning.

Supervised learning

In supervised learning, we are given a set of example pairs $\scriptstyle (x, y), x \in X, y \in Y$ and the aim is to find a function $\scriptstyle f : X \rightarrow Y$ in the allowed class of functions that matches the examples. In other words, we wish to infer the mapping implied by the data; the cost function is related to the mismatch between our mapping and the data and it implicitly contains prior knowledge about the problem domain.

A commonly used cost is the mean-squared error, which tries to minimize the average squared error between the network's output, f(x), and the target value y over all the example pairs. When one tries to minimize this cost using gradient descent for the class of neural networks called multilayer perceptrons, one obtains the common and well-known backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation). The supervised learning paradigm is also applicable to sequential data (e.g., for speech and gesture recognition). This can be thought of as learning with a "teacher," in the form of a function that provides continuous feedback on the quality of solutions obtained thus far.

Unsupervised learning

In unsupervised learning, some data $\scriptstyle x$ is given and the cost function to be minimized, that can be any function of the data $\scriptstyle x$ and the network's output, $\scriptstyle f$.

The cost function is dependent on the task (what we are trying to model) and our a priori assumptions (the implicit properties of our model, its parameters and the observed variables).

As a trivial example, consider the model $\scriptstyle f(x) = a$, where $\scriptstyle a$ is a constant and the cost $\scriptstyle C=E[(x - f(x))^2]$. Minimizing this cost will give us a value of $\scriptstyle a$ that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: for example, in compression it could be related to the mutual information between $\scriptstyle x$ and $\scriptstyle f(x)$, whereas in statistical modeling, it could be related to the posterior probability of the model given the data. (Note that in both of those examples those quantities would be maximized rather than minimized).

Tasks that fall within the paradigm of unsupervised learning are in general estimation problems; the applications include clustering, the estimation of statistical distributions, compression and filtering.

Reinforcement learning

In reinforcement learning, data $\scriptstyle x$ are usually not given, but generated by an agent's interactions with the environment. At each point in time $\scriptstyle t$, the agent performs an action $\scriptstyle y_t$ and the environment generates an observation $\scriptstyle x_t$ and an instantaneous cost $\scriptstyle c_t$, according to some (usually unknown) dynamics. The aim is to discover a policy for selecting actions that minimizes some measure of a long-term cost; i.e., the expected cumulative cost. The environment's dynamics and the long-term cost for each policy are usually unknown, but can be estimated.

More formally, the environment is modeled as a Markov decision process (MDP) with states $\scriptstyle {s_1,...,s_n}\in S$ and actions $\scriptstyle {a_1,...,a_m} \in A$ with the following probability distributions: the instantaneous cost distribution $\scriptstyle P(c_t|s_t)$, the observation distribution $\scriptstyle P(x_t|s_t)$ and the transition $\scriptstyle P(s_{t+1}|s_t, a_t)$, while a policy is defined as conditional distribution over actions given the observations. Taken together, the two define a Markov chain (MC). The aim is to discover the policy that minimizes the cost; i.e., the MC for which the cost is minimal.

ANNs are frequently used in reinforcement learning as part of the overall algorithm.

Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

See also: dynamic programming and stochastic control

Learning algorithms

Training a neural network model essentially means selecting one model from the set of allowed models (or, in a Bayesian framework, determining a distribution over the set of allowed models) that minimizes the cost criterion. There are numerous algorithms available for training neural network models; most of them can be viewed as a straightforward application of optimization theory and statistical estimation.

Most of the algorithms used in training artificial neural networks employ some form of gradient descent. This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction.

Evolutionary methods, simulated annealing, expectation-maximization, non-parametric methods and particle swarm optimization are some commonly used methods for training neural networks.

See also: machine learning

Employing artificial neural networks

Perhaps the greatest advantage of ANNs is their ability to be used as an arbitrary function approximation mechanism that 'learns' from observed data. However, using them is not so straightforward and a relatively good understanding of the underlying theory is essential.

• Choice of model: This will depend on the data representation and the application. Overly complex models tend to lead to problems with learning.
• Learning algorithm: There are numerous trade-offs between learning algorithms. Almost any algorithm will work well with the correct hyperparameters for training on a particular fixed data set. However selecting and tuning an algorithm for training on unseen data requires a significant amount of experimentation.
• Robustness: If the model, cost function and learning algorithm are selected appropriately the resulting ANN can be extremely robust.

With the correct implementation, ANNs can be used naturally in online learning and large data set applications. Their simple implementation and the existence of mostly local dependencies exhibited in the structure allows for fast, parallel implementations in hardware.

Applications

The utility of artificial neural network models lies in the fact that they can be used to infer a function from observations. This is particularly useful in applications where the complexity of the data or task makes the design of such a function by hand impractical.

Real-life applications

The tasks artificial neural networks are applied to tend to fall within the following broad categories:

• Function approximation, or regression analysis, including time series prediction, fitness approximation and modeling.
• Classification, including pattern and sequence recognition, novelty detection and sequential decision making.
• Data processing, including filtering, clustering, blind source separation and compression.
• Robotics, including directing manipulators, Computer numerical control.

Application areas include system identification and control (vehicle control, process control), quantum chemistry,^[2] game-playing and decision making (backgammon, chess, poker), pattern recognition (radar systems, face identification, object recognition and more), sequence recognition (gesture, speech, handwritten text recognition), medical diagnosis, financial applications (automated trading systems), data mining (or knowledge discovery in databases, "KDD"), visualization and e-mail spam filtering.

Neural networks and neuroscience

Theoretical and computational neuroscience is the field concerned with the theoretical analysis and computational modeling of biological neural systems. Since neural systems are intimately related to cognitive processes and behavior, the field is closely related to cognitive and behavioral modeling.

The aim of the field is to create models of biological neural systems in order to understand how biological systems work. To gain this understanding, neuroscientists strive to make a link between observed biological processes (data), biologically plausible mechanisms for neural processing and learning (biological neural network models) and theory (statistical learning theory and information theory).

Types of models

Many models are used in the field defined at different levels of abstraction and modeling different aspects of neural systems. They range from models of the short-term behavior of individual neurons, models of how the dynamics of neural circuitry arise from interactions between individual neurons and finally to models of how behavior can arise from abstract neural modules that represent complete subsystems. These include models of the long-term, and short-term plasticity, of neural systems and their relations to learning and memory from the individual neuron to the system level.

Current research

While initial research had been concerned mostly with the electrical characteristics of neurons, a particularly important part of the investigation in recent years has been the exploration of the role of neuromodulators such as dopamine, acetylcholine, and serotonin on behavior and learning.

Biophysical models, such as BCM theory, have been important in understanding mechanisms for synaptic plasticity, and have had applications in both computer science and neuroscience. Research is ongoing in understanding the computational algorithms used in the brain, with some recent biological evidence for radial basis networks and neural backpropagation as mechanisms for processing data.

Computational devices have been created in CMOS for both biophysical simulation and neuromorphic computing. More recent efforts show promise for creating nanodevices for very large scale principal components analyses and convolution. If successful, these effort could usher in a new era of neural computing that is a step beyond digital computing, because it depends on learning rather than programming and because it is fundamentally analog rather than digital even though the first instantiations may in fact be with CMOS digital devices.

Neural network software

Main article: Neural network software

Neural network software is used to simulate, research, develop and apply artificial neural networks, biological neural networks and in some cases a wider array of adaptive systems.

Types of artificial neural networks

Main article: Types of artificial neural networks

Artificial neural network types vary from those with only one or two layers of single direction logic, to complicated multi–input many directional feedback loop and layers. On the whole, these systems use algorithms in their programming to determine control and organization of their functions. Some may be as simple, one neuron layer with an input and an output, and others can mimic complex systems such as dANN, which can mimic chromosomal DNA through sizes at cellular level, into artificial organisms and simulate reproduction, mutation and population sizes.^[3]

Most systems use "weights" to change the parameters of the throughput and the varying connections to the neurons. Artificial neural networks can be autonomous and learn by input from outside "teachers" or even self-teaching from written in rules.

Theoretical properties

Computational power

The multi-layer perceptron (MLP) is a universal function approximator, as proven by the Cybenko theorem. However, the proof is not constructive regarding the number of neurons required or the settings of the weights.

Work by Hava Siegelmann and Eduardo D. Sontag has provided a proof that a specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a Universal Turing Machine^[4] using a finite number of neurons and standard linear connections. They have further shown that the use of irrational values for weights results in a machine with super-Turing power.^{[citation needed]}

Capacity

Artificial neural network models have a property called 'capacity', which roughly corresponds to their ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity.

Convergence

Nothing can be said in general about convergence since it depends on a number of factors. Firstly, there may exist many local minima. This depends on the cost function and the model. Secondly, the optimization method used might not be guaranteed to converge when far away from a local minimum. Thirdly, for a very large amount of data or parameters, some methods become impractical. In general, it has been found that theoretical guarantees regarding convergence are an unreliable guide to practical application.

Generalization and statistics

In applications where the goal is to create a system that generalizes well in unseen examples, the problem of over-training has emerged. This arises in convoluted or over-specified systems when the capacity of the network significantly exceeds the needed free parameters. There are two schools of thought for avoiding this problem: The first is to use cross-validation and similar techniques to check for the presence of overtraining and optimally select hyperparameters such as to minimize the generalization error. The second is to use some form of regularization. This is a concept that emerges naturally in a probabilistic (Bayesian) framework, where the 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 an 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 the output of the network, 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 on the output layer of the neural network (or a softmax component in a component-based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is very useful in classification as it gives a certainty measure on classifications.

The softmax activation function is:

$y_i=\frac{e^{x_i}}{\sum_{j=1}^c e^{x_j}}$

Dynamic properties

Various techniques originally developed for studying disordered magnetic systems (i.e., the spin glass) have been successfully applied to simple neural network architectures, such as the Hopfield network. Influential work by E. Gardner and B. Derrida has revealed many interesting properties about perceptrons with real-valued synaptic weights, while later work by W. Krauth and M. Mezard has extended these principles to binary-valued synapses.

Criticism

A common criticism of artificial neural networks, particularly in robotics, is that they require a large diversity of training for real-world operation. Dean Pomerleau, in his research presented in the paper "Knowledge-based Training of Artificial Neural Networks for Autonomous Robot Driving," uses a neural network to train a robotic vehicle to drive on multiple types of roads (single lane, multi-lane, dirt, etc.). A large amount of his research is devoted to (1) extrapolating multiple training scenarios from a single training experience, and (2) preserving past training diversity so that the system does not become overtrained (if, for example, it is presented with a series of right turns – it should not learn to always turn right). These issues are common in neural networks that must decide from amongst a wide variety of responses.

A. K. Dewdney, a former Scientific American columnist, wrote in 1997, "Although neural nets do solve a few toy problems, their powers of computation are so limited that I am surprised anyone takes them seriously as a general problem-solving tool." (Dewdney, p. 82)

Arguments for Dewdney's position are that to implement large and effective software neural networks, much processing and storage resources need to be committed. While the brain has hardware tailored to the task of processing signals through a graph of neurons, simulating even a most simplified form on Von Neumann technology may compel a NN designer to fill many millions of database rows for its connections - which can lead to excessive RAM and HD necessities. Furthermore, the designer of NN systems will often need to simulate the transmission of signals through many of these connections and their associated neurons - which must often be matched with incredible amounts of CPU processing power and time. While neural networks often yield effective programs, they too often do so at the cost of time and money efficiency.

Arguments against Dewdney's position are that neural nets have been successfully used to solve many complex and diverse tasks, ranging from autonomously flying aircraft^[5] to detecting credit card fraud.^[6] Technology writer Roger Bridgman commented on Dewdney's statements about neural nets:

Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be "an opaque, unreadable table...valueless as a scientific resource". In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers. An unreadable table that a useful machine could read would still be well worth having.^[7]

Some other criticisms came from believers of hybrid models (combining neural networks and symbolic approaches). They advocate the intermix of these two approaches and believe that hybrid models can better capture the mechanisms of the human mind (Sun and Bookman 1994).

Gallery

• 

  A single-layer feedforward artificial neural network. Arrows originating from $\scriptstyle x_2$ are omitted for clarity. There are p inputs to this network and q outputs. There is no activation function (or equivalently, the activation function is $\scriptstyle g(x)=x$). In this system, the value of the qth output, $\scriptstyle y_q$ would be calculated as $\scriptstyle y_q = \sum(x_i*w_{iq})$

• 

  A two-layer feedforward artificial neural network.

• 
• 

See also

• 20Q
• Adaptive resonance theory
• Artificial life
• Associative memory
• Autoencoder
• Biologically inspired computing
• Blue brain
• Cascade Correlation
• Clinical decision support system
• Connectionist expert system
• Decision tree
• Expert system
• Fuzzy logic
• Genetic algorithm
• In Situ Adaptive Tabulation
• Linear discriminant analysis
• Logistic regression
• Memristor
• Nearest neighbor (pattern recognition)
• Neuroevolution, NeuroEvolution of Augmented Topologies (NEAT), HyperNEAT
• Neural gas
• Ni1000 chip
• Optical neural network
• Predictive analytics
• Systolic array
• Time delay neural network (TDNN)

References

1. ^ "The Machine Learning Dictionary". http://www.cse.unsw.edu.au/~billw/mldict.html#activnfn. 
2. ^ Roman M. Balabin, Ekaterina I. Lomakina (2009). "Neural network approach to quantum-chemistry data: Accurate prediction of density functional theory energies". J. Chem. Phys. 131 (7): 074104. doi:10.1063/1.3206326. PMID 19708729. 
3. ^ "DANN:Genetic Wavelets". dANN project. http://wiki.syncleus.com/index.php/DANN:Genetic_Wavelets. Retrieved 12 July 2010. 
4. ^ Siegelmann, H.T.; Sontag, E.D. (1991). "Turing computability with neural nets". Appl. Math. Lett. 4 (6): 77–80. doi:10.1016/0893-9659(91)90080-F. http://www.math.rutgers.edu/~sontag/FTP_DIR/aml-turing.pdf. 
5. ^ "NASA NEURAL NETWORK PROJECT PASSES MILESTONE". NASA. http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html. Retrieved 12 July 2010. 
6. ^ "Counterfeit Fraud" (PDF). VISA. p. 1. http://www.visa.ca/en/personal/pdfs/counterfeit_fraud.pdf. Retrieved 12 July 2010. "Neural Networks (24/7 Monitoring):" 
7. ^ Roger Bridgman's defense of neural networks

Bibliography

• Bar-Yam, Yaneer (2003). Dynamics of Complex Systems, Chapter 2. http://necsi.edu/publications/dcs/Bar-YamChap2.pdf. 
• Bar-Yam, Yaneer (2003). Dynamics of Complex Systems, Chapter 3. http://necsi.edu/publications/dcs/Bar-YamChap3.pdf. 
• Bar-Yam, Yaneer (2005). Making Things Work. http://necsi.edu/publications/mtw/.  Please see Chapter 3
• Bhadeshia H. K. D. H. (1999). "Neural Networks in Materials Science". ISIJ International 39 (10): 966–979. doi:10.2355/isijinternational.39.966. http://www.msm.cam.ac.uk/phase-trans/abstracts/neural.review.pdf. 
• Bishop, C.M. (1995) Neural Networks for Pattern Recognition, Oxford: Oxford University Press. ISBN 0-19-853849-9 (hardback) or ISBN 0-19-853864-2 (paperback)
• Cybenko, G.V. (1989). Approximation by Superpositions of a Sigmoidal function, Mathematics of Control, Signals, and Systems, Vol. 2 pp. 303–314. electronic version
• Duda, R.O., Hart, P.E., Stork, D.G. (2001) Pattern classification (2nd edition), Wiley, ISBN 0-471-05669-3
• Egmont-Petersen, M., de Ridder, D., Handels, H. (2002). "Image processing with neural networks - a review". Pattern Recognition 35 (10): 2279–2301. doi:10.1016/S0031-3203(01)00178-9. 
• Gurney, K. (1997) An Introduction to Neural Networks London: Routledge. ISBN 1-85728-673-1 (hardback) or ISBN 1-85728-503-4 (paperback)
• Haykin, S. (1999) Neural Networks: A Comprehensive Foundation, Prentice Hall, ISBN 0-13-273350-1
• Fahlman, S, Lebiere, C (1991). The Cascade-Correlation Learning Architecture, created for National Science Foundation, Contract Number EET-8716324, and Defense Advanced Research Projects Agency (DOD), ARPA Order No. 4976 under Contract F33615-87-C-1499. electronic version
• Hertz, J., Palmer, R.G., Krogh. A.S. (1990) Introduction to the theory of neural computation, Perseus Books. ISBN 0-201-51560-1
• Lawrence, Jeanette (1994) Introduction to Neural Networks, California Scientific Software Press. ISBN 1-883157-00-5
• Masters, Timothy (1994) Signal and Image Processing with Neural Networks, John Wiley & Sons, Inc. ISBN 0-471-04963-8
• Ness, Erik. 2005. SPIDA-Web. Conservation in Practice 6(1):35-36. On the use of artificial neural networks in species taxonomy.
• Ripley, Brian D. (1996) Pattern Recognition and Neural Networks, Cambridge
• Siegelmann, H.T. and Sontag, E.D. (1994). Analog computation via neural networks, Theoretical Computer Science, v. 131, no. 2, pp. 331–360. electronic version
• Sergios Theodoridis, Konstantinos Koutroumbas (2009) "Pattern Recognition", 4th Edition, Academic Press, ISBN 978-1-59749-272-0.
• Smith, Murray (1993) Neural Networks for Statistical Modeling, Van Nostrand Reinhold, ISBN 0-442-01310-8
• Wasserman, Philip (1993) Advanced Methods in Neural Computing, Van Nostrand Reinhold, ISBN 0-442-00461-3

Further reading

• Dedicated issue of Philosophical Transactions B on Neural Networks and Perception. Some articles are freely available.

External links

• Neural Networks at the Open Directory Project
• A close view to Artificial Neural Networks Algorithms
• A Brief Introduction to Neural Networks (D. Kriesel) - Illustrated, bilingual manuscript about artificial neural networks; Topics so far: Perceptrons, Backpropagation, Radial Basis Functions, Recurrent Neural Networks, Self Organizing Maps, Hopfield Networks.
• Neural Networks in Materials Science
• A practical tutorial on Neural Networks
• Applications of neural networks