View source for Artificial neural network

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{{short description|Computational model used in machine learning, based on connected, hierarchical functions}}
{{Machine learning|Artificial neural network}}
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[[File:Colored neural network.svg|thumb|upright=1.15|An artificial neural network is an interconnected group of nodes, inspired by a simplification of [[w:en:neuron|neuron]]s in a [[w:en:brain|brain]]. Here, each circular node represents an [[w:en:artificial neuron|artificial neuron]] and an arrow represents a connection from the output of one artificial neuron to the input of another.]]
'''Artificial neural networks''' ('''ANNs'''), usually simply called '''neural networks''' ('''NNs''') or '''neural nets''',<ref>{{cite web|last=Hardesty|first=Larry|title=Explained: Neural networks|date=14 April 2017|url=https://news.mit.edu/2017/explained-neural-networks-deep-learning-0414|publisher=MIT News Office|access-date=2 June 2022}}</ref> are computing systems inspired by the [[w:en:biological neural network|biological neural network]]s that constitute animal [[w:en:brain|brain]]s.<ref>{{cite book |last1=Yang |first1=Z.R. |last2=Yang |first2=Z. |title=Comprehensive Biomedical Physics |date=2014 |publisher=Elsevier |location=Karolinska Institute, Stockholm, Sweden |isbn=978-0-444-53633-4 |page=1 |url=https://www.sciencedirect.com/topics/neuroscience/artificial-neural-network |access-date=28 July 2022 |archive-date=28 July 2022 |archive-url=https://web.archive.org/web/20220728183237/https://www.sciencedirect.com/topics/neuroscience/artificial-neural-network |url-status=live }}</ref>

An ANN is based on a collection of connected units or nodes called [[w:en:artificial neuron|artificial neuron]]s, which loosely model the [[w:en:neuron|neuron]]s in a biological brain. Each connection, like the [[w:en:synapse|synapse]]s 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 [[w:en:real number|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 ''[[w:en:Weighting|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.{{toclimit|3}}

==Learning Units ==
* '''[[/Training/]]:''' What is the basic idea of [[machine learning]] in the context of Artificial Neural Networks.
* '''[[/History/]]:''' Learn how the knowledge and application of neural networks evolved in the past.
* '''[[/Models/]]:''' What basic models of representing an Artificial Neural Network (ANN) with Mathematics and/or with a corresponding computer model.
** [[/Neuron/|Model of Neuron]]
** [[/Network/|Model of a Network of Neurons]]

==Models==
[[File:Neuron3.png|thumb|right|upright=1.35|Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals]]

=== Hyperparameter ===
{{Main|w:en:Hyperparameter (machine learning)}}
A [[w:en:hyperparameter|hyperparameter]] is a constant [[w:en:parameter|parameter]] whose value is set before the learning process begins. The values of [[w:en:parameter|parameter]]s are derived via learning. Examples of hyperparameters include [[w:en:learning rate|learning rate]], the number of hidden layers and batch size.<ref>{{Cite web|url=https://towardsdatascience.com/a-walkthrough-of-convolutional-neural-network-7f474f91d7bd|title=A Walkthrough of Convolutional Neural Network – Hyperparameter Tuning|last=Lau|first=Suki|date=10 July 2017|website=Medium|access-date=23 August 2019|archive-date=4 February 2023|archive-url=https://web.archive.org/web/20230204153904/https://towardsdatascience.com/a-walkthrough-of-convolutional-neural-network-7f474f91d7bd?gi=95062e80a5b5|url-status=live}}</ref> The values of some hyperparameters can be dependent on those of other hyperparameters. For example, the size of some layers can depend on the overall number of layers.

===Learning===
{{No footnotes|date=August 2019|section}}{{See also|Mathematical optimization|Estimation theory|Machine learning}}

Learning is the adaptation of the network to better handle a task by considering sample observations. Learning involves adjusting the weights (and optional thresholds) of the network to improve the accuracy of the result. This is done by minimizing the observed errors. Learning is complete when examining additional observations does not usefully reduce the error rate. Even after learning, the error rate typically does not reach 0. If after learning, the error rate is too high, the network typically must be redesigned. Practically this is done by defining a [[w:en:Loss function|cost function]] that is evaluated periodically during learning. As long as its output continues to decline, learning continues. The cost is frequently defined as a [[w:en:statistic|statistic]] whose value can only be approximated. The outputs are actually numbers, so when the error is low, the difference between the output (almost certainly a cat) and the correct answer (cat) is small. Learning attempts to reduce the total of the differences across the observations. Most learning models can be viewed as a straightforward application of [[w:en:Mathematical optimization|optimization]] theory and [[w:en:statistical estimation|statistical estimation]].<ref name="Zell1994ch5.2"/><ref>{{Cite book|last1=Kelleher|first1=John D. |last2=Mac Namee|first2=Brian|last3=D'Arcy|first3=Aoife |title=Fundamentals of machine learning for predictive data analytics: algorithms, worked examples, and case studies|date=2020|isbn=978-0-262-36110-1 |edition=2nd|location=Cambridge, MA|chapter=7-8|oclc=1162184998}}</ref>

==== Learning rate ====
The learning rate defines the size of the corrective steps that the model takes to adjust for errors in each observation.<ref>{{cite arXiv|last=Wei|first=Jiakai|date=2019-04-26|title=Forget the Learning Rate, Decay Loss|class=cs.LG|eprint=1905.00094}}</ref> A high learning rate shortens the training time, but with lower ultimate accuracy, while a lower learning rate takes longer, but with the potential for greater accuracy. Optimizations such as [[w:en:Quickprop|Quickprop]] are primarily aimed at speeding up error minimization, while other improvements mainly try to increase reliability. In order to avoid [[w:en:oscillation|oscillation]] inside the network such as alternating connection weights, and to improve the rate of convergence, refinements use an [[w:en:adaptive learning rate|adaptive learning rate]] that increases or decreases as appropriate.<ref>{{Cite book|title=The Improved Training Algorithm of Back Propagation Neural Network with Self-adaptive Learning Rate|last1=Li|first1=Y.|last2=Fu|first2=Y.|last3=Li|first3=H.|last4=Zhang|first4=S. W.|s2cid=10557754|date=1 June 2009|journal=2009 International Conference on Computational Intelligence and Natural Computing|isbn=978-0-7695-3645-3|volume=1|pages=73–76|doi=10.1109/CINC.2009.111}}</ref> The concept of momentum allows the balance between the gradient and the previous change to be weighted such that the weight adjustment depends to some degree on the previous change. A momentum close to 0 emphasizes the gradient, while a value close to 1 emphasizes the last change.

====Cost function====
While it is possible to define a cost function [[w:en:ad hoc|ad hoc]], frequently the choice is determined by the function's desirable properties (such as [[w:en:Convex function|convexity]]) or because it arises from the model (e.g. in a probabilistic model the model's [[w:en:posterior probability|posterior probability]] can be used as an inverse cost).

====Backpropagation====
{{Main|w:en:Backpropagation}}
Backpropagation is a method used to adjust the connection weights to compensate for each error found during learning. The error amount is effectively divided among the connections. Technically, backprop calculates the [[w:en:gradient|gradient]] (the derivative) of the [[w:en:loss function|cost function]] associated with a given state with respect to the weights. The weight updates can be done via [[w:en:stochastic gradient descent|stochastic gradient descent]] or other methods, such as [[w:en:Extreme Learning Machines|Extreme Learning Machines]],<ref>{{cite journal|last1=Huang|first1=Guang-Bin|last2=Zhu |first2=Qin-Yu|last3=Siew|first3=Chee-Kheong|year=2006|title=Extreme learning machine: theory and applications|journal=Neurocomputing|volume=70|issue=1 |pages=489–501|doi=10.1016/j.neucom.2005.12.126 |citeseerx=10.1.1.217.3692|s2cid=116858 }}</ref> "No-prop" networks,<ref>{{cite journal|year=2013|title=The no-prop algorithm: A new learning algorithm for multilayer neural networks |journal=Neural Networks|volume=37 |pages=182–188|doi=10.1016/j.neunet.2012.09.020|pmid=23140797|last1=Widrow|first1=Bernard|display-authors=etal}}</ref> training without backtracking,<ref>{{cite arXiv|eprint=1507.07680|first1=Yann |last1=Ollivier|first2=Guillaume|last2=Charpiat|title=Training recurrent networks without backtracking |year=2015|class=cs.NE}}</ref> "weightless" networks,<ref name="RBMTRAIN">{{Cite journal |last=Hinton |first=G. E. |date=2010 |title=A Practical Guide to Training Restricted Boltzmann Machines |url=https://www.researchgate.net/publication/221166159 |journal=Tech. Rep. UTML TR 2010-003 |access-date=27 June 2017 |archive-date=9 May 2021 |archive-url=https://web.archive.org/web/20210509123211/https://www.researchgate.net/publication/221166159_A_brief_introduction_to_Weightless_Neural_Systems |url-status=live }}</ref><ref>ESANN. 2009.{{full citation needed|date=June 2022}}</ref> and [[w:en:Holographic associative memory|non-connectionist neural networks]].{{citation needed|date=June 2022}}

===Learning paradigms===
{{No footnotes|date=August 2019|section}}
Machine learning is commonly separated into three main learning paradigms, [[w:en:supervised learning|supervised learning]], [[w:en:unsupervised learning|unsupervised learning]] and [[w:en:reinforcement learning|reinforcement learning]].<ref>{{cite book|url=https://www.wolfram.com/language/introduction-machine-learning/|title=Introduction to Machine Learning|first1=Etienne|publisher=Wolfram Media Inc|year=2021|isbn=978-1-579550-48-6|page=9|last1=Bernard}}</ref> Each corresponds to a particular learning task.

==== Supervised learning ====
[[w:en:Supervised learning|Supervised learning]] uses a set of paired inputs and desired outputs. The learning task is to produce the desired output for each input. In this case, the cost function is related to eliminating incorrect deductions.<ref>{{Cite journal|last1=Ojha|first1=Varun Kumar|last2=Abraham|first2=Ajith|last3=Snášel|first3=Václav|date=1 April 2017|title=Metaheuristic design of feedforward neural networks: A review of two decades of research|journal=Engineering Applications of Artificial Intelligence|volume=60|pages=97–116|doi=10.1016/j.engappai.2017.01.013|arxiv=1705.05584|bibcode=2017arXiv170505584O|s2cid=27910748}}</ref> A commonly used cost is the [[w:en:mean-squared error|mean-squared error]], which tries to minimize the average squared error between the network's output and the desired output. Tasks suited for supervised learning are [[w:en:pattern recognition|pattern recognition]] (also known as classification) and [[w:en:Regression analysis|regression]] (also known as function approximation). Supervised learning is also applicable to sequential data (e.g., for hand writing, speech and [[w:en:gesture recognition|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 [[w:en:unsupervised learning|unsupervised learning]], input data is given along with the cost function, some function of the data <math>\textstyle x</math> and the network's output. The cost function is dependent on the task (the model domain) and any ''[[w:en:A priori and a posteriori|a priori]]'' assumptions (the implicit properties of the model, its parameters and the observed variables). As a trivial example, consider the model <math>\textstyle f(x) = a</math> where <math>\textstyle a</math> is a constant and the cost <math>\textstyle C=E[(x - f(x))^2]</math>. Minimizing this cost produces a value of <math>\textstyle a</math> 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 [[w:en:Data compression|compression]] it could be related to the [[w:en:mutual information|mutual information]] between <math>\textstyle x</math> and <math>\textstyle f(x)</math>, whereas in statistical modeling, it could be related to the [[w:en:posterior probability|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 [[w:en:Approximation|estimation]] problems; the applications include [[w:en:Data clustering|clustering]], the estimation of [[w:en:statistical distributions|statistical distributions]], [[w:en:Data compression|compression]] and [[w:en:Bayesian spam filtering|filtering]].

====Reinforcement learning====
{{Main|w:en:Reinforcement learning}}
{{See also|w:en:Stochastic control}}

In applications such as playing video games, an actor takes a string of actions, receiving a generally unpredictable response from the environment after each one. The goal is to win the game, i.e., generate the most positive (lowest cost) responses. In [[w:en:reinforcement learning|reinforcement learning]], the aim is to weight the network (devise a policy) to perform actions that minimize long-term (expected cumulative) cost. At each point in time the agent performs an action and the environment generates an observation and an [[w:en:instant|instant]]aneous cost, according to some (usually unknown) rules. The rules and the long-term cost usually only can be estimated. At any juncture, the agent decides whether to explore new actions to uncover their costs or to exploit prior learning to proceed more quickly.

Formally the environment is modeled as a [[w:en:Markov decision process|Markov decision process]] (MDP) with states <math>\textstyle {s_1,...,s_n}\in S </math> and actions <math>\textstyle {a_1,...,a_m} \in A</math>. Because the state transitions are not known, probability distributions are used instead: the instantaneous cost distribution <math>\textstyle P(c_t|s_t)</math>, the observation distribution <math>\textstyle P(x_t|s_t)</math> and the transition distribution <math>\textstyle P(s_{t+1}|s_t, a_t)</math>, while a policy is defined as the conditional distribution over actions given the observations. Taken together, the two define a [[w:en:Markov chain|Markov chain]] (MC). The aim is to discover the lowest-cost MC.

ANNs serve as the learning component in such applications.<ref>{{cite conference | author = Dominic, S. | author2 = Das, R. | author3 = Whitley, D. | author4 = Anderson, C. | date = July 1991 | title = Genetic reinforcement learning for neural networks | pages = 71–76 | conference = IJCNN-91-Seattle International Joint Conference on Neural Networks | book-title = IJCNN-91-Seattle International Joint Conference on Neural Networks | publisher = IEEE | location = Seattle, Washington, USA | doi = 10.1109/IJCNN.1991.155315 | isbn = 0-7803-0164-1 | url-access = registration | url = https://archive.org/details/ijcnn91seattlein01ieee }}</ref><ref>{{cite journal |last=Hoskins |first=J.C. |author2=Himmelblau, D.M. |title=Process control via artificial neural networks and reinforcement learning |journal=Computers & Chemical Engineering |year=1992 |volume=16 |pages=241–251 |doi=10.1016/0098-1354(92)80045-B |issue=4}}</ref> [[w:en:Dynamic programming|Dynamic programming]] coupled with ANNs (giving [[w:en:Neural oscillation|neurodynamic]] programming)<ref>{{cite book|url=https://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images|title=Neuro-dynamic programming|first1=D.P.|first2=J.N.|publisher=Athena Scientific|year=1996|isbn=978-1-886529-10-6|page=512|last1=Bertsekas|last2=Tsitsiklis|access-date=17 June 2017|archive-date=29 June 2017|archive-url=https://web.archive.org/web/20170629172039/http://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images|url-status=live}}</ref> has been applied to problems such as those involved in [[w:en:vehicle routing|vehicle routing]],<ref>{{cite journal |last=Secomandi |first=Nicola |title=Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands |journal=Computers & Operations Research |year=2000 |volume=27 |pages=1201–1225 |doi=10.1016/S0305-0548(99)00146-X |issue=11–12|citeseerx=10.1.1.392.4034 }}</ref> video games, [[w:en:natural resource management|natural resource management]]<ref>{{cite conference | author = de Rigo, D. | author2 = Rizzoli, A. E. | author3 = Soncini-Sessa, R. | author4 = Weber, E. | author5 = Zenesi, P. | year = 2001 | title = Neuro-dynamic programming for the efficient management of reservoir networks | conference = MODSIM 2001, International Congress on Modelling and Simulation | url = http://www.mssanz.org.au/MODSIM01/MODSIM01.htm | book-title = Proceedings of MODSIM 2001, International Congress on Modelling and Simulation | publisher = Modelling and Simulation Society of Australia and New Zealand | location = Canberra, Australia | doi = 10.5281/zenodo.7481 | isbn = 0-86740-525-2 | access-date = 29 July 2013 | archive-date = 7 August 2013 | archive-url = https://web.archive.org/web/20130807223658/http://mssanz.org.au/MODSIM01/MODSIM01.htm | url-status = live }}</ref><ref>{{cite conference| author = Damas, M. |author2=Salmeron, M. |author3=Diaz, A. |author4=Ortega, J. |author5=Prieto, A. |author6=Olivares, G.| year = 2000 | title = Genetic algorithms and neuro-dynamic programming: application to water supply networks |volume=1 |pages=7–14 | conference = 2000 Congress on Evolutionary Computation | book-title = Proceedings of 2000 Congress on Evolutionary Computation | publisher = IEEE | location = La Jolla, California, USA | doi = 10.1109/CEC.2000.870269 | isbn = 0-7803-6375-2  }}</ref> and [[w:en:medicine|medicine]]<ref>{{Cite book |last=Deng |first=Geng |author2=Ferris, M.C. |title=Neuro-dynamic programming for fractionated radiotherapy planning |year=2008 |volume=12 |pages=47–70 |doi=10.1007/978-0-387-73299-2_3|citeseerx=10.1.1.137.8288 |series=Springer Optimization and Its Applications |isbn=978-0-387-73298-5 }}</ref> because of ANNs ability to mitigate losses of accuracy even when reducing the [[w:en:discretization|discretization]] grid density for numerically approximating the solution of control problems. Tasks that fall within the paradigm of reinforcement learning are control problems, [[w:en:game|game]]s and other sequential decision making tasks.

====Self-learning====

Self-learning in neural networks was introduced in 1982 along with a neural network capable of self-learning named Crossbar Adaptive Array (CAA).<ref>Bozinovski, S. (1982). "A self-learning system using secondary reinforcement". In R. Trappl (ed.) Cybernetics and Systems Research: Proceedings of the Sixth European Meeting on Cybernetics and Systems Research. North Holland. pp. 397–402. {{ISBN|978-0-444-86488-8}}.</ref> It is a system with only one input, situation s, and only one output, action (or behavior) a. It has neither external advice input nor external reinforcement input from the environment. The CAA computes, in a crossbar fashion, both decisions about actions and emotions (feelings) about encountered situations. The system is driven by the interaction between cognition and emotion.<ref>Bozinovski, S. (2014) "[https://core.ac.uk/download/pdf/81973924.pdf Modeling mechanisms of cognition-emotion interaction in artificial neural networks, since 1981] {{Webarchive|url=https://web.archive.org/web/20190323204838/https://core.ac.uk/download/pdf/81973924.pdf |date=23 March 2019 }}." Procedia Computer Science p. 255-263</ref> Given the memory matrix, W =||w(a,s)||, the crossbar self-learning algorithm in each iteration performs the following computation:
   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').

The backpropagated value (secondary reinforcement) is the emotion toward the consequence situation. The CAA exists in two environments, one is behavioral environment where it behaves, and the other is genetic environment, where from it initially and only once receives initial emotions about to be encountered situations in the behavioral environment. Having received the genome vector (species vector) from the genetic environment, the CAA will learn a goal-seeking behavior, in the behavioral environment that contains both desirable and undesirable situations.<ref>{{cite journal | last1 = Bozinovski | first1 = Stevo | last2 = Bozinovska | first2 = Liljana | year = 2001 | title = Self-learning agents: A connectionist theory of emotion based on crossbar value judgment | journal = Cybernetics and Systems | volume = 32 | issue = 6| pages = 637–667 | doi = 10.1080/01969720118145 | s2cid = 8944741 }}</ref>

==== Neuroevolution ====
{{Main|w:en:Neuroevolution}}

[[w:en:Neuroevolution|Neuroevolution]] can create neural network topologies and weights using [[w:en:evolutionary computation|evolutionary computation]]. It is competitive with sophisticated gradient descent approaches{{citation needed|date=July 2019}}. One advantage of neuroevolution is that it may be less prone to get caught in "dead ends".<ref>{{cite news|date=10 January 2018|title=Artificial intelligence can 'evolve' to solve problems|language=en|work=Science {{!}} AAAS|url=https://www.science.org/content/article/artificial-intelligence-can-evolve-solve-problems|access-date=7 February 2018|archive-date=9 December 2021|archive-url=https://web.archive.org/web/20211209231714/https://www.science.org/content/article/artificial-intelligence-can-evolve-solve-problems|url-status=live}}</ref>

===Stochastic neural network===

'''Stochastic neural networks''' originating from  [[w:en:Spin glass#Sherrington–Kirkpatrick model|Sherrington–Kirkpatrick model]]s  are a type of artificial neural network built by introducing random variations into the network, either by giving the network's [[w:en:artificial neuron|artificial neuron]]s [[w:en:Stochastic process|stochastic]] transfer functions, or by giving them stochastic weights. This makes them useful tools for [[w:en:Optimization (mathematics)|optimization]] problems, since the random fluctuations help the network escape from [[w:en:Maxima and minima|local minima]].<ref>{{citation|title=Stochastic Models of Neural Networks|volume=102|series=Frontiers in artificial intelligence and applications: Knowledge-based intelligent engineering systems|first=Claudio|last=Turchetti|publisher=IOS Press|year=2004|isbn=9781586033880}}</ref> Stochastic neural networks trained using a Bayesian approach are known as '''Bayesian neural networks'''.<ref>{{Cite journal |last1=Jospin |first1=Laurent Valentin |last2=Laga |first2=Hamid |last3=Boussaid |first3=Farid |last4=Buntine |first4=Wray |last5=Bennamoun |first5=Mohammed |date=2022 |title=Hands-On Bayesian Neural Networks—A Tutorial for Deep Learning Users |url=http://dx.doi.org/10.1109/mci.2022.3155327 |journal=IEEE Computational Intelligence Magazine |volume=17 |issue=2 |pages=29–48 |doi=10.1109/mci.2022.3155327 |arxiv=2007.06823 |s2cid=220514248 |issn=1556-603X |access-date=19 November 2022 |archive-date=4 February 2023 |archive-url=https://web.archive.org/web/20230204153904/https://ieeexplore.ieee.org/document/9756596/ |url-status=live }}</ref>

===Other===
In a [[w:en:Bayesian probability|Bayesian]] framework, a distribution over the set of allowed models is chosen to minimize the cost. [[w:en:Evolutionary methods|Evolutionary methods]],<ref>{{cite conference |author1=de Rigo, D. |author2=Castelletti, A. |author3=Rizzoli, A. E. |author4=Soncini-Sessa, R. |author5=Weber, E. |date=January 2005 |title=A selective improvement technique for fastening Neuro-Dynamic Programming in Water Resources Network Management |conference=16th IFAC World Congress |publisher=IFAC |location=Prague, Czech Republic |conference-url=http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Index.html |book-title=Proceedings of the 16th IFAC World Congress – IFAC-PapersOnLine |editor=Pavel Zítek |volume=16 |pages=7–12 |url=http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Papers/Paper4269.html |access-date=30 December 2011 |doi=10.3182/20050703-6-CZ-1902.02172 |isbn=978-3-902661-75-3 |hdl=11311/255236 |hdl-access=free |archive-date=26 April 2012 |archive-url=https://web.archive.org/web/20120426012450/http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Papers/Paper4269.html |url-status=live }}</ref> [[w:en:gene expression programming|gene expression programming]],<ref>{{cite book |last=Ferreira |first=C. |year=2006 |contribution=Designing Neural Networks Using Gene Expression Programming |url=http://www.gene-expression-programming.com/webpapers/Ferreira-ASCT2006.pdf |editor=A. Abraham |editor2=B. de Baets |editor3=M. Köppen |editor4=B. Nickolay |title=Applied Soft Computing Technologies: The Challenge of Complexity |pages=517–536 |publisher=Springer-Verlag |access-date=8 October 2012 |archive-date=19 December 2013 |archive-url=https://web.archive.org/web/20131219022806/http://www.gene-expression-programming.com/webpapers/Ferreira-ASCT2006.pdf |url-status=live }}</ref> [[w:en:simulated annealing|simulated annealing]],<ref>{{cite conference |author=Da, Y. |author2=Xiurun, G. |date=July 2005 |title=An improved PSO-based ANN with simulated annealing technique |volume=63 |pages=527–533 |editor=T. Villmann |book-title=New Aspects in Neurocomputing: 11th European Symposium on Artificial Neural Networks |url=http://www.dice.ucl.ac.be/esann/proceedings/electronicproceedings.htm |publisher=Elsevier |doi=10.1016/j.neucom.2004.07.002 |access-date=30 December 2011 |archive-date=25 April 2012 |archive-url=https://web.archive.org/web/20120425233611/http://www.dice.ucl.ac.be/esann/proceedings/electronicproceedings.htm}}</ref> [[w:en:expectation-maximization|expectation-maximization]], [[w:en:non-parametric methods|non-parametric methods]] and [[w:en:particle swarm optimization|particle swarm optimization]]<ref>{{cite conference |author=Wu, J. |author2=Chen, E. |date=May 2009 |title=A Novel Nonparametric Regression Ensemble for Rainfall Forecasting Using Particle Swarm Optimization Technique Coupled with Artificial Neural Network |series=Lecture Notes in Computer Science |volume=5553 |pages=49–58 |book-title=6th International Symposium on Neural Networks, ISNN 2009 |url=http://www2.mae.cuhk.edu.hk/~isnn2009/ |editor=Wang, H. |editor2=Shen, Y. |editor3=Huang, T. |editor4=Zeng, Z. |publisher=Springer |doi=10.1007/978-3-642-01513-7_6 |isbn=978-3-642-01215-0 |access-date=1 January 2012 |archive-date=31 December 2014 |archive-url=https://web.archive.org/web/20141231221755/http://www2.mae.cuhk.edu.hk/~isnn2009/ |url-status=dead}}</ref> are other learning algorithms. Convergent recursion is a learning algorithm for [[w:en:cerebellar model articulation controller|cerebellar model articulation controller]] (CMAC) neural networks.<ref name="Qin1">{{cite journal |author1=Ting Qin |author2=Zonghai Chen |author3=Haitao Zhang |author4=Sifu Li |author5=Wei Xiang |author6=Ming Li |url=http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_998.pdf |title=A learning algorithm of CMAC based on RLS |journal=Neural Processing Letters |volume=19 |issue=1 |date=2004 |pages=49–61 |doi=10.1023/B:NEPL.0000016847.18175.60 |s2cid=6233899 |access-date=30 January 2019 |archive-date=14 April 2021 |archive-url=https://web.archive.org/web/20210414103815/http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_998.pdf |url-status=live }}</ref><ref name="Qin2">{{cite journal |author1=Ting Qin |author2=Haitao Zhang |author3=Zonghai Chen |author4=Wei Xiang |url=http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_997.pdf |title=Continuous CMAC-QRLS and its systolic array |journal=Neural Processing Letters |volume=22 |issue=1 |date=2005 |pages=1–16 |doi=10.1007/s11063-004-2694-0 |s2cid=16095286 |access-date=30 January 2019 |archive-date=18 November 2018 |archive-url=https://web.archive.org/web/20181118122850/http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_997.pdf |url-status=live }}</ref>

==== Modes ====
{{No footnotes|date=August 2019|section}}
Two modes of learning are available: [[w:en:stochastic gradient descent|stochastic]] and batch. In stochastic learning, each input creates a weight adjustment. In batch learning weights are adjusted based on a batch of inputs, accumulating errors over the batch. Stochastic learning introduces "noise" into the process, using the local gradient calculated from one data point; this reduces the chance of the network getting stuck in local minima. However, batch learning typically yields a faster, more stable descent to a local minimum, since each update is performed in the direction of the batch's average error. A common compromise is to use "mini-batches", small batches with samples in each batch selected stochastically from the entire data set.

== Types ==
<!-- Split to [[w:en:Types of artificial neural networks|Types of artificial neural networks]] -->
{{Main|w:en:Types of artificial neural networks}}

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 [[w:en:topology|topology]]. Dynamic types allow one or more of these to evolve via learning. The latter are 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: [[w:en:convolutional neural network|convolutional neural network]]s that have proven particularly successful in processing visual and other two-dimensional data;<ref name="LECUN1989">{{cite journal |vauthors=LeCun Y, Boser B, Denker JS, Henderson D, Howard RE, Hubbard W, Jackel LD |title=Backpropagation Applied to Handwritten Zip Code Recognition |journal=Neural Computation |volume=1 |issue=4 |pages=541–551 |date=1989 |doi=10.1162/neco.1989.1.4.541|s2cid=41312633 }}</ref><ref name="lecun2016slides">[[w:en:Yann LeCun|Yann LeCun]] (2016). Slides on Deep Learning [https://indico.cern.ch/event/510372/ Online] {{Webarchive|url=https://web.archive.org/web/20160423021403/https://indico.cern.ch/event/510372/ |date=23 April 2016 }}</ref> long short-term memory avoid the [[w:en:vanishing gradient problem|vanishing gradient problem]]<ref name=":03">{{Cite journal |last1=Hochreiter|first1=Sepp|author-link=Sepp Hochreiter|last2=Schmidhuber|first2=Jürgen|s2cid=1915014|author-link2=Jürgen Schmidhuber|date=1 November 1997|title=Long Short-Term Memory|journal=Neural Computation|volume=9|issue=8 |pages=1735–1780 |doi=10.1162/neco.1997.9.8.1735|pmid=9377276|issn=0899-7667}}</ref> and can handle signals that have a mix of low and high frequency components aiding large-vocabulary speech recognition,<ref name="sak2014">{{Cite web|url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43905.pdf |title=Long Short-Term Memory recurrent neural network architectures for large scale acoustic modeling |last1=Sak|first1=Hasim |last2=Senior|first2=Andrew|date=2014|last3=Beaufays|first3=Francoise|archive-url=https://web.archive.org/web/20180424203806/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43905.pdf|archive-date=24 April 2018|url-status=dead}}</ref><ref name="liwu2015">{{cite arXiv|last1=Li|first1=Xiangang|last2=Wu|first2=Xihong|date=15 October 2014|title=Constructing Long Short-Term Memory based Deep Recurrent Neural Networks for Large Vocabulary Speech Recognition|eprint=1410.4281 |class=cs.CL}}</ref> text-to-speech synthesis,<ref>{{Cite journal|title=TTS synthesis with bidirectional LSTM based Recurrent Neural Networks|pages=1964–1968|last1=Fan|first1=Y. |last2=Qian|first2=Y.|date=2014 |journal=Proceedings of the Annual Conference of the International Speech Communication Association, Interspeech|url=https://www.researchgate.net/publication/287741874|access-date=13 June 2017 |last3=Xie |first3=F.|last4=Soong|first4=F. K.}}</ref><ref name="scholarpedia2"/><ref name="zen2015">{{Cite web|url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43266.pdf|title=Unidirectional Long Short-Term Memory Recurrent Neural Network with Recurrent Output Layer for Low-Latency Speech Synthesis|last1=Zen|first1=Heiga|last2=Sak|first2=Hasim|date=2015|website=Google.com|publisher=ICASSP|pages=4470–4474|access-date=27 June 2017|archive-date=9 May 2021|archive-url=https://web.archive.org/web/20210509123113/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43266.pdf|url-status=live}}</ref> and photo-real talking heads;<ref name="fan2015">{{Cite journal|last1=Fan|first1=Bo|last2=Wang|first2=Lijuan|last3=Soong|first3=Frank K.|last4=Xie|first4=Lei|date=2015|title=Photo-Real Talking Head with Deep Bidirectional LSTM|url=https://www.microsoft.com/en-us/research/wp-content/uploads/2015/04/icassp2015_fanbo_1009.pdf|journal=Proceedings of ICASSP|access-date=27 June 2017|archive-date=1 November 2017|archive-url=https://web.archive.org/web/20171101052317/https://www.microsoft.com/en-us/research/wp-content/uploads/2015/04/icassp2015_fanbo_1009.pdf|url-status=live}}</ref> competitive networks such as [[w:en:generative adversarial network|generative adversarial network]]s in which multiple networks (of varying structure) compete with each other, on tasks such as winning a game<ref name="preprint">{{Cite arXiv |eprint=1712.01815|class=cs.AI|first1=David|last1=Silver|first2=Thomas|last2=Hubert|author-link1=David Silver (programmer)|title=Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm|date=5 December 2017|first3=Julian|last3=Schrittwieser|first4=Ioannis|last4=Antonoglou |first5=Matthew|last5=Lai |first6=Arthur|last6=Guez|first7=Marc|last7=Lanctot|first8=Laurent|last8=Sifre |first9=Dharshan|last9=Kumaran|author-link9=Dharshan Kumaran|first10=Thore|last10=Graepel|first11=Timothy |last11=Lillicrap|first12=Karen |last12=Simonyan|first13=Demis|last13=Hassabis|author-link13=Demis Hassabis}}</ref> or on deceiving the opponent about the authenticity of an input.<ref name="GANnips">{{cite conference|last1=Goodfellow|first1=Ian|last2=Pouget-Abadie|first2=Jean|last3=Mirza|first3=Mehdi|last4=Xu|first4=Bing|last5=Warde-Farley|first5=David|last6=Ozair|first6=Sherjil|last7=Courville|first7=Aaron|last8=Bengio|first8=Yoshua|year=2014|title=Generative Adversarial Networks|url=https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf|conference=Proceedings of the International Conference on Neural Information Processing Systems (NIPS 2014)|pages=2672–2680|access-date=20 August 2019|archive-date=22 November 2019|archive-url=https://web.archive.org/web/20191122034612/http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf|url-status=live}}</ref>

== Network design ==
{{Main|w:en:Neural architecture search}}
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.<ref>{{cite arXiv|last1=Zoph|first1=Barret|last2=Le|first2=Quoc V.|date=4 November 2016|title=Neural Architecture Search with Reinforcement Learning|eprint=1611.01578|class=cs.LG}}</ref> Available systems include [[w:en:Automated machine learning|AutoML]] and AutoKeras.<ref>{{cite journal |author1=Haifeng Jin |author2=Qingquan Song |author3=Xia Hu |title=Auto-keras: An efficient neural architecture search system |journal=Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining |publisher=ACM |date=2019 |arxiv=1806.10282 |url=https://autokeras.com/ |via=autokeras.com |access-date=21 August 2019 |archive-date=21 August 2019 |archive-url=https://web.archive.org/web/20190821163310/https://autokeras.com/ |url-status=live }}</ref>

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, ...).

[[w:en:Hyperparameter|Hyperparameter]]s 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.<ref name="abs1502.02127">{{cite arXiv|eprint=1502.02127|last1=Claesen|first1=Marc|last2=De Moor|first2=Bart |title=Hyperparameter Search in Machine Learning |date=2015|class=cs.LG }} {{bibcode|2015arXiv150202127C}}</ref>

== Use ==
{{Unreferenced section|date=November 2020}}
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 [[w:en:hyperparameter|hyperparameter]]s 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:{{Citation needed|date=June 2017}}

* [[w:en:Function approximation|Function approximation]], or [[w:en:regression analysis|regression analysis]], including [[w:en:Time series#Prediction and forecasting|time series prediction]], [[w:en:fitness approximation|fitness approximation]] and modeling.
* [[w:en:Statistical classification|Classification]], including [[w:en:Pattern recognition|pattern]] and sequence recognition, [[w:en:novelty detection|novelty detection]] and sequential decision making.<ref name ="TurekNeuralNet">{{cite journal|author=Turek, Fred D.|title=Introduction to Neural Net Machine Vision|url=http://www.vision-systems.com/articles/print/volume-12/issue-3/features/introduction-to-neural-net-machine-vision.html|access-date=5 March 2013|journal=Vision Systems Design|date=March 2007|volume=12|number=3|archive-date=16 May 2013|archive-url=https://web.archive.org/web/20130516124148/http://www.vision-systems.com/articles/print/volume-12/issue-3/features/introduction-to-neural-net-machine-vision.html|url-status=live}}</ref>
* [[w:en:Data processing|Data processing]], including filtering, clustering, [[w:en:blind source separation|blind source separation]] and compression.
* [[w:en:Robotics|Robotics]], including directing manipulators and [[w:en:prosthesis|prostheses]].

==Applications==
Because of their ability to reproduce and model nonlinear processes, artificial neural networks have found applications in many disciplines. Application areas include [[w:en:system identification|system identification]] and control (vehicle control, trajectory prediction,<ref>{{cite journal|last1=Zissis|first1=Dimitrios|title=A cloud based architecture capable of perceiving and predicting multiple vessel behaviour|journal=Applied Soft Computing|date=October 2015|volume=35|doi=10.1016/j.asoc.2015.07.002|pages=652–661|url=https://zenodo.org/record/848743|access-date=18 July 2019|archive-date=26 July 2020|archive-url=https://web.archive.org/web/20200726091505/https://zenodo.org/record/848743|url-status=live}}</ref> [[w:en:process control|process control]], [[w:en:natural resource management|natural resource management]]), [[w:en:quantum chemistry|quantum chemistry]],<ref name="Balabin_2009">{{Cite journal|journal=[[w:en:J. Chem. Phys.|J. Chem. Phys.]] |volume=131 |issue=7 |page=074104 |doi=10.1063/1.3206326 |title=Neural network approach to quantum-chemistry data: Accurate prediction of density functional theory energies |year=2009 |author1=Roman M. Balabin |author2=Ekaterina I. Lomakina |pmid=19708729|bibcode=2009JChPh.131g4104B}}</ref> [[w:en:general game playing|general game playing]],<ref>{{cite journal |last1=Silver |first1=David |display-authors=etal |year=2016 |title=Mastering the game of Go with deep neural networks and tree search |url=http://web.iitd.ac.in/~sumeet/Silver16.pdf |journal=Nature |volume=529 |issue=7587 |pages=484–489 |doi=10.1038/nature16961 |pmid=26819042 |bibcode=2016Natur.529..484S |s2cid=515925 |access-date=31 January 2019 |archive-date=23 November 2018 |archive-url=https://web.archive.org/web/20181123112812/http://web.iitd.ac.in/~sumeet/Silver16.pdf |url-status=live }}</ref> [[w:en:pattern recognition|pattern recognition]] (radar systems, [[w:en:Facial recognition system|face identification]], signal classification,<ref>{{cite journal|last=Sengupta |first=Nandini|author2=Sahidullah, Md|author3=Saha, Goutam|title=Lung sound classification using cepstral-based statistical features|journal=Computers in Biology and Medicine|date=August 2016|volume=75|issue=1 |pages=118–129|doi=10.1016/j.compbiomed.2016.05.013 |pmid=27286184}}</ref> [[w:en:3D reconstruction|3D reconstruction]],<ref>Choy, Christopher B., et al. "[https://arxiv.org/abs/1604.00449 3d-r2n2: A unified approach for single and multi-view 3d object reconstruction] {{Webarchive|url=https://web.archive.org/web/20200726091721/https://arxiv.org/abs/1604.00449 |date=26 July 2020 }}." European conference on computer vision. Springer, Cham, 2016.</ref> object recognition and more), sensor data analysis,<ref>{{cite journal|last=Gessler|first=Josef|title=Sensor for food analysis applying impedance spectroscopy and artificial neural networks|journal=RiuNet UPV|date=August 2021|issue=1|pages=8–12|url=https://riunet.upv.es/handle/10251/174498|access-date=21 October 2021|archive-date=21 October 2021|archive-url=https://web.archive.org/web/20211021115443/https://riunet.upv.es/handle/10251/174498|url-status=live}}</ref> sequence recognition (gesture, speech, [[w:en:handwriting recognition|handwritten]] and printed text recognition<ref>{{Cite journal|last1=Maitra|first1=D. S. |last2=Bhattacharya|first2=U.|last3=Parui|first3=S. K.|date=August 2015|title=CNN based common approach to handwritten character recognition of multiple scripts |url=https://ieeexplore.ieee.org/document/7333916|journal=2015 13th International Conference on Document Analysis and Recognition (ICDAR)|pages=1021–1025|doi=10.1109/ICDAR.2015.7333916|isbn=978-1-4799-1805-8 |s2cid=25739012}}</ref>), [[w:en:medical diagnosis|medical diagnosis]], finance<ref>{{cite journal|last1=French |first1=Jordan |title=The time traveller's CAPM|journal=Investment Analysts Journal|volume=46|issue=2|pages=81–96 |doi=10.1080/10293523.2016.1255469|year=2016|s2cid=157962452}}</ref> (e.g. ex-ante models for specific financial long-run forecasts and [[w:en:artificial financial market|artificial financial market]]s), [[w:en:data mining|data mining]], visualization, [[w:en:machine translation|machine translation]], social network filtering<ref>{{Cite news|url=https://www.wsj.com/articles/facebook-boosts-a-i-to-block-terrorist-propaganda-1497546000 |title=Facebook Boosts A.I. to Block Terrorist Propaganda|last=Schechner|first=Sam|date=15 June 2017 |work=[[w:en:The Wall Street Journal|The Wall Street Journal]]|access-date=16 June 2017|issn=0099-9660}}</ref> and [[w:en:e-mail spam|e-mail spam]] filtering. ANNs have been used to diagnose several types of cancers<ref>{{cite journal|last=Ganesan|first=N |title=Application of Neural Networks in Diagnosing Cancer Disease Using Demographic Data |journal=International Journal of Computer Applications|volume=1|issue=26|pages=81–97 |bibcode=2010IJCA....1z..81G|year=2010|doi=10.5120/476-783|doi-access=free}}</ref><ref>{{cite journal |url=http://www.lcc.uma.es/~jja/recidiva/042.pdf|title=Artificial Neural Networks Applied to Outcome Prediction for Colorectal Cancer Patients in Separate Institutions|journal=Lancet|volume=350|issue=9076 |pages=469–72|last=Bottaci|first=Leonardo|publisher=The Lancet|pmid=9274582|year=1997|doi=10.1016/S0140-6736(96)11196-X|s2cid=18182063|access-date=2 May 2012|url-status=dead|archive-date=23 November 2018|archive-url=https://web.archive.org/web/20181123170444/http://www.lcc.uma.es/~jja/recidiva/042.pdf}}</ref> and to distinguish highly invasive cancer cell lines from less invasive lines using only cell shape information.<ref>{{cite journal|last1=Alizadeh|first1=Elaheh|last2=Lyons|first2=Samanthe M|last3=Castle|first3=Jordan M |last4=Prasad|first4=Ashok|date=2016|title=Measuring systematic changes in invasive cancer cell shape using Zernike moments|url=http://pubs.rsc.org/en/Content/ArticleLanding/2016/IB/C6IB00100A|journal=Integrative Biology|volume=8|issue=11 |pages=1183–1193|doi=10.1039/C6IB00100A|pmid=27735002}}</ref><ref>{{cite journal |last1=Lyons|first1=Samanthe|date=2016|title=Changes in cell shape are correlated with metastatic potential in murine|journal=Biology Open|volume=5|issue=3|pages=289–299|doi=10.1242/bio.013409|pmid=26873952 |pmc=4810736}}</ref>

ANNs have been used to accelerate reliability analysis of infrastructures subject to natural disasters<ref>{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=28 August 2017|title=Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks|journal=Computer-Aided Civil and Infrastructure Engineering|volume=33|issue=6|pages=443–458|arxiv=1708.08551|doi=10.1111/mice.12359 |bibcode=2017arXiv170808551N |s2cid=36661983}}</ref><ref>{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2018|title=Accelerating Stochastic Assessment of Post-Earthquake Transportation Network Connectivity via Machine-Learning-Based Surrogates|url=https://trid.trb.org/view/1496617|journal=Transportation Research Board 97th Annual Meeting|access-date=14 March 2018|archive-date=9 March 2018|archive-url=https://web.archive.org/web/20180309120108/https://trid.trb.org/view/1496617|url-status=live}}</ref> and to predict foundation settlements.<ref>{{Cite journal|last1=Díaz|first1=E.|last2=Brotons|first2=V. |last3=Tomás|first3=R.|date=September 2018|title=Use of artificial neural networks to predict 3-D elastic settlement of foundations on soils with inclined bedrock|journal=Soils and Foundations|volume=58|issue=6 |pages=1414–1422 |doi=10.1016/j.sandf.2018.08.001|issn=0038-0806|hdl=10045/81208|doi-access=free}}</ref> ANNs have also been used for building black-box models in [[w:en:geoscience|geoscience]]: [[w:en:hydrology|hydrology]],<ref>{{Cite journal |first=Rao S.|last=Govindaraju |date=1 April 2000|title=Artificial Neural Networks in Hydrology. I: Preliminary Concepts|journal=Journal of Hydrologic Engineering|volume=5|issue=2|pages=115–123|doi=10.1061/(ASCE)1084-0699(2000)5:2(115)|citeseerx=<!--10.1.1.127.3861-->}}</ref><ref>{{Cite journal|first=Rao S.|last=Govindaraju|date=1 April 2000|title=Artificial Neural Networks in Hydrology. II: Hydrologic Applications|journal=Journal of Hydrologic Engineering|volume=5|issue=2 |pages=124–137 |doi=10.1061/(ASCE)1084-0699(2000)5:2(124)}}</ref> ocean modelling and [[w:en:coastal engineering|coastal engineering]],<ref>{{Cite journal|last1=Peres|first1=D. J.|last2=Iuppa|first2=C.|last3=Cavallaro|first3=L.|last4=Cancelliere |first4=A. |last5=Foti|first5=E.|date=1 October 2015|title=Significant wave height record extension by neural networks and reanalysis wind data|journal=Ocean Modelling|volume=94|pages=128–140 |doi=10.1016/j.ocemod.2015.08.002 |bibcode=2015OcMod..94..128P}}</ref><ref>{{Cite journal|last1=Dwarakish|first1=G. S.|last2=Rakshith|first2=Shetty|last3=Natesan|first3=Usha|date=2013|title=Review on Applications of Neural Network in Coastal Engineering|journal=Artificial Intelligent Systems and Machine Learning|url=http://www.ciitresearch.org/dl/index.php/aiml/article/view/AIML072013007|volume=5|issue=7|pages=324–331|access-date=5 July 2017|archive-date=15 August 2017|archive-url=https://web.archive.org/web/20170815185634/http://www.ciitresearch.org/dl/index.php/aiml/article/view/AIML072013007|url-status=live}}</ref> and [[w:en:geomorphology|geomorphology]].<ref>{{Cite journal |last1=Ermini|first1=Leonardo|last2=Catani |first2=Filippo|last3=Casagli|first3=Nicola|date=1 March 2005|title=Artificial Neural Networks applied to landslide susceptibility assessment|journal=Geomorphology|series=Geomorphological hazard and human impact in mountain environments|volume=66|issue=1|pages=327–343|doi=10.1016/j.geomorph.2004.09.025 |bibcode=2005Geomo..66..327E}}</ref> ANNs have been employed in [[w:en:Computer security|cybersecurity]], with the objective to discriminate between legitimate activities and malicious ones. For example, machine learning has been used for classifying Android malware,<ref>{{Cite journal|last1=Nix|first1=R.|last2=Zhang |first2=J.|date=May 2017 |title=Classification of Android apps and malware using deep neural networks |journal=2017 International Joint Conference on Neural Networks (IJCNN)|pages=1871–1878|s2cid=8838479 |doi=10.1109/IJCNN.2017.7966078|isbn=978-1-5090-6182-2}}</ref> for identifying domains belonging to threat actors and for detecting URLs posing a security risk.<ref>{{Cite web|title=Detecting Malicious URLs |website=The systems and networking group at UCSD |url=http://www.sysnet.ucsd.edu/projects/url/|access-date=15 February 2019|url-status=dead|archive-date=14 July 2019|archive-url=https://web.archive.org/web/20190714201955/http://www.sysnet.ucsd.edu/projects/url/}}</ref> Research is underway on ANN systems designed for penetration testing, for detecting botnets,<ref>{{Citation |last1=Homayoun|first1=Sajad|title=BoTShark: A Deep Learning Approach for Botnet Traffic Detection |date=2018|work=Cyber Threat Intelligence|pages=137–153|editor-last=Dehghantanha|editor-first=Ali |series=Advances in Information Security|publisher=Springer International Publishing|doi=10.1007/978-3-319-73951-9_7|isbn=978-3-319-73951-9|last2=Ahmadzadeh |first2=Marzieh|last3=Hashemi|first3=Sattar |last4=Dehghantanha|first4=Ali|last5=Khayami|first5=Raouf|editor2-last=Conti|editor2-first=Mauro|editor3-last=Dargahi|editor3-first=Tooska}}</ref> credit cards frauds<ref>{{Cite journal |last1=Ghosh|last2=Reilly |name-list-style=and|s2cid=13260377 |date=January 1994|title=Credit card fraud detection with a neural-network|journal=1994 Proceedings of the Twenty-Seventh Hawaii International Conference on System Sciences |volume=3|pages=621–630|doi=10.1109/HICSS.1994.323314|isbn=978-0-8186-5090-1}}</ref> and network intrusions.

ANNs have been proposed as a tool to solve [[w:en:partial differential equation|partial differential equation]]s in physics<ref>{{Cite web|last=Ananthaswamy|first=Anil|date=19 April 2021|title=Latest Neural Nets Solve World's Hardest Equations Faster Than Ever Before|url=https://www.quantamagazine.org/new-neural-networks-solve-hardest-equations-faster-than-ever-20210419/|access-date=12 May 2021|website=Quanta Magazine|language=en}}</ref><ref>{{Cite web|title=AI has cracked a key mathematical puzzle for understanding our world|url=https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/|access-date=19 November 2020|website=MIT Technology Review|language=en}}</ref><ref>{{Cite web|title=Caltech Open-Sources AI for Solving Partial Differential Equations|url=https://www.infoq.com/news/2020/12/caltech-ai-pde/|access-date=20 January 2021|website=InfoQ|language=en|archive-date=25 January 2021|archive-url=https://web.archive.org/web/20210125233952/https://www.infoq.com/news/2020/12/caltech-ai-pde/|url-status=live}}</ref> and simulate the properties of many-body [[w:en:open quantum system|open quantum system]]s.<ref>{{cite journal |last1=Nagy |first1=Alexandra |title=Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems |journal=[[w:en:Physical Review Letters|Physical Review Letters]] |volume=122 |issue=25 |pages=250501 |date=28 June 2019 |doi=10.1103/PhysRevLett.122.250501 |pmid=31347886 |bibcode=2019PhRvL.122y0501N |arxiv=1902.09483 |s2cid=119074378 }}</ref><ref>{{Cite journal|last1=Yoshioka|first1=Nobuyuki|last2=Hamazaki|first2=Ryusuke|date=28 June 2019|title=Constructing neural stationary states for open quantum many-body systems|journal=Physical Review B|volume=99|issue=21 |pages=214306|doi=10.1103/PhysRevB.99.214306|bibcode=2019PhRvB..99u4306Y|arxiv=1902.07006|s2cid=119470636}}</ref><ref>{{Cite journal|last1=Hartmann|first1=Michael J.|last2=Carleo|first2=Giuseppe |date=28 June 2019|title=Neural-Network Approach to Dissipative Quantum Many-Body Dynamics|journal=Physical Review Letters|volume=122|issue=25|pages=250502|doi=10.1103/PhysRevLett.122.250502|pmid=31347862 |bibcode=2019PhRvL.122y0502H|arxiv=1902.05131|s2cid=119357494}}</ref><ref>{{Cite journal|last1=Vicentini |first1=Filippo|last2=Biella|first2=Alberto|last3=Regnault|first3=Nicolas|last4=Ciuti|first4=Cristiano|date=28 June 2019 |title=Variational Neural-Network Ansatz for Steady States in Open Quantum Systems |journal=Physical Review Letters|volume=122|issue=25|pages=250503|doi=10.1103/PhysRevLett.122.250503 |pmid=31347877 |bibcode=2019PhRvL.122y0503V |arxiv=1902.10104|s2cid=119504484}}</ref> In brain research ANNs have studied short-term behavior of [[w:en:biological neuron models|individual neurons]],<ref>{{cite journal |author=Forrest MD |title=Simulation of alcohol action upon a detailed Purkinje neuron model and a simpler surrogate model that runs >400 times faster |journal=BMC Neuroscience |volume=16 |issue=27 |pages=27 |date=April 2015 |doi=10.1186/s12868-015-0162-6 |pmid=25928094 |pmc=4417229}}</ref> 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==
===Computational power===

The [[w:en:multilayer perceptron|multilayer perceptron]] is a [[w:en:UTM theorem|universal function]] approximator, as proven by the [[w:en:universal approximation theorem|universal approximation theorem]]. However, the proof is not constructive regarding the number of neurons required, the network topology, the weights and the learning parameters.

A specific recurrent architecture with [[w:en:Rational number|rational]]-valued weights (as opposed to full precision [[w:en:real number|real number]]-valued weights) has the power of a [[w:en:Universal Turing Machine|universal Turing machine]],<ref>{{Cite journal| title =  Turing computability with neural nets | url = http://www.math.rutgers.edu/~sontag/FTPDIR/aml-turing.pdf | year = 1991 | journal = Appl. Math. Lett. | pages = 77–80 | volume = 4 | issue = 6 | last1 = Siegelmann | first1 =  H.T. | last2 =  Sontag | first2 =  E.D. | doi =  10.1016/0893-9659(91)90080-F }}</ref> using a finite number of neurons and standard linear connections. Further, the use of [[w:en:Irrational number|irrational]] values for weights results in a machine with [[w:en:Hypercomputation|super-Turing]] power.<ref>{{cite journal |last1=Balcázar |first1=José |title=Computational Power of Neural Networks: A Kolmogorov Complexity Characterization |journal=IEEE Transactions on Information Theory|date=Jul 1997 |volume=43 |issue=4 |pages=1175–1183 |doi=10.1109/18.605580 |citeseerx=10.1.1.411.7782 }}</ref>

===Capacity===
A model's "capacity" property corresponds to its 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.
Two notions of capacity are known by the community. The information capacity and the VC Dimension. The information capacity of a perceptron is intensively discussed in Sir David MacKay's book<ref name="auto">{{cite book| last=MacKay| first=David, J.C.| author-link=David J.C. MacKay| year=2003| publisher=[[w:en:Cambridge University Press|Cambridge University Press]]| isbn=978-0-521-64298-9| title=Information Theory, Inference, and Learning Algorithms| url=http://www.inference.phy.cam.ac.uk/itprnn/book.pdf| access-date=11 June 2016| archive-date=19 October 2016| archive-url=https://web.archive.org/web/20161019163258/http://www.inference.phy.cam.ac.uk/itprnn/book.pdf| url-status=live}}</ref> which summarizes work by Thomas Cover.<ref>{{cite journal|last=Cover|first=Thomas|author-link=Thomas M. Cover|year=1965|publisher=[[w:en:IEEE|IEEE]]|url=http://www-isl.stanford.edu/people/cover/papers/paper2.pdf|title=Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition|journal=IEEE Transactions on Electronic Computers|issue=3|pages=326–334|volume=EC-14|doi=10.1109/PGEC.1965.264137|access-date=10 March 2020|archive-date=5 March 2016|archive-url=https://web.archive.org/web/20160305031348/http://www-isl.stanford.edu/people/cover/papers/paper2.pdf|url-status=live}}</ref> The capacity of a network of standard neurons (not convolutional) can be derived by four rules<ref>{{cite journal| last=Gerald | first=Friedland| author-link=Gerald Friedland|year=2019|publisher=[[w:en:Association for Computing Machinery|ACM]]|title=Reproducibility and Experimental Design for Machine Learning on Audio and Multimedia Data|journal=MM '19: Proceedings of the 27th ACM International Conference on Multimedia| pages=2709–2710| doi=10.1145/3343031.3350545| isbn=978-1-4503-6889-6| s2cid=204837170}}</ref> that derive from understanding a neuron as an [[w:en:ADALINE|electrical element]]. The information capacity captures the functions modelable by the network given any data as input. The second notion, is the [[w:en:VC dimension|VC dimension]]. VC Dimension uses the principles of [[w:en:measure theory|measure theory]] and finds the maximum capacity under the best possible circumstances. This is, given input data in a specific form.  As noted in,<ref name="auto"/> the VC Dimension for arbitrary inputs is half the information capacity of a Perceptron. The VC Dimension for arbitrary points is sometimes referred to as Memory Capacity.<ref>{{cite web| url=http://tfmeter.icsi.berkeley.edu/| title=Stop tinkering, start measuring! Predictable experimental design of Neural Network experiments| website=The Tensorflow Meter| access-date=10 March 2020| archive-date=18 April 2022| archive-url=https://web.archive.org/web/20220418025904/http://tfmeter.icsi.berkeley.edu/| url-status=dead}}</ref>

===Convergence===
Models may not consistently converge on a single solution, firstly because local minima may exist, depending on the cost function and the model. Secondly, the optimization method used might not guarantee to converge when it begins far from any local minimum. Thirdly, for sufficiently large data or parameters, some methods become impractical.

Another issue worthy to mention is that training may cross some [[w:en:Saddle point|Saddle point]] which may lead the convergence to the wrong direction.

The convergence behavior of certain types of ANN architectures are more understood than others. When the width of network approaches to infinity, the ANN is well described by its first order Taylor expansion throughout training, and so inherits the convergence behavior of [[w:en:Linear model|affine models]].<ref>{{Cite journal|last1=Lee|first1=Jaehoon|last2=Xiao|first2=Lechao|last3=Schoenholz|first3=Samuel S.|last4=Bahri |first4=Yasaman|last5=Novak |first5=Roman|last6=Sohl-Dickstein|first6=Jascha|last7=Pennington |first7=Jeffrey|title=Wide neural networks of any depth evolve as linear models under gradient descent |journal=Journal of Statistical Mechanics: Theory and Experiment|year=2020|volume=2020|issue=12|page=124002 |doi=10.1088/1742-5468/abc62b|arxiv=1902.06720|bibcode=2020JSMTE2020l4002L|s2cid=62841516}}</ref><ref>{{cite conference |conference=32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montreal, Canada |author1=Arthur Jacot |author2=Franck Gabriel |author3=Clement Hongler |date=2018 |url=https://proceedings.neurips.cc/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf |title=Neural Tangent Kernel: Convergence and Generalization in Neural Networks |access-date=4 June 2022 |archive-date=22 June 2022 |archive-url=https://web.archive.org/web/20220622033100/https://proceedings.neurips.cc/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf |url-status=live }}</ref> Another example is when parameters are small, it is observed that ANNs often fits target functions from low to high frequencies. This behavior is referred to as the spectral bias, or frequency principle, of neural networks.<ref>{{cite book |vauthors=Xu ZJ, Zhang Y, Xiao Y |date=2019 |veditors=Gedeon T, Wong K, Lee M |title=Neural Information Processing. ICONIP 2019. |series=Lecture Notes in Computer Science |volume=11953 |publisher=Springer, Cham |doi=10.1007/978-3-030-36708-4_22 |chapter=Training Behavior of Deep Neural Network in Frequency Domain |pages=264–274 |arxiv=1807.01251 |isbn=978-3-030-36707-7 |s2cid=49562099 }}</ref><ref>{{cite journal |author1=Nasim Rahaman |author2=Aristide Baratin |author3=Devansh Arpit |author4=Felix Draxler |author5=Min Lin |author6=Fred Hamprecht |author7=Yoshua Bengio |author8=Aaron Courville |journal=Proceedings of the 36th International Conference on Machine Learning |volume=97 |pages=5301–5310 |date=2019 |title=On the Spectral Bias of Neural Networks |arxiv=1806.08734 |url=http://proceedings.mlr.press/v97/rahaman19a/rahaman19a.pdf |access-date=4 June 2022 |archive-date=22 October 2022 |archive-url=https://web.archive.org/web/20221022155951/http://proceedings.mlr.press/v97/rahaman19a/rahaman19a.pdf |url-status=live }}</ref><ref>{{cite journal |arxiv=1901.06523 |author1=Zhi-Qin John Xu |author2=Yaoyu Zhang |author3=Tao Luo |author4=Yanyang Xiao |author5=Zheng Ma |title=Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks|journal=Communications in Computational Physics |year=2020 |volume=28 |issue=5 |pages=1746–1767 |doi=10.4208/cicp.OA-2020-0085 |bibcode=2020CCoPh..28.1746X |s2cid=58981616 }}</ref><ref>{{cite arXiv |eprint=1906.09235 |author1=Tao Luo |author2=Zheng Ma |author3=Zhi-Qin John Xu |author4=Yaoyu Zhang |date=2019 |title=Theory of the Frequency Principle for General Deep Neural Networks|class=cs.LG }}</ref> This phenomenon is the opposite to the behavior of some well studied iterative numerical schemes such as [[w:en:Jacobi method|Jacobi method]]. Deeper neural networks have been observed to be more biased towards low frequency functions.<ref>{{Cite journal|last1=Xu|first1=Zhiqin John|last2=Zhou|first2=Hanxu|date=2021-05-18|title=Deep Frequency Principle Towards Understanding Why Deeper Learning Is Faster|url=https://ojs.aaai.org/index.php/AAAI/article/view/17261|journal=Proceedings of the AAAI Conference on Artificial Intelligence|volume=35|issue=12|pages=10541–10550|doi=10.1609/aaai.v35i12.17261|arxiv=2007.14313|s2cid=220831156|issn=2374-3468|access-date=5 October 2021|archive-date=5 October 2021|archive-url=https://web.archive.org/web/20211005142300/https://ojs.aaai.org/index.php/AAAI/article/view/17261|url-status=live}}</ref>

===Generalization and statistics===
{{No footnotes|date=August 2019|section}}
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 [[w:en:cross-validation (statistics)|cross-validation]] and similar techniques to check for the presence of over-training and to select [[w:en:hyperparameters|hyperparameters]] to minimize the generalization error.

The second is to use some form of ''[[w:en:regularization (mathematics)|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.

[[File:Synapse deployment.jpg|thumb|right|upright=1.15|Confidence analysis of a neural network]]
Supervised neural networks that use a [[w:en:mean squared error|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 [[w:en:confidence interval|confidence interval]] of network output, assuming a [[w:en:normal distribution|normal distribution]]. A confidence analysis made this way is statistically valid as long as the output [[w:en:probability distribution|probability distribution]] stays the same and the network is not modified.

By assigning a [[w:en:softmax activation function|softmax activation function]], a generalization of the [[w:en:logistic function|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:

:<math>y_i=\frac{e^{x_i}}{\sum_{j=1}^c e^{x_j}}</math>
<section end="theory" />

==Criticism==

===Training ===
A common criticism of neural networks, particularly in robotics, is that they require too much training for real-world operation.<ref>{{Cite journal |last=Parisi |first=German I. |last2=Kemker |first2=Ronald |last3=Part |first3=Jose L. |last4=Kanan |first4=Christopher |last5=Wermter |first5=Stefan |date=2019-05-01 |title=Continual lifelong learning with neural networks: A review |url=https://www.sciencedirect.com/science/article/pii/S0893608019300231 |journal=Neural Networks |language=en |volume=113 |pages=54–71 |doi=10.1016/j.neunet.2019.01.012 |issn=0893-6080}}</ref> Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example, grouping examples in so-called mini-batches and/or introducing a recursive least squares algorithm for [[w:en:cerebellar model articulation controller|CMAC]].<ref name="Qin1"/>

===Theory===

A central claim{{cn|date=January 2023}} of ANNs is that they embody new and powerful general principles for processing information. These principles are ill-defined. It is often claimed{{by whom?|date=January 2023}} that they are [[w:en:Emergent properties|emergent]] from the network itself. This allows simple statistical association (the basic function of artificial neural networks) to be described as learning or recognition. In 1997, [[w:en:Alexander Dewdney|Alexander Dewdney]] commented that, as a result, artificial neural networks have a "something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are. No human hand (or mind) intervenes; solutions are found as if by magic; and no one, it seems, has learned anything".<ref>{{cite book|url={{google books |plainurl=y |id=KcHaAAAAMAAJ|page=82}}|title=Yes, we have no neutrons: an eye-opening tour through the twists and turns of bad science|last=Dewdney|first=A. K.|date=1 April 1997|publisher=Wiley|isbn=978-0-471-10806-1|pages=82}}</ref> One response to Dewdney is that neural networks handle many complex and diverse tasks, ranging from autonomously flying aircraft<ref>[http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html NASA – Dryden Flight Research Center – News Room: News Releases: NASA NEURAL NETWORK PROJECT PASSES MILESTONE] {{Webarchive|url=https://web.archive.org/web/20100402065100/http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html |date=2 April 2010 }}. Nasa.gov. Retrieved on 20 November 2013.</ref> to detecting credit card fraud to mastering the game of [[w:en:Go (game)|Go]].

Technology writer Roger Bridgman commented:

{{blockquote|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.<ref>{{Cite web |url=http://members.fortunecity.com/templarseries/popper.html |title=Roger Bridgman's defence of neural networks |access-date=12 July 2010 |archive-url=https://web.archive.org/web/20120319163352/http://members.fortunecity.com/templarseries/popper.html |archive-date=19 March 2012 |url-status=dead}}</ref>
}}

Biological brains use both shallow and deep circuits as reported by brain anatomy,<ref name="VanEssen1991">D. J. Felleman and D. C. Van Essen, "[https://archive.today/20150120022056/http://cercor.oxfordjournals.org/content/1/1/1.1.full.pdf+html Distributed hierarchical processing in the primate cerebral cortex]," ''Cerebral Cortex'', 1, pp. 1–47, 1991.</ref> displaying a wide variety of invariance. Weng<ref name="Weng2012">J. Weng, "[https://www.amazon.com/Natural-Artificial-Intelligence-Introduction-Computational/dp/0985875720 Natural and Artificial Intelligence: Introduction to Computational Brain-Mind]," BMI Press, {{ISBN|978-0-9858757-2-5}}, 2012.</ref> argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies.

===Hardware===
Large and effective neural networks require considerable computing resources.<ref name=":0">{{cite journal|last1=Edwards|first1=Chris|s2cid=11026540|title=Growing pains for deep learning|journal=Communications of the ACM|date=25 June 2015|volume=58|issue=7|pages=14–16|doi=10.1145/2771283}}</ref> While the brain has hardware tailored to the task of processing signals through a [[w:en:Graph (discrete mathematics)|graph]] of neurons, simulating even a simplified neuron on [[w:en:von Neumann architecture|von Neumann architecture]] may consume vast amounts of [[w:en:Random-access memory|memory]] and storage. Furthermore, the designer often needs to transmit signals through many of these connections and their associated neurons{{snd}} which require enormous [[w:en:Central processing unit|CPU]] power and time.

[[w:en:Jürgen Schmidhuber|Schmidhuber]] noted that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by [[w:en:General-purpose computing on graphics processing units|GPGPUs]] (on [[w:en:Graphics processing unit|GPUs]]), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before.<ref name="SCHIDHUB2" /> The use of accelerators such as [[w:en:Field-programmable gate array|FPGA]]s and GPUs can reduce training times from months to days.{{r|:0}}

[[w:en:Neuromorphic engineering|Neuromorphic engineering]] or a [[w:en:physical neural network|physical neural network]] addresses the hardware difficulty directly, by constructing non-von-Neumann chips to directly implement neural networks in circuitry. Another type of chip optimized for neural network processing is called a [[w:en:Tensor Processing Unit|Tensor Processing Unit]], or TPU.<ref>{{cite news |url=https://www.wired.com/2016/05/google-tpu-custom-chips/ |author=Cade Metz |newspaper=Wired |date=18 May 2016 |title=Google Built Its Very Own Chips to Power Its AI Bots |access-date=5 March 2017 |archive-date=13 January 2018 |archive-url=https://web.archive.org/web/20180113150305/https://www.wired.com/2016/05/google-tpu-custom-chips/ |url-status=live }}</ref>

===Practical counterexamples ===
Analyzing what has been learned by an ANN is much easier than analyzing what has been learned by a biological neural network. Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful. For example, local vs. non-local learning and shallow vs. deep architecture.<ref>{{Cite web|title=Scaling Learning Algorithms towards AI|url=http://yann.lecun.com/exdb/publis/pdf/bengio-lecun-07.pdf|access-date=6 July 2022|archive-date=12 August 2022|archive-url=https://web.archive.org/web/20220812081157/http://yann.lecun.com/exdb/publis/pdf/bengio-lecun-07.pdf|url-status=live}}</ref>

===Hybrid approaches===
Advocates of [[w:en:Hybrid neural network|hybrid]] models (combining neural networks and symbolic approaches) say that such a mixture can better capture the mechanisms of the human mind.<ref>{{Cite journal| last1=Tahmasebi| last2=Hezarkhani| title=A hybrid neural networks-fuzzy logic-genetic algorithm for grade estimation| year=2012| journal=Computers & Geosciences| pages=18–27 |volume=42| doi=10.1016/j.cageo.2012.02.004| pmid=25540468| pmc=4268588| bibcode=2012CG.....42...18T}}</ref>

==Gallery==
<gallery widths="220">
File:Single layer ann.svg|A single-layer feedforward artificial neural network. Arrows originating from <math>\scriptstyle x_2</math> are omitted for clarity. There are p inputs to this network and q outputs. In this system, the value of the qth output, <math>\scriptstyle y_q</math> would be calculated as <math>\scriptstyle y_q = K*(\sum(x_i*w_{iq})-b_q) </math>.
File:Two layer ann.svg|A two-layer feedforward artificial neural network
File:Artificial neural network.svg|An artificial neural network
File:Ann dependency (graph).svg|An ANN dependency graph
File:Single-layer feedforward artificial neural network.png|A single-layer feedforward artificial neural network with 4 inputs, 6 hidden and 2 outputs. Given position state and direction outputs wheel based control values.
File:Two-layer feedforward artificial neural network.png|A two-layer feedforward artificial neural network with 8 inputs, 2x8 hidden and 2 outputs. Given position state, direction and other environment values outputs thruster based control values.
File:Cmac.jpg|Parallel pipeline structure of CMAC neural network. This learning algorithm can converge in one step.
</gallery>

== See also ==
{{cols|colwidth=18em}}
* [[w:en:ADALINE|ADALINE]]
* [[w:en:Autoencoder|Autoencoder]]
* [[w:en:Bio-inspired computing|Bio-inspired computing]]
* [[w:en:Blue Brain Project|Blue Brain Project]]
* [[w:en:Catastrophic interference|Catastrophic interference]]
* [[w:en:Cognitive architecture|Cognitive architecture]]
* [[w:en:Connectionist expert system|Connectionist expert system]]
* [[w:en:Connectomics|Connectomics]]
* [[w:en:Large width limits of neural networks|Large width limits of neural networks]]
* [[w:en:List of machine learning concepts|List of machine learning concepts]]
* [[w:en:Neural gas|Neural gas]]
* [[w:en:Neural network software|Neural network software]]
* [[w:en:Optical neural network|Optical neural network]]
* [[w:en:Parallel distributed processing|Parallel distributed processing]]
* [[w:en:Philosophy of artificial intelligence|Philosophy of artificial intelligence]]
* [[w:en:Recurrent neural networks|Recurrent neural networks]]
* [[w:en:Spiking neural network|Spiking neural network]] 
* [[w:en:Tensor product network|Tensor product network]]
{{colend}}

==Notes==
{{notelist}}

==References==
{{Reflist|30em}}

==Bibliography==
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{{colend}}

{{Complex systems topics}}
{{Control theory}}
{{Differentiable computing}}
{{Neuroscience}}
{{Self-driving cars and enabling technologies}}
{{Authority control}}

{{DEFAULTSORT:Artificial Neural Network}}
[[Category:Statistics]]
[[Category:Artificial neural networks| ]]
[[Category:Algorithms]]
[[Category:Neuroscience]]
[[Category:Psychology]]


== Page Information ==
This page was based on the following [https://en.wikipedia.org/wiki/Artficial%20neural%20network wikipedia-source page]:
* [https://en.wikipedia.org/wiki/Artificial%20neural%20network Artificial neural network] https://en.wikipedia.org/wiki/Artificial%20neural%20network
* Datum: 3/11/2023
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