View source for Neural network (machine learning)

{{short description|Computational model used in machine learning, based on connected, hierarchical functions}}{{cs1 config|name-list-style=vanc|display-authors=6}}
{{about|the computational models used for artificial intelligence||Neural network (disambiguation)}}
{{Use dmy dates|date=March 2023}}
[[File:Colored neural network.svg|thumb|upright=1.15|An artificial neural network is an interconnected group of nodes, inspired by a simplification of [[neuron]]s in a [[brain]]. Here, each circular node represents an [[artificial neuron]] and an arrow represents a connection from the output of one artificial neuron to the input of another.]]
{{Machine learning|Artificial neural network}}

In [[machine learning]], a '''neural network''' or '''neural net''' ('''NN'''), also called '''artificial neural network''' ('''ANN'''), is a [[computational model]] inspired by the structure and functions of [[biological neural networks]].<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|archive-date=18 March 2024|archive-url=https://web.archive.org/web/20240318120205/https://news.mit.edu/2017/explained-neural-networks-deep-learning-0414|url-status=live}}</ref><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>

A neural network consists of connected units or nodes called ''[[artificial neuron]]s'', which loosely model the [[neuron]]s in the brain. Artificial neuron models that mimic biological neurons more closely have also been recently investigated and shown to significantly improve performance. These are connected by ''edges'', which model the [[synapse]]s in the brain. Each artificial neuron receives signals from connected neurons, then processes them and sends a signal to other connected neurons. The "signal" is a [[real number]], and the output of each neuron is computed by some non-linear function of the totality of its inputs, called the ''[[activation function]]''. The strength of the signal at each connection is determined by a ''weight'', which adjusts during the learning process.

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 passing through multiple intermediate layers (''[[hidden layer]]s''). A network is typically called a deep neural network if it has at least two hidden layers.<ref>{{Cite book |last=Bishop |first=Christopher M. |title=Pattern Recognition and Machine Learning |date=17 August 2006 |publisher=Springer |isbn=978-0-387-31073-2 |location=New York |language=en}}</ref>

Artificial neural networks are used for various tasks, including [[predictive modeling]], [[adaptive control]], and solving problems in [[artificial intelligence]]. They can learn from experience, and can derive conclusions from a complex and seemingly unrelated set of information.

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==Training==
Neural networks are typically trained through [[empirical risk minimization]], which is based on the idea of optimizing, the network's parameters to minimize the difference, or empirical risk, between the predicted output and the actual target values in a given dataset.<ref name=":2">{{Cite book |last1=Vapnik |first1=Vladimir N. |title=The nature of statistical learning theory |last2=Vapnik |first2=Vladimir Naumovich |year=1998 |publisher=Springer |isbn=978-0-387-94559-0 |edition=Corrected 2nd print. |location=New York Berlin Heidelberg}}</ref> Gradient-based methods such as [[backpropagation]] are usually used to estimate the parameters of the network.<ref name=":2" /> During the training phase, ANNs learn from [[Labeled data|labeled]] training data by iteratively updating their parameters to minimize a defined [[Loss functions for classification|loss function]].<ref name=":4">{{cite book |author=Ian Goodfellow and Yoshua Bengio and Aaron Courville |url=http://www.deeplearningbook.org/ |title=Deep Learning |publisher=MIT Press |year=2016 |access-date=1 June 2016 |archive-url=https://web.archive.org/web/20160416111010/http://www.deeplearningbook.org/ |archive-date=16 April 2016 |url-status=live}}</ref> This method allows the network to generalize to unseen data.{{multiple image
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 | image1    = Simplified neural network training example.svg
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 | caption1  = Simplified example of training a neural network in object detection: The network is trained by multiple images that are known to depict [[starfish]] and [[sea urchin]]s, which are correlated with "nodes" that represent visual [[Feature (computer vision)|features]]. The starfish match with a ringed texture and a star outline, whereas most sea urchins match with a striped texture and oval shape. However, the instance of a ring textured sea urchin creates a weakly weighted association between them.
 | image2    = Simplified neural network example.svg
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 | caption2  = Subsequent run of the network on an input image (left):<ref>{{cite book |author=Ferrie, C. |author2=Kaiser, S. |year=2019|title=Neural Networks for Babies|publisher=Sourcebooks|isbn=978-1-4926-7120-6}}</ref> The network correctly detects the starfish. However, the weakly weighted association between ringed texture and sea urchin also confers a weak signal to the latter from one of two intermediate nodes. In addition, a shell that was not included in the training gives a weak signal for the oval shape, also resulting in a weak signal for the sea urchin output. These weak signals may result in a [[false positive]] result for sea urchin.<br />In reality, textures and outlines would not be represented by single nodes, but rather by associated weight patterns of multiple nodes.}}

==History==
{{main|History of artificial neural networks}}

=== Early work ===

Today's deep neural networks are based on early work in [[statistics]] over 200 years ago. The simplest kind of [[feedforward neural network]] (FNN) is a linear network, which consists of a single layer of output nodes with linear activation functions; the inputs are fed directly to the outputs via a series of weights. The sum of the products of the weights and the inputs is calculated at each node. The [[mean squared error]]s between these calculated outputs and the given target values are minimized by creating an adjustment to the weights. This technique has been known for over two centuries as the [[method of least squares]] or [[linear regression]]. It was used as a means of finding a good rough linear fit to a set of points by [[Adrien-Marie Legendre|Legendre]] (1805) and [[Gauss]] (1795) for the prediction of planetary movement.<ref name="legendre1805">Mansfield Merriman, "A List of Writings Relating to the Method of Least Squares"</ref><ref name="gauss1795">{{cite journal |first=Stephen M. |last=Stigler |year=1981 |title=Gauss and the Invention of Least Squares |journal=Ann. Stat. |volume=9 |issue=3 |pages=465–474 |doi=10.1214/aos/1176345451 |doi-access=free }}</ref><ref name=brertscher>{{cite book |last=Bretscher |first=Otto |title=Linear Algebra With Applications |edition=3rd |publisher=Prentice Hall |year=1995 |location=Upper Saddle River, NJ}}</ref><ref name=DLhistory/><ref name=stigler>
{{cite book |last=Stigler |first=Stephen M. |author-link=Stephen Stigler |year=1986 |title=The History of Statistics: The Measurement of Uncertainty before 1900 |location=Cambridge |publisher=Harvard |isbn=0-674-40340-1 |url-access=registration |url=https://archive.org/details/historyofstatist00stig}}</ref>

Historically, digital computers such as the [[von Neumann model]] operate via the execution of explicit instructions with access to memory by a number of processors. Some neural networks, on the other hand, originated from efforts to model information processing in biological systems through the framework of [[connectionism]]. Unlike the von Neumann model, connectionist computing does not separate memory and processing.

[[Warren McCulloch]] and [[Walter Pitts]]<ref name=WM /> (1943) considered a non-learning computational model for neural networks.<ref>{{Cite news |last=Kleene |first=S.C. |year=1956 |title=Representation of Events in Nerve Nets and Finite Automata |url=https://www.degruyter.com/view/books/9781400882618/9781400882618-002/9781400882618-002.xml |access-date=17 June 2017 |work=Annals of Mathematics Studies |publisher=Princeton University Press |pages=3–41 |issue=34 |archive-date=19 May 2024 |archive-url=https://web.archive.org/web/20240519081121/https://www.degruyter.com/view/books/9781400882618/9781400882618-002/9781400882618-002.xml |url-status=live }}</ref> This model paved the way for research to split into two approaches. One approach focused on biological processes while the other focused on the application of neural networks to artificial intelligence.

In the late 1940s, [[Donald O. Hebb|D. O. Hebb]]<ref>{{cite book|url={{google books |plainurl=y |id=ddB4AgAAQBAJ}}|title=The Organization of Behavior|last=Hebb|first=Donald|publisher=Taylor & Francis|orig-date=1st Pub. 1949|year=2005|isbn=978-1-135-63190-1}}</ref> proposed a learning [[hypothesis]] based on the mechanism of [[Neuroplasticity|neural plasticity]] that became known as [[Hebbian learning]]. It was used in many early neural networks, such as Rosenblatt's [[perceptron]] and the [[Hopfield network]]. Farley and [[Wesley A. Clark|Clark]]<ref>{{cite journal|last=Farley|first=B.G.|author2=W.A. Clark|year=1954|title=Simulation of Self-Organizing Systems by Digital Computer|journal=IRE Transactions on Information Theory|volume=4|issue=4|pages=76–84|doi=10.1109/TIT.1954.1057468 |bibcode=1954TIPIT...4...76F }}</ref> (1954) used computational machines to simulate a Hebbian network. Other neural network computational machines were created by [[Nathaniel Rochester (computer scientist)|Rochester]], Holland, Habit and Duda (1956).<ref>{{cite journal|last=Rochester|first=N.|author2=J.H. Holland|author3=L.H. Habit|author4=W.L. Duda|year=1956|title=Tests on a cell assembly theory of the action of the brain, using a large digital computer|journal=IRE Transactions on Information Theory|volume=2|issue=3|pages=80–93|doi=10.1109/TIT.1956.1056810 |bibcode=1956IRTIT...2...80R }}</ref> 

In 1958, psychologist [[Frank Rosenblatt]] described the perceptron, one of the first implemented artificial neural networks,<ref>Haykin (2008) Neural Networks and Learning Machines, 3rd edition</ref><ref>{{cite journal|last=Rosenblatt|first=F.|title=The Perceptron: A Probabilistic Model For Information Storage And Organization in the Brain|journal=Psychological Review|year=1958|volume=65|pages=386–408|doi=10.1037/h0042519|pmid=13602029|issue=6|citeseerx=10.1.1.588.3775|s2cid=12781225 }}</ref><ref name="Werbos 1975">{{cite book|url={{google books |plainurl=y |id=z81XmgEACAAJ}}|title=Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences|last=Werbos|first=P.J.|year=1975}}</ref><ref>{{cite journal |last=Rosenblatt |first=Frank |year=1957 |title=The Perceptron—a perceiving and recognizing automaton |journal=Report 85-460-1 |publisher=Cornell Aeronautical Laboratory }}</ref> funded by the United States [[Office of Naval Research]].<ref name="Olazaran">{{cite journal |first=Mikel |last=Olazaran |title=A Sociological Study of the Official History of the Perceptrons Controversy |journal=Social Studies of Science |volume=26 |issue=3 |year=1996 |jstor=285702|doi=10.1177/030631296026003005 |pages=611–659|s2cid=16786738 }}</ref>
R. D. Joseph (1960)<ref name="joseph1960">{{cite book |last=Joseph |first=R. D. |title=Contributions to Perceptron Theory, Cornell Aeronautical Laboratory Report No. VG-11 96--G-7, Buffalo |year=1960}}</ref> mentions an even earlier perceptron-like device by Farley and Clark:<ref name="DLhistory"/> "Farley and Clark of MIT Lincoln Laboratory actually preceded Rosenblatt in the development of a perceptron-like device." However, "they dropped the subject."
The perceptron raised public excitement for research in Artificial Neural Networks, causing the US government to drastically increase funding. This contributed to "the Golden Age of AI" fueled by the optimistic claims made by computer scientists regarding the ability of perceptrons to emulate human intelligence.<ref name=":08">{{Cite book |author=Russel, Stuart |author2=Norvig, Peter |url=https://people.engr.tamu.edu/guni/csce421/files/AI_Russell_Norvig.pdf |title=Artificial Intelligence A Modern Approach |publisher=Pearson Education |year=2010 |isbn=978-0-13-604259-4 |edition=3rd |location=United States of America |pages=16–28 |language=en |archive-date=17 January 2023 |access-date=9 December 2023 |archive-url=https://archive.today/20230117223845/https://people.engr.tamu.edu/guni/csce421/files/AI_Russell_Norvig.pdf }}</ref>

The first perceptrons did not have adaptive hidden units. However, Joseph (1960)<ref name="joseph1960"/> also discussed [[multilayer perceptrons]] with an adaptive hidden layer. Rosenblatt (1962)<ref name="rosenblatt1962">{{cite book |last=Rosenblatt |first=Frank |author-link=Frank Rosenblatt |title=Principles of Neurodynamics |publisher=Spartan, New York |year=1962}}</ref>{{rp|section 16}} cited and adopted these ideas, also crediting work by H. D. Block and B. W. Knight. Unfortunately, these early efforts did not lead to a working learning algorithm for hidden units, i.e., [[deep learning]].

=== Deep learning breakthroughs in the 1960s and 1970s===
Fundamental research was conducted on ANNs in the 1960s and 1970s. The first working deep learning algorithm was the [[Group method of data handling]], a method to train arbitrarily deep neural networks, published by [[Alexey Ivakhnenko]] and Lapa in the [[Soviet Union]] (1965). They regarded it as a form of polynomial regression,<ref name="ivak1965">{{cite book|first1=A. G. |last1=Ivakhnenko |first2=V. G. |last2=Lapa |title=Cybernetics and Forecasting Techniques|url={{google books |plainurl=y |id=rGFgAAAAMAAJ}}|year=1967|publisher=American Elsevier Publishing Co.|isbn=978-0-444-00020-0}}</ref> or a generalization of Rosenblatt's perceptron.<ref>{{Cite journal |last=Ivakhnenko |first=A.G. |date=March 1970 |title=Heuristic self-organization in problems of engineering cybernetics |url=https://linkinghub.elsevier.com/retrieve/pii/0005109870900920 |journal=Automatica |language=en |volume=6 |issue=2 |pages=207–219 |doi=10.1016/0005-1098(70)90092-0 |archive-date=12 August 2024 |access-date=7 August 2024 |archive-url=https://web.archive.org/web/20240812123448/https://linkinghub.elsevier.com/retrieve/pii/0005109870900920 |url-status=live }}</ref> A 1971 paper described a deep network with eight layers trained by this method,<ref name="ivak1971">{{Cite journal|last=Ivakhnenko|first=Alexey|date=1971|title=Polynomial theory of complex systems|url=http://gmdh.net/articles/history/polynomial.pdf|journal=IEEE Transactions on Systems, Man, and Cybernetics|pages=364–378|doi=10.1109/TSMC.1971.4308320|volume=SMC-1|issue=4|bibcode=1971ITSMC...1..364I |access-date=5 November 2019|archive-date=29 August 2017|archive-url=https://web.archive.org/web/20170829230621/http://www.gmdh.net/articles/history/polynomial.pdf|url-status=live}}</ref> which is based on layer by layer training through regression analysis. Superfluous hidden units are pruned using a separate validation set. Since the activation functions of the nodes are Kolmogorov-Gabor polynomials, these were also the first deep networks with multiplicative units or "gates."<ref name="DLhistory">{{cite arXiv |eprint=2212.11279 |class=cs.NE |first=Jürgen |last=Schmidhuber |author-link=Jürgen Schmidhuber |title=Annotated History of Modern AI and Deep Learning |date=2022}}</ref>

The first deep learning [[multilayer perceptron]] trained by [[stochastic gradient descent]]<ref name="robbins1951">{{Cite journal | last1 = Robbins | first1 = H. | author-link = Herbert Robbins| last2 = Monro | first2 = S. | doi = 10.1214/aoms/1177729586 | title = A Stochastic Approximation Method | journal = The Annals of Mathematical Statistics | volume = 22 | issue = 3 | page = 400 | year = 1951 | doi-access = free }}</ref> was published in 1967 by [[Shun'ichi Amari]].<ref name="Amari1967">{{cite journal |last1=Amari |first1=Shun'ichi |author-link=Shun'ichi Amari|title=A theory of adaptive pattern classifier|journal= IEEE Transactions |date=1967 |volume=EC |issue=16 |pages=279–307}}</ref> In computer experiments conducted by Amari's student Saito, a five layer MLP with two modifiable layers learned [[Knowledge representation|internal representations]] to classify non-linearily separable pattern classes.<ref name="DLhistory"/> Subsequent developments in hardware and hyperparameter tunings have made end-to-end stochastic gradient descent the currently dominant training technique.

In 1969, [[Kunihiko Fukushima]] introduced the [[rectifier (neural networks)|ReLU]] (rectified linear unit) activation function.<ref name="DLhistory" /><ref name="Fukushima1969">{{cite journal |last1=Fukushima |first1=K. |date=1969 |title=Visual feature extraction by a multilayered network of analog threshold elements |journal=IEEE Transactions on Systems Science and Cybernetics |volume=5 |issue=4 |pages=322–333 |doi=10.1109/TSSC.1969.300225 |bibcode=1969ITSSC...5..322F }}</ref><ref name=sonoda17>{{cite journal | last1 = Sonoda | first1 = Sho | last2=Murata | first2=Noboru | s2cid = 12149203 | year = 2017 | title = Neural network with unbounded activation functions is universal approximator | journal = Applied and Computational Harmonic Analysis | volume = 43 | issue = 2 | pages = 233–268 | doi = 10.1016/j.acha.2015.12.005| arxiv = 1505.03654 }}</ref> The rectifier has become the most popular activation function for deep learning.<ref>{{cite arXiv |eprint=1710.05941 |class=cs.NE |first1=Prajit |last1=Ramachandran |first2=Zoph |last2=Barret |title=Searching for Activation Functions |date=16 October 2017 |last3=Quoc |first3=V. Le}}</ref>

Nevertheless, research stagnated in the United States following the work of [[Marvin Minsky|Minsky]] and [[Seymour Papert|Papert]] (1969),<ref name=":132">{{cite book |last1=Minsky |first1=Marvin |url={{google books |plainurl=y |id=Ow1OAQAAIAAJ}} |title=Perceptrons: An Introduction to Computational Geometry |last2=Papert |first2=Seymour |publisher=MIT Press |year=1969 |isbn=978-0-262-63022-1}}</ref> who emphasized that basic perceptrons were incapable of processing the exclusive-or circuit. This insight was irrelevant for the deep networks of Ivakhnenko (1965) and Amari (1967).

In 1976 transfer learning was introduced in neural networks learning.<ref>Bozinovski S. and Fulgosi A. (1976). "The influence of pattern similarity and transfer learning on the base perceptron training" (original in Croatian) Proceedings of Symposium Informatica 3-121-5, Bled.</ref><ref>Bozinovski S.(2020) "Reminder of the first paper on transfer learning in neural networks, 1976". Informatica 44: 291–302.</ref>

Deep learning architectures for [[convolutional neural network]]s (CNNs) with convolutional layers and downsampling layers and weight replication began with the [[Neocognitron]] introduced by Kunihiko Fukushima in 1979, though not trained by backpropagation.<ref name="FUKU1979">{{cite journal |last1=Fukushima |first1=K. |year=1979 |title=Neural network model for a mechanism of pattern recognition unaffected by shift in position—Neocognitron |journal=Trans. IECE (In Japanese)|volume= J62-A |issue=10 |pages=658–665 |doi=10.1007/bf00344251 |pmid=7370364 |s2cid=206775608 |url=http://ci.nii.ac.jp/naid/40000025975/ }}</ref><ref name="FUKU1980">{{cite journal |last1=Fukushima |first1=K. |year=1980 |title=Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position |journal=Biol. Cybern. |volume=36 |issue=4 |pages=193–202 |doi=10.1007/bf00344251 |pmid=7370364 |s2cid=206775608 |url=http://ci.nii.ac.jp/naid/40000025975/ }}</ref><ref name="SCHIDHUB4"/>

=== Backpropagation ===
[[Backpropagation]] is an efficient application of the [[chain rule]] derived by [[Gottfried Wilhelm Leibniz]] in 1673<ref name="leibniz16762">{{Cite book |last=Leibniz |first=Gottfried Wilhelm Freiherr von |url=https://books.google.com/books?id=bOIGAAAAYAAJ&q=leibniz+altered+manuscripts&pg=PA90 |title=The Early Mathematical Manuscripts of Leibniz: Translated from the Latin Texts Published by Carl Immanuel Gerhardt with Critical and Historical Notes (Leibniz published the chain rule in a 1676 memoir) |date=1920 |publisher=Open court publishing Company |isbn=978-0-598-81846-1 |language=en}}</ref> to networks of differentiable nodes. The terminology "back-propagating errors" was actually introduced in 1962 by Rosenblatt,<ref name="rosenblatt1962"/> but he did not know how to implement this, although [[Henry J. Kelley]] had a continuous precursor of backpropagation in 1960 in the context of [[control theory]].<ref name="kelley19602">{{cite journal |last1=Kelley |first1=Henry J. |author-link=Henry J. Kelley |year=1960 |title=Gradient theory of optimal flight paths |journal=ARS Journal |volume=30 |issue=10 |pages=947–954 |doi=10.2514/8.5282}}</ref> In 1970, [[Seppo Linnainmaa]] published the modern form of backpropagation in his Master's [[thesis]] (1970).<ref name="lin19703">{{cite thesis |first=Seppo |last=Linnainmaa |author-link=Seppo Linnainmaa |year=1970 |type=Masters |title=The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors |language=fi |publisher=University of Helsinki |page=6–7}}</ref><ref name="lin19763">{{cite journal |last1=Linnainmaa |first1=Seppo |author-link=Seppo Linnainmaa |year=1976 |title=Taylor expansion of the accumulated rounding error |journal=BIT Numerical Mathematics |volume=16 |issue=2 |pages=146–160 |doi=10.1007/bf01931367 |s2cid=122357351}}</ref><ref name="DLhistory" /> G.M. Ostrovski et al. republished it in 1971.<ref name="ostrowski1971">Ostrovski, G.M., Volin,Y.M., and Boris, W.W. (1971). On the computation of derivatives. Wiss. Z. Tech. Hochschule for Chemistry, 13:382–384.</ref><ref name="backprop"/> [[Paul Werbos]] applied backpropagation to neural networks in 1982<ref name="werbos1982">{{cite book |last=Werbos |first=Paul |author-link=Paul Werbos |title=System modeling and optimization |publisher=Springer |year=1982 |pages=762–770 |chapter=Applications of advances in nonlinear sensitivity analysis |access-date=2 July 2017 |chapter-url=http://werbos.com/Neural/SensitivityIFIPSeptember1981.pdf |archive-url=https://web.archive.org/web/20160414055503/http://werbos.com/Neural/SensitivityIFIPSeptember1981.pdf |archive-date=14 April 2016 |url-status=live}}</ref><ref name=":1">{{Cite book |url=https://direct.mit.edu/books/book/4886/Talking-NetsAn-Oral-History-of-Neural-Networks |title=Talking Nets: An Oral History of Neural Networks |date=2000 |publisher=The MIT Press |isbn=978-0-262-26715-1 |editor-last=Anderson |editor-first=James A. |language=en |doi=10.7551/mitpress/6626.003.0016 |editor-last2=Rosenfeld |editor-first2=Edward |archive-date=12 October 2024 |access-date=7 August 2024 |archive-url=https://archive.today/20241012223136/https://direct.mit.edu/books/book/4886/Talking-NetsAn-Oral-History-of-Neural-Networks |url-status=live }}</ref> (his 1974 PhD thesis, reprinted in a 1994 book,<ref name="werbos1974">{{cite book |last=Werbos |first=Paul J. |title=The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Political Forecasting |location=New York |publisher=John Wiley & Sons |year=1994 |isbn=0-471-59897-6 }}</ref> did not yet describe the algorithm<ref name="backprop">{{cite web | last = Schmidhuber | first = Juergen | title = Who Invented Backpropagation? | author-link = Juergen Schmidhuber | publisher = IDSIA, Switzerland | url = https://people.idsia.ch/~juergen/who-invented-backpropagation.html | date = 25 October 2014 | access-date = 14 September 2024 | archive-url = https://web.archive.org/web/20240730110408/https://people.idsia.ch/~juergen/who-invented-backpropagation.html | archive-date = 30 July 2024 | quote =  | url-status = live }}</ref>). In 1986, [[David E. Rumelhart]] et al. popularised backpropagation but did not cite the original work.<ref>{{Cite journal |last1=Rumelhart |first1=David E. |last2=Hinton |first2=Geoffrey E. |last3=Williams |first3=Ronald J. |date=October 1986 |title=Learning representations by back-propagating errors |url=https://www.nature.com/articles/323533a0 |journal=Nature |language=en |volume=323 |issue=6088 |pages=533–536 |doi=10.1038/323533a0 |bibcode=1986Natur.323..533R |issn=1476-4687 |archive-date=8 March 2021 |access-date=17 March 2021 |archive-url=https://web.archive.org/web/20210308045630/https://www.nature.com/articles/323533a0 |url-status=live }}</ref>

=== Convolutional neural networks ===
Kunihiko Fukushima's [[convolutional neural network]] (CNN) architecture of 1979<ref name="FUKU1979"/> also introduced [[max pooling]],<ref>{{Cite journal |last1=Fukushima |first1=Kunihiko |last2=Miyake |first2=Sei |date=1 January 1982 |title=Neocognitron: A new algorithm for pattern recognition tolerant of deformations and shifts in position |url=https://www.sciencedirect.com/science/article/abs/pii/0031320382900243 |journal=Pattern Recognition |volume=15 |issue=6 |pages=455–469 |doi=10.1016/0031-3203(82)90024-3 |bibcode=1982PatRe..15..455F |issn=0031-3203 |archive-date=12 October 2024 |access-date=9 September 2024 |archive-url=https://archive.today/20241012232918/https://www.sciencedirect.com/science/article/abs/pii/0031320382900243 |url-status=live }}</ref> a popular downsampling procedure for CNNs. CNNs have become an essential tool for [[computer vision]].

The [[time delay neural network]] (TDNN) was introduced in 1987 by [[Alex Waibel]] to apply CNN to phoneme recognition. It used convolutions, weight sharing, and backpropagation.<ref name=Waibel1987>{{cite conference |title=Phoneme Recognition Using Time-Delay Neural Networks |last1=Waibel |first1=Alex |date=December 1987 |location=Tokyo, Japan |conference=Meeting of the Institute of Electrical, Information and Communication Engineers (IEICE) |url=https://isl.anthropomatik.kit.edu/pdf/Waibel1987a.pdf |access-date=20 September 2024 |archive-date=17 September 2024 |archive-url=https://web.archive.org/web/20240917173146/https://isl.anthropomatik.kit.edu/pdf/Waibel1987a.pdf |url-status=live }}</ref><ref name="speechsignal">[[Alex Waibel|Alexander Waibel]] et al., ''[http://www.inf.ufrgs.br/~engel/data/media/file/cmp121/waibel89_TDNN.pdf Phoneme Recognition Using Time-Delay Neural Networks] {{Webarchive|url=https://web.archive.org/web/20241211184304/https://www.inf.ufrgs.br/~engel/data/media/file/cmp121/waibel89_TDNN.pdf |date=11 December 2024 }}'' IEEE Transactions on Acoustics, Speech, and Signal Processing, Volume 37, No. 3, pp. 328. – 339 March 1989.</ref> In 1988, Wei Zhang applied a backpropagation-trained CNN to alphabet recognition.<ref name="wz1988">{{cite journal |last=Zhang |first=Wei |date=1988 |title=Shift-invariant pattern recognition neural network and its optical architecture |url=https://drive.google.com/file/d/1nN_5odSG_QVae54EsQN_qSz-0ZsX6wA0/view?usp=sharing |journal=Proceedings of Annual Conference of the Japan Society of Applied Physics |archive-date=23 June 2020 |access-date=12 April 2023 |archive-url=https://web.archive.org/web/20200623051222/https://drive.google.com/file/d/1nN_5odSG_QVae54EsQN_qSz-0ZsX6wA0/view?usp=sharing |url-status=live }}</ref>
In 1989, [[Yann LeCun]] et al. created a CNN called [[LeNet]] for [[Handwriting recognition|recognizing handwritten ZIP code]]s on mail. Training required 3 days.<ref name="LECUN1989">LeCun ''et al.'', "Backpropagation Applied to Handwritten Zip Code Recognition", ''Neural Computation'', 1, pp. 541–551, 1989.</ref> In 1990, Wei Zhang implemented a CNN on [[optical computing]] hardware.<ref name="wz1990">{{cite journal |last=Zhang |first=Wei |date=1990 |title=Parallel distributed processing model with local space-invariant interconnections and its optical architecture |url=https://drive.google.com/file/d/0B65v6Wo67Tk5ODRzZmhSR29VeDg/view?usp=sharing |journal=Applied Optics |volume=29 |issue=32 |pages=4790–7 |bibcode=1990ApOpt..29.4790Z |doi=10.1364/AO.29.004790 |pmid=20577468 |archive-date=6 February 2017 |access-date=12 April 2023 |archive-url=https://web.archive.org/web/20170206111407/https://drive.google.com/file/d/0B65v6Wo67Tk5ODRzZmhSR29VeDg/view?usp=sharing |url-status=live }}</ref> In 1991, a CNN was applied to medical image object segmentation<ref>{{cite journal |last=Zhang |first=Wei |date=1991 |title=Image processing of human corneal endothelium based on a learning network |url=https://drive.google.com/file/d/0B65v6Wo67Tk5cm5DTlNGd0NPUmM/view?usp=sharing |journal=Applied Optics |volume=30 |issue=29 |pages=4211–7 |bibcode=1991ApOpt..30.4211Z |doi=10.1364/AO.30.004211 |pmid=20706526 |archive-date=19 June 2024 |access-date=20 September 2024 |archive-url=https://web.archive.org/web/20240619084309/https://drive.google.com/file/d/0B65v6Wo67Tk5cm5DTlNGd0NPUmM/view?usp=sharing |url-status=live }}</ref> and breast cancer detection in mammograms.<ref>{{cite journal |last=Zhang |first=Wei |date=1994 |title=Computerized detection of clustered microcalcifications in digital mammograms using a shift-invariant artificial neural network |url=https://drive.google.com/file/d/0B65v6Wo67Tk5Ml9qeW5nQ3poVTQ/view?usp=sharing |journal=Medical Physics |volume=21 |issue=4 |pages=517–24 |bibcode=1994MedPh..21..517Z |doi=10.1118/1.597177 |pmid=8058017 |archive-date=20 June 2024 |access-date=20 September 2024 |archive-url=https://web.archive.org/web/20240620055642/https://drive.google.com/file/d/0B65v6Wo67Tk5Ml9qeW5nQ3poVTQ/view?usp=sharing |url-status=live }}</ref> [[LeNet]]-5 (1998), a 7-level CNN by Yann LeCun et al., that classifies digits, was applied by several banks to recognize hand-written numbers on checks digitized in 32×32 pixel images.<ref name="lecun98">{{cite journal |last=LeCun |first=Yann |author2=Léon Bottou |author3=Yoshua Bengio |author4=Patrick Haffner |year=1998 |title=Gradient-based learning applied to document recognition |url=http://yann.lecun.com/exdb/publis/pdf/lecun-01a.pdf |journal=Proceedings of the IEEE |volume=86 |issue=11 |pages=2278–2324 |citeseerx=10.1.1.32.9552 |doi=10.1109/5.726791 |bibcode=1998IEEEP..86.2278L |s2cid=14542261 |access-date=7 October 2016 |archive-date=30 October 2023 |archive-url=https://web.archive.org/web/20231030100650/http://yann.lecun.com/exdb/publis/pdf/lecun-01a.pdf }}</ref>

From 1988 onward,<ref name="Qian1988">Qian, Ning, and Terrence J. Sejnowski. "Predicting the secondary structure of globular proteins using neural network models." ''Journal of molecular biology'' 202, no. 4 (1988): 865–884.</ref><ref name="Bohr1988">Bohr, Henrik, Jakob Bohr, Søren Brunak, Rodney MJ Cotterill, Benny Lautrup, Leif Nørskov, Ole H. Olsen, and Steffen B. Petersen. "Protein secondary structure and homology by neural networks The α-helices in rhodopsin." ''FEBS letters'' 241, (1988): 223–228</ref> the use of neural networks transformed the field of [[protein structure prediction]], in particular when the first cascading networks were trained on ''profiles'' (matrices) produced by multiple [[sequence alignment]]s.<ref name="Rost1993">Rost, Burkhard, and Chris Sander. "Prediction of protein secondary structure at better than 70% accuracy." ''Journal of molecular biology'' 232, no. 2 (1993): 584–599.</ref>

=== Recurrent neural networks ===
One origin of RNN was [[statistical mechanics]]. In 1972, [[Shun'ichi Amari]] proposed to modify the weights of an [[Ising model]] by [[Hebbian theory|Hebbian learning]] rule as a model of [[Hopfield network|associative memory]], adding in the component of learning.<ref>{{Cite journal |last=Amari |first=S.-I. |date=November 1972 |title=Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements |journal=IEEE Transactions on Computers |volume=C-21 |issue=11 |pages=1197–1206 |doi=10.1109/T-C.1972.223477 |bibcode=1972ITCmp.100.1197A |issn=0018-9340 }}</ref> This was popularized as the Hopfield network by [[John Hopfield]] (1982).<ref name="Hopfield19822">{{cite journal |last1=Hopfield |first1=J. J. |date=1982 |title=Neural networks and physical systems with emergent collective computational abilities |journal=Proceedings of the National Academy of Sciences |volume=79 |issue=8 |pages=2554–2558 |bibcode=1982PNAS...79.2554H |doi=10.1073/pnas.79.8.2554 |pmc=346238 |pmid=6953413 |doi-access=free}}</ref> Another origin of RNN was neuroscience. The word "recurrent" is used to describe loop-like structures in anatomy. In 1901, [[Santiago Ramón y Cajal|Cajal]] observed "recurrent semicircles" in the [[Cerebellum|cerebellar cortex]].<ref>{{Cite journal |last1=Espinosa-Sanchez |first1=Juan Manuel |last2=Gomez-Marin |first2=Alex |last3=de Castro |first3=Fernando |date=5 July 2023 |title=The Importance of Cajal's and Lorente de Nó's Neuroscience to the Birth of Cybernetics |url=http://journals.sagepub.com/doi/10.1177/10738584231179932 |journal=The Neuroscientist |volume=31 |issue=1 |pages=14–30 |language=en |doi=10.1177/10738584231179932 |issn=1073-8584 |pmid=37403768 |hdl=10261/348372 |hdl-access=free |archive-date=12 October 2024 |access-date=7 August 2024 |archive-url=https://archive.today/20241012221924/http://journals.sagepub.com/doi/10.1177/10738584231179932 |url-status=live }}</ref> [[Donald O. Hebb|Hebb]] considered "reverberating circuit" as an explanation for short-term memory.<ref>{{Cite web |title=reverberating circuit |url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100417461 |access-date=27 July 2024 |website=Oxford Reference |archive-date=12 October 2024 |archive-url=https://archive.today/20241012222600/https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100417461 |url-status=live }}</ref> The McCulloch and Pitts paper (1943) considered neural networks that contain cycles, and noted that the current activity of such networks can be affected by activity indefinitely far in the past.<ref name=WM>{{Cite journal |last1=McCulloch |first1=Warren S. |last2=Pitts |first2=Walter |date=December 1943 |title=A logical calculus of the ideas immanent in nervous activity |url=http://link.springer.com/10.1007/BF02478259 |journal=The Bulletin of Mathematical Biophysics |volume=5 |issue=4 |pages=115–133 |doi=10.1007/BF02478259 |issn=0007-4985 |archive-date=12 October 2024 |access-date=7 August 2024 |archive-url=https://archive.today/20241012221923/http://link.springer.com/10.1007/BF02478259 |url-status=live }}</ref>

In 1982 a recurrent neural network with an array architecture (rather than a multilayer perceptron architecture), namely a Crossbar Adaptive Array,<ref name="CAA1982"> Bozinovski, S. (1982). "A self-learning system using secondary reinforcement". In Trappl, Robert (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><ref name="" "caa1995"="">Bozinovski S. (1995) "Neuro genetic agents and structural theory of self-reinforcement learning systems". CMPSCI Technical Report 95-107, University of Massachusetts at Amherst [https://web.cs.umass.edu/publication/docs/1995/UM-CS-1995-107.pdf] {{Webarchive|url=https://web.archive.org/web/20241008120651/https://web.cs.umass.edu/publication/docs/1995/UM-CS-1995-107.pdf|date=8 October 2024}}</ref> used direct recurrent connections from the output to the supervisor (teaching) inputs. In addition of computing actions (decisions), it computed internal state evaluations (emotions) of the consequence situations. Eliminating the external supervisor, it introduced the self-learning method in neural networks.  

In cognitive psychology, the journal American Psychologist in early 1980's carried out a debate on the relation between cognition and emotion. Zajonc in 1980 stated that emotion is computed first and is independent from cognition, while Lazarus in 1982 stated that cognition is computed first and is inseparable from emotion.<ref>{{Cite journal |first=R. |last=Zajonc |year=1980 |title=Feeling and thinking: Preferences need no inferences |journal=American Psychologist |volume=35 |issue=2 |pages=151–175 |doi=10.1037/0003-066X.35.2.151 |url=https://repositorio.unal.edu.co/handle/unal/77993 }}</ref><ref>Lazarus R. (1982) "Thoughts on the relations between emotion and cognition" American Psychologist 37 (9): 1019-1024</ref> In 1982 the Crossbar Adaptive Array gave a neural network model of cognition-emotion relation.<ref name = "CAA1982" /><ref>Bozinovski, S. (2014) "Modeling mechanisms of cognition-emotion interaction in artificial neural networks, since 1981" Procedia Computer Science p. 255-263 (https://core.ac.uk/download/pdf/81973924.pdf {{Webarchive|url=https://web.archive.org/web/20190323204838/https://core.ac.uk/download/pdf/81973924.pdf |date=23 March 2019 }})</ref> It was an example of a debate where an AI system, a recurrent neural network, contributed to an issue in the same time addressed by cognitive psychology.

Two early influential works were the [[Recurrent neural network#Jordan network|Jordan network]] (1986) and the [[Recurrent neural network#Elman network|Elman network]] (1990), which applied RNN to study [[cognitive psychology]]. 

In the 1980s, backpropagation did not work well for deep RNNs. To overcome this problem, in 1991, [[Jürgen Schmidhuber]] proposed the "neural sequence chunker" or "neural history compressor"<ref name="chunker1991">{{cite journal |last1=Schmidhuber |first1=Jürgen |date=April 1991 |title=Neural Sequence Chunkers |author-link=Jürgen Schmidhuber |url=https://people.idsia.ch/~juergen/FKI-148-91ocr.pdf |journal=TR FKI-148, TU Munich |archive-date=14 September 2024 |access-date=21 September 2024 |archive-url=https://web.archive.org/web/20240914162750/https://people.idsia.ch/~juergen/FKI-148-91ocr.pdf |url-status=live }}</ref><ref name="schmidhuber1992">{{cite journal |last1=Schmidhuber |first1=Jürgen |year=1992 |title=Learning complex, extended sequences using the principle of history compression (based on TR FKI-148, 1991) |url=https://sferics.idsia.ch/pub/juergen/chunker.pdf |journal=Neural Computation |volume=4 |issue=2 |pages=234–242 |doi=10.1162/neco.1992.4.2.234 |s2cid=18271205 |archive-date=14 September 2024 |access-date=21 September 2024 |archive-url=https://web.archive.org/web/20240914162750/https://sferics.idsia.ch/pub/juergen/chunker.pdf |url-status=live }}</ref> which introduced the important concepts of self-supervised pre-training (the "P" in [[ChatGPT]]) and neural [[knowledge distillation]].<ref name=DLhistory/> In 1993, a neural history compressor system solved a "Very Deep Learning" task that required more than 1000 subsequent [[Layer (deep learning)|layers]] in an RNN unfolded in time.<ref name="schmidhuber19932">{{Cite book |last=Schmidhuber |first=Jürgen |url=https://sferics.idsia.ch/pub/juergen/habilitation.pdf |title=Habilitation thesis: System modeling and optimization |year=1993 |archive-date=7 August 2024 |access-date=21 September 2024 |archive-url=https://web.archive.org/web/20240807084323/https://sferics.idsia.ch/pub/juergen/habilitation.pdf |url-status=live }} Page 150 ff demonstrates credit assignment across the equivalent of 1,200 layers in an unfolded RNN.</ref>

In 1991, [[Sepp Hochreiter]]'s diploma thesis<ref name="HOCH1991">S. Hochreiter., "[http://people.idsia.ch/~juergen/SeppHochreiter1991ThesisAdvisorSchmidhuber.pdf Untersuchungen zu dynamischen neuronalen Netzen]", {{Webarchive|url=https://web.archive.org/web/20150306075401/http://people.idsia.ch/~juergen/SeppHochreiter1991ThesisAdvisorSchmidhuber.pdf|date=6 March 2015}}, ''Diploma thesis. Institut f. Informatik, Technische Univ. Munich. Advisor: J. Schmidhuber'', 1991.</ref> identified and analyzed the [[vanishing gradient problem]]<ref name="HOCH1991" /><ref name="HOCH2001">{{cite book |last=Hochreiter |first=S. |title=A Field Guide to Dynamical Recurrent Networks |date=15 January 2001 |publisher=John Wiley & Sons |isbn=978-0-7803-5369-5 |editor-last1=Kolen |editor-first1=John F. |chapter=Gradient flow in recurrent nets: the difficulty of learning long-term dependencies |display-authors=etal |editor-last2=Kremer |editor-first2=Stefan C. |chapter-url=https://books.google.com/books?id=NWOcMVA64aAC |access-date=26 June 2017 |archive-date=19 May 2024 |archive-url=https://web.archive.org/web/20240519081124/https://books.google.com/books?id=NWOcMVA64aAC |url-status=live }}</ref> and proposed recurrent [[Residual neural network|residual]] connections to solve it. He and Schmidhuber introduced [[long short-term memory]] (LSTM), which set accuracy records in multiple applications domains.<ref>{{Cite Q|Q98967430}}</ref><ref name="lstm2">{{Cite journal |last1=Hochreiter |first1=Sepp |author-link=Sepp Hochreiter |last2=Schmidhuber |first2=Jürgen |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 |s2cid=1915014}}</ref> This was not yet the modern version of LSTM, which required the forget gate, which was introduced in 1999.<ref name="lstm1999">{{Cite book |last1=Gers |first1=Felix |title=9th International Conference on Artificial Neural Networks: ICANN '99 |last2=Schmidhuber |first2=Jürgen |last3=Cummins |first3=Fred |year=1999 |isbn=0-85296-721-7 |volume=1999 |pages=850–855 |chapter=Learning to forget: Continual prediction with LSTM |doi=10.1049/cp:19991218}}</ref> It became the default choice for RNN architecture.

During 1985–1995, inspired by statistical mechanics, several architectures and methods were developed by [[Terry Sejnowski]], [[Peter Dayan]], [[Geoffrey Hinton]], etc., including the [[Boltzmann machine]],<ref>{{Cite journal |last1=Ackley |first1=David H. |last2=Hinton |first2=Geoffrey E. |last3=Sejnowski |first3=Terrence J. |date=1 January 1985 |title=A learning algorithm for boltzmann machines |url=https://www.sciencedirect.com/science/article/pii/S0364021385800124 |journal=Cognitive Science |volume=9 |issue=1 |pages=147–169 |doi=10.1016/S0364-0213(85)80012-4 |issn=0364-0213 |archive-date=17 September 2024 |access-date=7 August 2024 |archive-url=https://web.archive.org/web/20240917124802/https://www.sciencedirect.com/science/article/pii/S0364021385800124 |url-status=live }}</ref> [[restricted Boltzmann machine]],<ref>{{cite book |last=Smolensky |first=Paul |title=Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Volume 1: Foundations |title-link=Connectionism |publisher=MIT Press |year=1986 |isbn=0-262-68053-X |editor1-last=Rumelhart |editor1-first=David E. |pages=[https://archive.org/details/paralleldistribu00rume/page/194 194–281] |chapter=Chapter 6: Information Processing in Dynamical Systems: Foundations of Harmony Theory |editor2-last=McLelland |editor2-first=James L. |chapter-url=https://stanford.edu/~jlmcc/papers/PDP/Volume%201/Chap6_PDP86.pdf |archive-date=14 July 2023 |access-date=7 August 2024 |archive-url=https://web.archive.org/web/20230714174222/https://stanford.edu/~jlmcc/papers/PDP/Volume%201/Chap6_PDP86.pdf |url-status=live }}</ref> [[Helmholtz machine]],<ref name=""nc95"">{{Cite journal |last1=Peter |first1=Dayan |author-link1=Peter Dayan |last2=Hinton |first2=Geoffrey E. |author-link2=Geoffrey Hinton |last3=Neal |first3=Radford M. |author-link3=Radford M. Neal |last4=Zemel |first4=Richard S. |author-link4=Richard Zemel |date=1995 |title=The Helmholtz machine. |journal=Neural Computation |volume=7 |issue=5 |pages=889–904 |doi=10.1162/neco.1995.7.5.889 |pmid=7584891 |s2cid=1890561 |hdl-access=free |hdl=21.11116/0000-0002-D6D3-E}} {{closed access}}</ref> and the [[wake-sleep algorithm]].<ref name=":13">{{Cite journal |last1=Hinton |first1=Geoffrey E. |author-link=Geoffrey Hinton |last2=Dayan |first2=Peter |author-link2=Peter Dayan |last3=Frey |first3=Brendan J. |author-link3=Brendan Frey |last4=Neal |first4=Radford |date=26 May 1995 |title=The wake-sleep algorithm for unsupervised neural networks |journal=Science |volume=268 |issue=5214 |pages=1158–1161 |bibcode=1995Sci...268.1158H |doi=10.1126/science.7761831 |pmid=7761831 |s2cid=871473}}</ref> These were designed for unsupervised learning of deep generative models.

=== Deep learning ===
Between 2009 and 2012, ANNs began winning prizes in image recognition contests, approaching human level performance on various tasks, initially in [[pattern recognition]] and [[handwriting recognition]].<ref>[http://www.kurzweilai.net/how-bio-inspired-deep-learning-keeps-winning-competitions 2012 Kurzweil AI Interview] {{Webarchive|url=https://web.archive.org/web/20180831075249/http://www.kurzweilai.net/how-bio-inspired-deep-learning-keeps-winning-competitions |date=31 August 2018 }} with Juergen Schmidhuber on the eight competitions won by his Deep Learning team 2009–2012</ref><ref>{{Cite web|url=http://www.kurzweilai.net/how-bio-inspired-deep-learning-keeps-winning-competitions|title=How bio-inspired deep learning keeps winning competitions {{!}} KurzweilAI|website=kurzweilai.net|access-date=16 June 2017|archive-url=https://web.archive.org/web/20180831075249/http://www.kurzweilai.net/how-bio-inspired-deep-learning-keeps-winning-competitions|archive-date=31 August 2018}}</ref> In 2011, a CNN named ''DanNet<ref name=":32">{{Cite journal |last1=Cireşan |first1=Dan Claudiu |last2=Meier |first2=Ueli |last3=Gambardella |first3=Luca Maria |last4=Schmidhuber |first4=Jürgen |date=21 September 2010 |title=Deep, Big, Simple Neural Nets for Handwritten Digit Recognition |journal=Neural Computation |volume=22 |issue=12 |pages=3207–3220 |arxiv=1003.0358 |doi=10.1162/neco_a_00052 |issn=0899-7667 |pmid=20858131 |s2cid=1918673}}</ref>''<ref name=":62">{{Cite journal |last1=Ciresan |first1=D. C. |last2=Meier |first2=U. |last3=Masci |first3=J. |last4=Gambardella |first4=L.M. |last5=Schmidhuber |first5=J. |date=2011 |title=Flexible, High Performance Convolutional Neural Networks for Image Classification |url=http://ijcai.org/papers11/Papers/IJCAI11-210.pdf |url-status=live |journal=International Joint Conference on Artificial Intelligence |doi=10.5591/978-1-57735-516-8/ijcai11-210 |archive-url=https://web.archive.org/web/20140929094040/http://ijcai.org/papers11/Papers/IJCAI11-210.pdf |archive-date=29 September 2014 |access-date=13 June 2017}}</ref> by Dan Ciresan, Ueli Meier, Jonathan Masci, [[Luca Maria Gambardella]], and Jürgen Schmidhuber achieved for the first time superhuman performance in a visual pattern recognition contest, outperforming traditional methods by a factor of 3.<ref name="SCHIDHUB4">{{cite journal |last=Schmidhuber |first=J. |year=2015 |title=Deep Learning in Neural Networks: An Overview |journal=Neural Networks |volume=61 |pages=85–117 |arxiv=1404.7828 |doi=10.1016/j.neunet.2014.09.003 |pmid=25462637 |s2cid=11715509}}</ref> It then won more contests.<ref name=":82">{{Cite book |last1=Ciresan |first1=Dan |url=http://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images.pdf |title=Advances in Neural Information Processing Systems 25 |last2=Giusti |first2=Alessandro |last3=Gambardella |first3=Luca M. |last4=Schmidhuber |first4=Jürgen |date=2012 |publisher=Curran Associates, Inc. |editor-last=Pereira |editor-first=F. |pages=2843–2851 |access-date=13 June 2017 |editor-last2=Burges |editor-first2=C. J. C. |editor-last3=Bottou |editor-first3=L. |editor-last4=Weinberger |editor-first4=K. Q. |archive-url=https://web.archive.org/web/20170809081713/http://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images.pdf |archive-date=9 August 2017 |url-status=live}}</ref><ref name="ciresan2013miccai">{{Cite book |last1=Ciresan |first1=D. |title=Medical Image Computing and Computer-Assisted Intervention – MICCAI 2013 |last2=Giusti |first2=A. |last3=Gambardella |first3=L.M. |last4=Schmidhuber |first4=J. |date=2013 |isbn=978-3-642-38708-1 |series=Lecture Notes in Computer Science |volume=7908 |pages=411–418 |chapter=Mitosis Detection in Breast Cancer Histology Images with Deep Neural Networks |doi=10.1007/978-3-642-40763-5_51 |pmid=24579167 |issue=Pt 2}}</ref> They also showed how [[Max pooling|max-pooling]] CNNs on GPU improved performance significantly.<ref name=":9">{{Cite book |last1=Ciresan |first1=D. |title=2012 IEEE Conference on Computer Vision and Pattern Recognition |last2=Meier |first2=U. |last3=Schmidhuber |first3=J. |year=2012 |isbn=978-1-4673-1228-8 |pages=3642–3649 |chapter=Multi-column deep neural networks for image classification |doi=10.1109/cvpr.2012.6248110 |arxiv=1202.2745 |s2cid=2161592}}</ref>

In October 2012, [[AlexNet]] by [[Alex Krizhevsky]], [[Ilya Sutskever]], and Geoffrey Hinton<ref name="krizhevsky20122">{{cite journal |last1=Krizhevsky |first1=Alex |last2=Sutskever |first2=Ilya |last3=Hinton |first3=Geoffrey |date=2012 |title=ImageNet Classification with Deep Convolutional Neural Networks |url=https://www.cs.toronto.edu/~kriz/imagenet_classification_with_deep_convolutional.pdf |url-status=live |journal=NIPS 2012: Neural Information Processing Systems, Lake Tahoe, Nevada |archive-url=https://web.archive.org/web/20170110123024/http://www.cs.toronto.edu/~kriz/imagenet_classification_with_deep_convolutional.pdf |archive-date=10 January 2017 |access-date=24 May 2017}}</ref> won the large-scale [[ImageNet competition]] by a significant margin over shallow machine learning methods. Further incremental improvements included the VGG-16 network by Karen Simonyan and [[Andrew Zisserman]]<ref name="VGG">{{cite arXiv |eprint=1409.1556 |class=cs.CV |first1=Karen |last1=Simonyan |first2=Zisserman |last2=Andrew |title=Very Deep Convolution Networks for Large Scale Image Recognition |year=2014}}</ref> and Google's [[Inceptionv3]].<ref name="szegedy">{{Cite journal |last=Szegedy |first=Christian |date=2015 |title=Going deeper with convolutions |url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43022.pdf |journal=Cvpr2015 |arxiv=1409.4842 |archive-date=30 September 2024 |access-date=7 August 2024 |archive-url=https://web.archive.org/web/20240930225513/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43022.pdf |url-status=live }}</ref>

In 2012, [[Andrew Ng|Ng]] and [[Jeff Dean (computer scientist)|Dean]] created a network that learned to recognize higher-level concepts, such as cats, only from watching unlabeled images.<ref name="ng2012">{{cite arXiv |eprint=1112.6209 |class=cs.LG |first1=Andrew |last1=Ng |first2=Jeff |last2=Dean |title=Building High-level Features Using Large Scale Unsupervised Learning |year=2012}}</ref> Unsupervised pre-training and increased computing power from [[GPU]]s and [[distributed computing]] allowed the use of larger networks, particularly in image and visual recognition problems, which became known as "deep learning".<ref name=":4" />

[[Radial basis function network|Radial basis function]] and wavelet networks were introduced in 2013. These can be shown to offer best approximation properties and have been applied in [[nonlinear system identification]] and classification applications.<ref name="SAB1" />

[[Generative adversarial network]] (GAN) ([[Ian Goodfellow]] et al., 2014)<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 |archive-url=https://web.archive.org/web/20191122034612/http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf |archive-date=22 November 2019 |access-date=20 August 2019 |url-status=live}}</ref> became state of the art in generative modeling during 2014–2018 period. The GAN principle was originally published in 1991 by Jürgen Schmidhuber who called it "artificial curiosity": two neural networks contest with each other in the form of a [[zero-sum game]], where one network's gain is the other network's loss.<ref name="curiosity1991">{{cite conference| title = A possibility for implementing curiosity and boredom in model-building neural controllers | last1 = Schmidhuber | first1 = Jürgen | author-link = Jürgen Schmidhuber | date = 1991 | publisher = MIT Press/Bradford Books| book-title = Proc. SAB'1991| pages = 222–227}}</ref><ref name="gancurpm2020">{{Cite journal|last=Schmidhuber|first=Jürgen| author-link = Jürgen Schmidhuber |date=2020|title=Generative Adversarial Networks are Special Cases of Artificial Curiosity (1990) and also Closely Related to Predictability Minimization (1991)|journal=Neural Networks |language=en|volume=127|pages=58–66|doi=10.1016/j.neunet.2020.04.008 |pmid=32334341 |arxiv=1906.04493 |s2cid=216056336 }}</ref> The first network is a [[generative model]] that models a [[probability distribution]] over output patterns. The second network learns by [[gradient descent]] to predict the reactions of the environment to these patterns. Excellent image quality is achieved by [[Nvidia]]'s [[StyleGAN]] (2018)<ref name="SyncedReview201822">{{Cite web |date=14 December 2018 |title=GAN 2.0: NVIDIA's Hyperrealistic Face Generator |url=https://syncedreview.com/2018/12/14/gan-2-0-nvidias-hyperrealistic-face-generator/ |access-date=3 October 2019 |website=SyncedReview.com |archive-date=12 September 2024 |archive-url=https://web.archive.org/web/20240912080503/https://syncedreview.com/2018/12/14/gan-2-0-nvidias-hyperrealistic-face-generator/ |url-status=live }}</ref> based on the Progressive GAN by Tero Karras et al.<ref name="progressiveGAN201722">{{cite arXiv |eprint=1710.10196 |class=cs.NE |first1=T. |last1=Karras |first2=T. |last2=Aila |title=Progressive Growing of GANs for Improved Quality, Stability, and Variation |date=26 February 2018 |last3=Laine |first3=S. |last4=Lehtinen |first4=J.}}</ref> Here, the GAN generator is grown from small to large scale in a pyramidal fashion. Image generation by GAN reached popular success, and provoked discussions concerning [[Deepfake|deepfakes]].<ref>{{Cite web |title=Prepare, Don't Panic: Synthetic Media and Deepfakes |url=https://lab.witness.org/projects/synthetic-media-and-deep-fakes/ |url-status=live |archive-url=https://web.archive.org/web/20201202231744/https://lab.witness.org/projects/synthetic-media-and-deep-fakes/ |archive-date=2 December 2020 |access-date=25 November 2020 |publisher=witness.org}}</ref> [[Diffusion model|Diffusion models]] (2015)<ref>{{Cite journal |last1=Sohl-Dickstein |first1=Jascha |last2=Weiss |first2=Eric |last3=Maheswaranathan |first3=Niru |last4=Ganguli |first4=Surya |date=1 June 2015 |title=Deep Unsupervised Learning using Nonequilibrium Thermodynamics |url=http://proceedings.mlr.press/v37/sohl-dickstein15.pdf |journal=Proceedings of the 32nd International Conference on Machine Learning |language=en |publisher=PMLR |volume=37 |pages=2256–2265 |arxiv=1503.03585 |archive-date=21 September 2024 |access-date=7 August 2024 |archive-url=https://web.archive.org/web/20240921065319/http://proceedings.mlr.press/v37/sohl-dickstein15.pdf |url-status=live }}</ref> eclipsed GANs in generative modeling since then, with systems such as [[DALL·E 2]] (2022) and [[Stable Diffusion]] (2022).

In 2014, the state of the art was training "very deep neural network" with 20 to 30 layers.<ref>{{Citation |last1=Simonyan |first1=Karen |title=Very Deep Convolutional Networks for Large-Scale Image Recognition |date=10 April 2015 |arxiv=1409.1556 |last2=Zisserman |first2=Andrew}}</ref> Stacking too many layers led to a steep reduction in [[Training, validation, and test data sets|training]] accuracy,<ref name="prelu2">{{cite arXiv |eprint=1502.01852 |class=cs.CV |first1=Kaiming |last1=He |first2=Xiangyu |last2=Zhang |title=Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification |last3=Ren |first3=Shaoqing |last4=Sun |first4=Jian |year=2016}}</ref> known as the "degradation" problem.<ref name="resnet2">{{Cite conference |last1=He |first1=Kaiming |last2=Zhang |first2=Xiangyu |last3=Ren |first3=Shaoqing |last4=Sun |first4=Jian |date=10 December 2015 |title=Deep Residual Learning for Image Recognition |arxiv=1512.03385}}</ref> In 2015, two techniques were developed to train very deep networks: the [[highway network]] was published in May 2015,<ref name="highway20153">{{cite arXiv |eprint=1505.00387 |class=cs.LG |first1=Rupesh Kumar |last1=Srivastava |first2=Klaus |last2=Greff |title=Highway Networks |date=2 May 2015 |last3=Schmidhuber |first3=Jürgen}}</ref> and the residual neural network (ResNet) in December 2015.<ref name="resnet20153">{{Cite book |last1=He |first1=Kaiming |last2=Zhang |first2=Xiangyu |last3=Ren |first3=Shaoqing |last4=Sun |first4=Jian |title=2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) |chapter=Deep Residual Learning for Image Recognition |date=2016 |publisher=IEEE |pages=770–778 |arxiv=1512.03385 |doi=10.1109/CVPR.2016.90 |isbn=978-1-4673-8851-1  }}</ref><ref>{{Cite web |last=Linn |first=Allison |date=10 December 2015 |title=Microsoft researchers win ImageNet computer vision challenge |url=https://blogs.microsoft.com/ai/microsoft-researchers-win-imagenet-computer-vision-challenge/ |access-date=29 June 2024 |website=The AI Blog |language=en-US |archive-date=21 May 2023 |archive-url=https://archive.today/20230521191955/https://blogs.microsoft.com/ai/microsoft-researchers-win-imagenet-computer-vision-challenge/ |url-status=live }}</ref> ResNet behaves like an open-gated Highway Net. 

{{Main|Transformer (deep learning architecture)#History}}

During the 2010s, the [[seq2seq]] model was developed, and attention mechanisms were added. It led to the modern Transformer architecture in 2017 in ''[[Attention Is All You Need]]''.<ref name="vaswani2017">{{cite arXiv |eprint=1706.03762 |class=cs.CL |first1=Ashish |last1=Vaswani |first2=Noam |last2=Shazeer |title=Attention Is All You Need |date=12 June 2017 |last8=Polosukhin |first8=Illia |last7=Kaiser |first7=Lukasz |last6=Gomez |first6=Aidan N. |last5=Jones |first5=Llion |last4=Uszkoreit |first4=Jakob |last3=Parmar |first3=Niki}}</ref>
It requires computation time that is quadratic in the size of the context window. Jürgen Schmidhuber's fast weight controller (1992)<ref name="transform19922">{{cite journal |last1=Schmidhuber |first1=Jürgen |author-link1=Jürgen Schmidhuber |date=1992 |title=Learning to control fast-weight memories: an alternative to recurrent nets. |url=https://archive.org/download/wikipedia-scholarly-sources-corpus/10.1162.zip/10.1162%252Fneco.1992.4.1.131.pdf |journal=Neural Computation |volume=4 |issue=1 |pages=131–139 |doi=10.1162/neco.1992.4.1.131 |s2cid=16683347}}</ref> scales linearly and was later shown to be equivalent to the unnormalized linear Transformer.<ref name="fastlinear20202">{{cite conference |last1=Katharopoulos |first1=Angelos |last2=Vyas |first2=Apoorv |last3=Pappas |first3=Nikolaos |last4=Fleuret |first4=François |date=2020 |title=Transformers are RNNs: Fast autoregressive Transformers with linear attention |url=https://paperswithcode.com/paper/a-decomposable-attention-model-for-natural |publisher=PMLR |pages=5156–5165 |book-title=ICML 2020 |access-date=21 September 2024 |archive-date=11 July 2023 |archive-url=https://web.archive.org/web/20230711021546/https://paperswithcode.com/paper/a-decomposable-attention-model-for-natural |url-status=live }}</ref><ref name="schlag20212">{{cite conference |last1=Schlag |first1=Imanol |last2=Irie |first2=Kazuki |last3=Schmidhuber |first3=Jürgen |author-link3=Juergen Schmidhuber |date=2021 |title=Linear Transformers Are Secretly Fast Weight Programmers |publisher=Springer |pages=9355–9366 |book-title=ICML 2021}}</ref><ref name="DLhistory" />
Transformers have increasingly become the model of choice for [[natural language processing]].<ref name="wolf2020">{{cite book |last1=Wolf |first1=Thomas |title=Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations |last2=Debut |first2=Lysandre |last3=Sanh |first3=Victor |last4=Chaumond |first4=Julien |last5=Delangue |first5=Clement |last6=Moi |first6=Anthony |last7=Cistac |first7=Pierric |last8=Rault |first8=Tim |last9=Louf |first9=Remi |year=2020 |pages=38–45 |chapter=Transformers: State-of-the-Art Natural Language Processing |doi=10.18653/v1/2020.emnlp-demos.6 |last10=Funtowicz |first10=Morgan |last11=Davison |first11=Joe |last12=Shleifer |first12=Sam |last13=von Platen |first13=Patrick |last14=Ma |first14=Clara |last15=Jernite |first15=Yacine |last16=Plu |first16=Julien |last17=Xu |first17=Canwen |last18=Le Scao |first18=Teven |last19=Gugger |first19=Sylvain |last20=Drame |first20=Mariama |last21=Lhoest |first21=Quentin |last22=Rush |first22=Alexander |s2cid=208117506}}</ref> Many modern [[large language model]]s such as [[ChatGPT]], [[GPT-4]], and [[BERT (language model)|BERT]] use this architecture.

==Models==
{{Confusing|section|date=April 2017}}{{Further|Mathematics of artificial neural networks}}[[File:Neuron3.png|thumb|right|upright=1.35|Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals]]
ANNs began as an attempt to exploit the architecture of the human brain to perform tasks that conventional algorithms had little success with. They soon reoriented towards improving empirical results, abandoning attempts to remain true to their biological precursors. ANNs have the ability to learn and model non-linearities and complex relationships. This is achieved by neurons being connected in various patterns, allowing the output of some neurons to become the input of others. The network forms a [[Directed graph|directed]], [[weighted graph]].<ref name="Zell1994ch5.2">{{Cite book|title=Simulation neuronaler Netze|last=Zell |first=Andreas|date=2003|publisher=Addison-Wesley|isbn=978-3-89319-554-1|oclc=249017987|trans-title=Simulation of Neural Networks |language=de |edition=1st |chapter=chapter 5.2 }}</ref>

An artificial neural network consists of simulated neurons. Each neuron is connected to other [[Vertex (graph theory)|nodes]] via [[Glossary of graph theory terms#edge|links]] like a biological axon-synapse-dendrite connection. All the nodes connected by links take in some data and use it to perform specific operations and tasks on the data. Each link has a weight, determining the strength of one node's influence on another,<ref name='Winston'>{{cite book |title=Artificial intelligence |publisher=Addison-Wesley Pub. Co |isbn=0-201-53377-4 |edition=3rd|year=1992 }}</ref> allowing weights to choose the signal between neurons.

===Artificial neurons ===
{{main|Artificial neuron}}
ANNs are composed of [[artificial neurons]] which are conceptually derived from biological [[neuron]]s. Each artificial neuron has inputs and produces a single output which can be sent to multiple other neurons.<ref name="Abbod2007">{{cite journal|last1=Abbod|first1=Maysam F.|year=2007|title=Application of Artificial Intelligence to the Management of Urological Cancer|journal=The Journal of Urology|volume=178|issue=4|pages=1150–1156|doi=10.1016/j.juro.2007.05.122|pmid=17698099}}</ref> The inputs can be the feature values of a sample of external data, such as images or documents, or they can be the outputs of other neurons. The outputs of the final ''output neurons'' of the neural net accomplish the task, such as recognizing an object in an image.{{citation needed|date=October 2024}}

To find the output of the neuron we take the weighted sum of all the inputs, weighted by the ''weights'' of the ''connections'' from the inputs to the neuron. We add a ''bias'' term to this sum.<ref name="DAWSON1998">{{cite journal|last1=Dawson|first1=Christian W.|year=1998|title=An artificial neural network approach to rainfall-runoff modelling|journal=Hydrological Sciences Journal|volume=43|issue=1|pages=47–66|doi=10.1080/02626669809492102|bibcode=1998HydSJ..43...47D |doi-access=free}}</ref> This weighted sum is sometimes called the ''activation''. This weighted sum is then passed through a (usually nonlinear) activation function to produce the output. The initial inputs are external data, such as images and documents. The ultimate outputs accomplish the task, such as recognizing an object in an image.<ref>{{Cite web|url=http://www.cse.unsw.edu.au/~billw/mldict.html#activnfn|title=The Machine Learning Dictionary|website=cse.unsw.edu.au|access-date=4 November 2009|archive-url=https://web.archive.org/web/20180826151959/http://www.cse.unsw.edu.au/~billw/mldict.html#activnfn|archive-date=26 August 2018}}</ref>

=== Organization ===
The neurons are typically organized into multiple layers, especially in deep learning. Neurons of one layer connect only to neurons of the immediately preceding and immediately following layers. The layer that receives external data is the ''input layer''. The layer that produces the ultimate result is the ''output layer''. In between them are zero or more ''hidden layers''. Single layer and unlayered networks are also used. Between two layers, multiple connection patterns are possible. They can be 'fully connected', with every neuron in one layer connecting to every neuron in the next layer. They can be ''pooling'', where a group of neurons in one layer connects to a single neuron in the next layer, thereby reducing the number of neurons in that layer.<ref name="flexible">{{cite journal|last=Ciresan|first=Dan|author2=Ueli Meier|author3=Jonathan Masci|author4=Luca M. Gambardella|author5=Jurgen Schmidhuber|year=2011|title=Flexible, High Performance Convolutional Neural Networks for Image Classification|url=https://people.idsia.ch/~juergen/ijcai2011.pdf|journal=Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence-Volume Volume Two|volume=2|pages=1237–1242|access-date=7 July 2022|url-status=live|archive-url=https://web.archive.org/web/20220405190128/https://people.idsia.ch/~juergen/ijcai2011.pdf|archive-date=5 April 2022}}</ref> Neurons with only such connections form a [[directed acyclic graph]] and are known as [[feedforward neural network|''feedforward networks'']].<ref name="Zell1994p73">{{cite book|title=Simulation Neuronaler Netze|last=Zell|first=Andreas|publisher=Addison-Wesley|year=1994|isbn=3-89319-554-8|edition=1st|page=73|language=de|trans-title=Simulation of Neural Networks}}</ref> Alternatively, networks that allow connections between neurons in the same or previous layers are known as [[Recurrent neural network|''recurrent networks'']].<ref>{{Cite journal|last=Miljanovic|first=Milos|date=February–March 2012|title=Comparative analysis of Recurrent and Finite Impulse Response Neural Networks in Time Series Prediction|url=http://www.ijcse.com/docs/INDJCSE12-03-01-028.pdf|journal=Indian Journal of Computer and Engineering|volume=3|issue=1|access-date=21 August 2019|archive-date=19 May 2024|archive-url=https://web.archive.org/web/20240519081156/http://www.ijcse.com/docs/INDJCSE12-03-01-028.pdf|url-status=live}}</ref>

=== Hyperparameter ===
{{Main|Hyperparameter (machine learning)}}
A [[hyperparameter (machine learning)|hyperparameter]] is a constant [[parameter]] defining any configurable part of the learning process, whose value is set prior to training.<ref>{{cite web |title=What Is Hyperparameter Tuning? IBM |url=https://www.ibm.com/think/topics/hyperparameter-tuning |website=www.ibm.com |language=en |date=23 July 2024}}</ref> Examples of hyperparameters include [[learning rate]], batch size and regularization parameters.<ref>{{cite book |last1=Goodfellow |first1=Ian |last2=Bengio |first2=Yoshua |last3=Courville |first3=Aaron |title=Deep learning |date=2016 |publisher=The MIT press |location=Cambridge, Mass |isbn=9780262035613 |pages=271–274 |url=http://www.deeplearningbook.org |access-date=26 October 2025 |chapter=8}}</ref>. The performance of a neural network is strongly influenced by the choice of hyperparameter values, and thus the hyperparameters are often optimized as part of the training process, a process called hyperparameter tuning or hyperparameter optimization.<ref>{{cite journal |last1=Bischl |first1=Bernd |last2=Binder |first2=Martin |last3=Lang |first3=Michel |last4=Pielok |first4=Tobias |last5=Richter |first5=Jakob |last6=Coors |first6=Stefan |last7=Thomas |first7=Janek |last8=Ullmann |first8=Theresa |last9=Becker |first9=Marc |last10=Boulesteix |first10=Anne-Laure |last11=Deng |first11=Difan |last12=Lindauer |first12=Marius |title=Hyperparameter optimization: Foundations, algorithms, best practices, and open challenges |journal=WIREs Data Mining and Knowledge Discovery |date=March 2023 |volume=13 |issue=2 |article-number=e1484 |doi=10.1002/widm.1484 }}</ref>

===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 [[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 [[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 [[Mathematical optimization|optimization]] theory and [[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 |publisher=The MIT Press |chapter=7-8|oclc=1162184998}}</ref>

==== Learning rate ====
{{main|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=26 April 2019|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 [[Quickprop]] are primarily aimed at speeding up error minimization, while other improvements mainly try to increase reliability. In order to avoid [[oscillation]] inside the network such as alternating connection weights, and to improve the rate of convergence, refinements use an [[adaptive learning rate]] that increases or decreases as appropriate.<ref>{{Cite book|last1=Li|first1=Y.|last2=Fu|first2=Y.|last3=Li|first3=H.|last4=Zhang|first4=S. W.|title=2009 International Conference on Computational Intelligence and Natural Computing |chapter=The Improved Training Algorithm of Back Propagation Neural Network with Self-adaptive Learning Rate |s2cid=10557754|date=1 June 2009|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.{{citation needed|date=October 2024}}

====Cost function====
While it is possible to define a cost function [[ad hoc]], frequently the choice is determined by the function's desirable properties (such as [[Convex function|convexity]]) because it arises from the model (e.g. in a probabilistic model, the model's [[posterior probability]] can be used as an inverse cost).{{citation needed|date=October 2024}}

====Backpropagation====
{{Main|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, backpropagation calculates the [[gradient]] (the derivative) of the [[loss function|cost function]] associated with a given state with respect to the weights. The weight updates can be done via stochastic gradient descent or other methods, such as ''[[extreme learning machine]]s'',<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 [[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, [[supervised learning]],<ref>{{cite book |last1=Bernard |first1=Etienne |title=Introduction to machine learning |date=2021 |location=Champaign |publisher=Wolfram Media |isbn=978-1-57955-048-6 |page=9 |url=https://www.wolfram.com/language/introduction-machine-learning/machine-learning-paradigms/#p-9 |access-date=22 March 2023 |language=en |archive-date=19 May 2024 |archive-url=https://web.archive.org/web/20240519081126/https://www.wolfram.com/language/introduction-machine-learning/machine-learning-paradigms/#p-9 |url-status=live }}</ref> [[unsupervised learning]]<ref>{{cite book |last1=Bernard |first1=Etienne |title=Introduction to machine learning |date=2021 |location=Champaign |publisher=Wolfram Media |isbn=978-1-57955-048-6 |page=12 |url=https://www.wolfram.com/language/introduction-machine-learning/machine-learning-paradigms/#p-9 |access-date=22 March 2023 |language=en |archive-date=19 May 2024 |archive-url=https://web.archive.org/web/20240519081126/https://www.wolfram.com/language/introduction-machine-learning/machine-learning-paradigms/#p-9 |url-status=live }}</ref> and [[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=978-1-57955-048-6|page=9|last1=Bernard|access-date=28 July 2022|archive-date=19 May 2024|archive-url=https://web.archive.org/web/20240519081126/https://www.wolfram.com/language/introduction-machine-learning/|url-status=live}}</ref> Each corresponds to a particular learning task.

==== 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 [[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 [[pattern recognition]] (also known as classification) and [[Regression analysis|regression]] (also known as function approximation). Supervised learning is also applicable to sequential data (e.g., for handwriting, 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]], 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 ''[[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 [[Data compression|compression]] it could be related to the [[mutual information]] between <math>\textstyle x</math> and <math>\textstyle f(x)</math>, 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 [[Approximation|estimation]] problems; the applications include [[Data clustering|clustering]], the estimation of [[statistical distributions]], [[Data compression|compression]] and [[Bayesian spam filtering|filtering]].

====Reinforcement learning====
{{main|Reinforcement learning}}
{{See also|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 [[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 [[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 [[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 [[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, US | 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> [[Dynamic programming]] coupled with ANNs (giving [[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 [[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, [[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, US | doi = 10.1109/CEC.2000.870269 | isbn = 0-7803-6375-2 }}</ref> and [[medicine]]<ref>{{Cite book |last=Deng |first=Geng |author2=Ferris, M.C. |title=Optimization in Medicine |chapter=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 [[discretization]] grid density for numerically approximating the solution of control problems. Tasks that fall within the paradigm of reinforcement learning are control problems, [[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 receives initial emotions (only once) 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 = 637–669 | doi = 10.1080/01969720118145 | s2cid = 8944741 }}</ref>

==== Neuroevolution ====
{{Main|Neuroevolution}}

[[Neuroevolution]] can create neural network topologies and weights using [[evolutionary computation]]. It is competitive with sophisticated gradient descent approaches.<ref>{{cite arXiv |last1=Salimans |first1=Tim |title=Evolution Strategies as a Scalable Alternative to Reinforcement Learning |date=7 September 2017 |eprint=1703.03864 |last2=Ho |first2=Jonathan |last3=Chen |first3=Xi |last4=Sidor |first4=Szymon |last5=Sutskever |first5=Ilya|class=stat.ML }}</ref><ref>{{cite arXiv|last1=Such |first1=Felipe Petroski |title=Deep Neuroevolution: Genetic Algorithms Are a Competitive Alternative for Training Deep Neural Networks for Reinforcement Learning |date=20 April 2018 |eprint=1712.06567 |last2=Madhavan |first2=Vashisht |last3=Conti |first3=Edoardo |last4=Lehman |first4=Joel |last5=Stanley |first5=Kenneth O. |last6=Clune |first6=Jeff|class=cs.NE }}</ref> 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|
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 [[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 artificial neurons [[Stochastic process|stochastic]] transfer functions {{Citation needed|date=September 2024}}, or by giving them stochastic weights. This makes them useful tools for [[Optimization (mathematics)|optimization]] problems, since the random fluctuations help the network escape from [[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=978-1-58603-388-0}}</ref> Stochastic neural networks trained using a [[Bayes' theorem|Bayesian]] approach are known as '''Bayesian neural networks'''.<ref>{{Cite magazine |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 |magazine=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}}</ref>

===Topological deep learning===
[[Topological deep learning]], first introduced in 2017,<ref>{{Cite journal |last1=Cang |first1=Zixuan |last2=Wei |first2=Guo-Wei |date=27 July 2017 |title=TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions |journal=PLOS Computational Biology |language=en |volume=13 |issue=7 |article-number=e1005690 |doi=10.1371/journal.pcbi.1005690 |doi-access=free |issn=1553-7358 |pmc=5549771 |pmid=28749969|arxiv=1704.00063 |bibcode=2017PLSCB..13E5690C }}</ref> is an emerging approach in [[machine learning]] that integrates topology with deep neural networks to address highly intricate and high-order data. Initially rooted in [[algebraic topology]], TDL has since evolved into a versatile framework incorporating tools from other mathematical disciplines, such as [[differential topology]] and [[geometric topology]]. As a successful example of mathematical deep learning, TDL continues to inspire advancements in mathematical [[artificial intelligence]], fostering a mutually beneficial relationship between AI and [[mathematics]].  

===Other===
In a [[Bayesian probability|Bayesian]] framework, a distribution over the set of allowed models is chosen to minimize the cost. [[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> [[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> [[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> [[expectation–maximization algorithm|expectation–maximization]], [[non-parametric methods]] and [[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/ }}</ref> are other learning algorithms. Convergent recursion is a learning algorithm for [[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: 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 [[Types of artificial neural networks]] -->
{{Main|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 [[topology]]. Dynamic types allow one or more of these to evolve via learning. The latter is much more complicated but can shorten learning periods and produce better results. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers.

Some of the main breakthroughs include: 

* Convolutional neural networks that have proven particularly successful in processing visual and other two-dimensional data;<ref>{{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">[[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> where long short-term memory avoids the [[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}}</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> [[Speech synthesis|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">{{cite journal |last1=Schmidhuber |first1=Jürgen |author-link=Jürgen Schmidhuber |year=2015 |title=Deep Learning |journal=Scholarpedia |volume=10 |issue=11 |pages=85–117 |bibcode=2015SchpJ..1032832S |doi=10.4249/scholarpedia.32832 |doi-access=free}}</ref><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>Convolutional Neural Networks have also been applied to fraud detection<ref>{{Citation |last1=Qu |first1=Bo |title=2024 IEEE International Conference on Data Mining Workshops (ICDMW) |date=2024-11-29 |url=http://arxiv.org/abs/2411.19457 |access-date=2025-11-07 |arxiv=2411.19457 |last2=Wang |first2=Zhurong |last3=Gu |first3=Minghao |last4=Yagi |first4=Daisuke |last5=Zhao |first5=Yang |last6=Shan |first6=Yinan |last7=Zahradnik |first7=Frank |chapter=Multi-task CNN Behavioral Embedding Model for Transaction Fraud Detection |pages=286–292 |doi=10.1109/ICDMW65004.2024.00043 |isbn=979-8-3315-3063-1 }}</ref><ref>{{Citation |last1=Chagahi |first1=Mehdi Hosseini |title=Explainable AI for Fraud Detection: An Attention-Based Ensemble of CNNs, GNNs, and A Confidence-Driven Gating Mechanism |date=2025-02-22 |url=http://arxiv.org/abs/2410.09069 |access-date=2025-11-07 |arxiv=2410.09069 |last2=Delfan |first2=Niloufar |last3=Dashtaki |first3=Saeed Mohammadi |last4=Moshiri |first4=Behzad |last5=Piran |first5=Md Jalil}}</ref>. 
* Competitive networks such as [[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"/>

== Network design ==
Using artificial neural networks requires an understanding of their characteristics.
* Choice of model: This depends on the data representation and the application. Model parameters include the number, type, and connectedness of network layers, as well as the size of each and the connection type (full, pooling, etc.). Overly complex models learn slowly.
* [[Machine learning|Learning algorithm]]: Numerous trade-offs exist between learning algorithms. Almost any algorithm will work well with the correct hyperparameters<ref>{{cite journal |last1=Probst |first1=Philipp |last2=Boulesteix |first2=Anne-Laure |last3=Bischl |first3=Bernd |title=Tunability: Importance of Hyperparameters of Machine Learning Algorithms |journal=J. Mach. Learn. Res. |date=26 February 2018 |volume=20 |page=53:1–53:32 |s2cid=88515435 }}</ref> 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.

[[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 [[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> [[Scikit-learn|scikit-learn library]] provides functions to help with building a deep network from scratch. We can then implement a deep network with [[TensorFlow]] or [[Keras]].

Hyperparameters must also be defined as part of the design (they are not learned), governing matters such as how many neurons are in each layer, learning rate, step, stride, depth, receptive field and padding (for CNNs), etc.<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> {{citation needed span|date=July 2023|The [[Python (programming language)|Python]] code snippet provides an overview of the training function, which uses the training dataset, number of hidden layer units, learning rate, and number of iterations as parameters:}}
<syntaxhighlight lang="python3" line="1">
def train(X, y, n_hidden, learning_rate, n_iter):
    """Training function.

    Args:
      X: Argument X.
      y: Argument y.
      n_hidden: The number of hidden layer units.
      learning_rate: The learning rate.
      n_iter: The number of iterations.

    Returns:
      dict: A dictionary.
    """
    m, n_input = X.shape

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

    # 2. in each iteration, feed all layers with the latest weights and biases
    for i in range(n_iter + 1):
        z2 = np.dot(X, w1) + b1
        a2 = sigmoid(z2)
        z3 = np.dot(a2, w2) + b2
        a3 = z3
        dz3 = a3 - y
        dw2 = np.dot(a2.T, dz3)
        db2 = np.sum(dz3, axis=0, keepdims=True)
        dz2 = np.dot(dz3, w2.T) * sigmoid_derivative(z2)
        dw1 = np.dot(X.Y, dz2)
        db1 = np.sum(dz2, axis=0)

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

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

    model = {"w1": w1, "b1": b1, "w2": w2, "b2": b2}
    return model
</syntaxhighlight>


== Monitoring and concept drift detection of ANNs ==
When neural networks are deployed in real-world applications, the statistical properties of the input data may change over time, a phenomenon known as [[concept drift]] or [[stationary process|non-stationarity]]. Drift can reduce predictive accuracy and lead to unreliable or biased decisions if it is not detected and corrected. In practice, this means that the model's accuracy in deployment may differ substantially from the levels observed during training or cross-validation.  

Several strategies have been developed to monitor neural networks for drift and degradation:  

* '''Error-based monitoring''': comparing current predictions against ground-truth labels when they become available. This approach directly quantifies predictive performance but may be impractical when labels are delayed or costly to obtain.  
* '''Data distribution monitoring''': detecting changes in the input data distribution using statistical tests, divergence measures, or density-ratio estimation.  
* '''Representation monitoring''': tracking the distribution of internal ''embeddings'' or hidden-layer features. Shifts in the latent representation can indicate nonstationarity even when labels are unavailable. Statistical methods such as ''[[statistical process control]]'' charts have been adapted for this purpose.<ref>{{cite journal 
 |last1=Malinovskaya 
 |first1=Anna 
 |last2=Mozharovskyi 
 |first2=Pavlo 
 |last3=Otto 
 |first3=Philipp 
 |title=Statistical process monitoring of artificial neural networks 
 |journal=Technometrics 
 |date=January 2024 
 |volume=66 
 |issue=1 
 |pages=104–117 
 |doi=10.1080/00401706.2023.2239886 
 |doi-access=free}}</ref>  

== Applications ==
Because of their ability to model and reproduce nonlinear processes, artificial neural networks have found applications in many disciplines. These include:
* [[Function approximation]],<ref>{{cite book |last1=Esch |first1=Robin |title=Handbook of Applied Mathematics |chapter=Functional Approximation |date=1990 |publisher=Springer US |location=Boston, MA |isbn=978-1-4684-1423-3 |pages=928–987 |doi=10.1007/978-1-4684-1423-3_17 |edition=Springer US}}</ref> or [[regression analysis]],<ref>{{cite book |last1=Sarstedt |first1=Marko |last2=Moo |first2=Erik |title=A Concise Guide to Market Research |chapter=Regression Analysis |series=Springer Texts in Business and Economics |date=2019 |publisher=Springer Berlin Heidelberg |pages=209–256 |doi=10.1007/978-3-662-56707-4_7 |isbn=978-3-662-56706-7 |s2cid=240396965 |chapter-url=https://link.springer.com/chapter/10.1007/978-3-662-56707-4_7#Sec1 |access-date=20 March 2023 |archive-date=20 March 2023 |archive-url=https://web.archive.org/web/20230320212723/https://link.springer.com/chapter/10.1007/978-3-662-56707-4_7#Sec1 |url-status=live }}</ref> (including [[Time series#Prediction and forecasting|time series prediction]], [[fitness approximation]],<ref>{{cite book |last1=Tian |first1=Jie |last2=Tan |first2=Yin |last3=Sun |first3=Chaoli |last4=Zeng |first4=Jianchao |last5=Jin |first5=Yaochu |title=2016 IEEE Symposium Series on Computational Intelligence (SSCI) |chapter=A self-adaptive similarity-based fitness approximation for evolutionary optimization |date=December 2016 |pages=1–8 |doi=10.1109/SSCI.2016.7850209 |isbn=978-1-5090-4240-1 |s2cid=14948018 }}</ref> and modeling)
* [[Data processing]]<ref>{{cite book |last1=Alaloul |first1=Wesam Salah |last2=Qureshi |first2=Abdul Hannan |title=Dynamic Data Assimilation – Beating the Uncertainties |chapter=Data Processing Using Artificial Neural Networks |date=2019 |doi=10.5772/intechopen.91935 |isbn=978-1-83968-083-0 |s2cid=219735060 |chapter-url=https://www.intechopen.com/chapters/71673 |access-date=20 March 2023 |archive-date=20 March 2023 |archive-url=https://web.archive.org/web/20230320212722/https://www.intechopen.com/chapters/71673 |url-status=live }}</ref> (including filtering, clustering, [[blind source separation]],<ref>{{cite book |last1=Pal |first1=Madhab |last2=Roy |first2=Rajib |last3=Basu |first3=Joyanta |last4=Bepari |first4=Milton S. |title=2013 International Conference Oriental COCOSDA held jointly with 2013 Conference on Asian Spoken Language Research and Evaluation (O-COCOSDA/CASLRE) |chapter=Blind source separation: A review and analysis |date=2013 |publisher=IEEE |pages=1–5 |doi=10.1109/ICSDA.2013.6709849 |isbn=978-1-4799-2378-6 |s2cid=37566823 }}</ref> and compression)
* [[Nonlinear system identification]]<ref name="SAB1">{{cite book |last=Billings |first=S. A. |title=Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains |publisher=Wiley |year=2013 |isbn=978-1-119-94359-4 }}</ref> and control (including 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> [[adaptive control]], [[process control]], and [[natural resource management]])
* [[Pattern recognition]] (including radar systems, [[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> [[novelty detection]], [[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 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>)
* Sequence recognition (including [[Gesture recognition|gesture]], [[Speech recognition|speech]], and handwritten and printed text recognition<ref>{{Cite book|last1=Maitra|first1=Durjoy S.|last2=Bhattacharya|first2=Ujjwal|last3=Parui|first3=Swapan K.|title=2015 13th International Conference on Document Analysis and Recognition (ICDAR)|chapter=CNN based common approach to handwritten character recognition of multiple scripts|date=August 2015|pages=1021–1025|doi=10.1109/ICDAR.2015.7333916|isbn=978-1-4799-1805-8|s2cid=25739012}}</ref>)
* 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> (including [[image analysis]])
* [[Robotics]] (including directing manipulators and [[prosthesis|prostheses]])
* [[Data mining]] (including [[knowledge discovery in databases]])
* 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> (such as [[ex-ante]] models for specific financial long-run forecasts and [[artificial financial market]]s)
* [[Quantum chemistry]]<ref name="Balabin_2009">{{Cite journal|journal=[[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>
* [[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>
* [[Generative AI]]<ref>{{Cite news |last=Pasick |first=Adam |date=27 March 2023 |title=Artificial Intelligence Glossary: Neural Networks and Other Terms Explained |language=en-US |work=The New York Times |url=https://www.nytimes.com/article/ai-artificial-intelligence-glossary.html |access-date=22 April 2023 |issn=0362-4331 |archive-date=1 September 2023 |archive-url=https://web.archive.org/web/20230901183440/https://www.nytimes.com/article/ai-artificial-intelligence-glossary.html |url-status=live }}</ref>
* [[Data visualization]]
* [[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=[[The Wall Street Journal]]|access-date=16 June 2017|issn=0099-9660|archive-date=19 May 2024|archive-url=https://web.archive.org/web/20240519082135/https://www.wsj.com/articles/facebook-boosts-a-i-to-block-terrorist-propaganda-1497546000|url-status=live}}</ref>
* [[E-mail spam]] filtering
* [[Medical diagnosis]]<ref name=Ciaramella>{{cite book|last1=Ciaramella|first1=Alberto|author-link=Alberto Ciaramella|last2=Ciaramella|first2=Marco|title=Introduction to Artificial Intelligence: from data analysis to generative AI|date=2024|publisher=Intellisemantic Editions|isbn=978-8-8947-8760-3}}</ref>

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|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|access-date=28 March 2017|archive-date=19 May 2024|archive-url=https://web.archive.org/web/20240519082133/https://pubs.rsc.org/en/Content/ArticleLanding/2016/IB/C6IB00100A|url-status=live}}</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|bibcode=2018SoFou..58.1414D |issn=0038-0806|hdl=10045/81208|doi-access=free|hdl-access=free}}</ref> It can also be useful to mitigate flood by the use of ANNs for modelling rainfall-runoff.<ref>{{Cite journal |last1=Tayebiyan |first1=A. |last2=Mohammad |first2=T. A. |last3=Ghazali |first3=A. H. |last4=Mashohor |first4=S. |title=Artificial Neural Network for Modelling Rainfall-Runoff |url=http://www.pertanika.upm.edu.my/pjtas/browse/regular-issue?article=JST-0566-2015 |journal=Pertanika Journal of Science & Technology |volume=24 |issue=2 |pages=319–330 |access-date=17 May 2023 |archive-date=17 May 2023 |archive-url=https://web.archive.org/web/20230517014047/http://www.pertanika.upm.edu.my/pjtas/browse/regular-issue?article=JST-0566-2015 |url-status=live }}</ref> ANNs have also been used for building black-box models in [[geoscience]]: [[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 [[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 [[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 [[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 book|last1=Nix|first1=R.|last2=Zhang |first2=J.|title=2017 International Joint Conference on Neural Networks (IJCNN) |chapter=Classification of Android apps and malware using deep neural networks |date=May 2017 |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|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|volume=70 |editor2-last=Conti|editor2-first=Mauro|editor3-last=Dargahi|editor3-first=Tooska}}</ref> credit cards frauds<ref>{{Cite book |last1=Ghosh|last2=Reilly |title=Proceedings of the Twenty-Seventh Hawaii International Conference on System Sciences HICSS-94 |chapter=Credit card fraud detection with a neural-network |s2cid=13260377 |date=January 1994|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 [[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|archive-date=19 May 2024|archive-url=https://web.archive.org/web/20240519082138/https://www.quantamagazine.org/new-neural-networks-solve-hardest-equations-faster-than-ever-20210419/|url-status=live}}</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|archive-date=19 May 2024|archive-url=https://web.archive.org/web/20240519082138/https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/|url-status=live}}</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/|first1=Anthony|last1=Alford|access-date=24 August 2025|website=InfoQ|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 [[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=[[Physical Review Letters]] |volume=122 |issue=25 |article-number=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 |article-number=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|article-number=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|article-number=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 [[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 |article-number=27 |date=April 2015 |doi=10.1186/s12868-015-0162-6 |pmid=25928094 |pmc=4417229 |doi-access=free }}</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.

It is possible to create a profile of a user's interests from pictures, using artificial neural networks trained for object recognition.<ref>{{cite journal | url=https://www.researchgate.net/publication/328964756 | doi=10.3233/978-1-61499-894-5-179 | last1=Wieczorek | first1=Szymon | last2=Filipiak | first2=Dominik | last3=Filipowska | first3=Agata | title=Semantic Image-Based Profiling of Users' Interests with Neural Networks | journal=Studies on the Semantic Web | volume=36 | issue=Emerging Topics in Semantic Technologies | year=2018 | access-date=20 January 2024 | archive-date=19 May 2024 | archive-url=https://web.archive.org/web/20240519082144/https://www.researchgate.net/publication/328964756_Semantic_Image-Based_Profiling_of_Users%27_Interests_with_Neural_Networks | url-status=live }}</ref>

Beyond their traditional applications, artificial neural networks are increasingly being utilized in interdisciplinary research, such as materials science. For instance, graph neural networks (GNNs) have demonstrated their capability in scaling deep learning for the discovery of new stable materials by efficiently predicting the total energy of crystals. This application underscores the adaptability and potential of ANNs in tackling complex problems beyond the realms of predictive modeling and artificial intelligence, opening new pathways for scientific discovery and innovation.<ref>{{Cite journal |last1=Merchant |first1=Amil |last2=Batzner |first2=Simon |last3=Schoenholz |first3=Samuel S. |last4=Aykol |first4=Muratahan |last5=Cheon |first5=Gowoon |last6=Cubuk |first6=Ekin Dogus |date=December 2023 |title=Scaling deep learning for materials discovery |journal=Nature |language=en |volume=624 |issue=7990 |pages=80–85 |doi=10.1038/s41586-023-06735-9 |issn=1476-4687 |pmc=10700131 |pmid=38030720|bibcode=2023Natur.624...80M }}</ref>

==Theoretical properties==
===Computational power===
The [[multilayer perceptron]] is a [[UTM theorem|universal function]] approximator, as proven by the [[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 [[rational number|rational]]-valued weights (as opposed to full precision real number-valued weights) has the power of a [[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 | access-date = 10 January 2017 | archive-date = 19 May 2024 | archive-url = https://web.archive.org/web/20240519082138/http://www.math.rutgers.edu/~sontag/FTPDIR/aml-turing.pdf | url-status = live }}</ref> using a finite number of neurons and standard linear connections. Further, the use of [[Irrational number|irrational]] values for weights results in a machine with [[Hypercomputation|super-Turing]] power.<ref>{{cite news |title=Analog computer trumps Turing model |first=Sunny |last=Bains |date=3 November 1998 |work=EE Times |url=https://www.eetimes.com/analog-computer-trumps-turing-model/ |access-date=11 May 2023 |archive-date=11 May 2023 |archive-url=https://web.archive.org/web/20230511152308/https://www.eetimes.com/analog-computer-trumps-turing-model/ |url-status=live }}</ref><ref>{{cite journal |last1=Balcázar |first1=José |title=Computational Power of Neural Networks: A Kolmogorov Complexity Characterization |journal=IEEE Transactions on Information Theory|date=July 1997 |volume=43 |issue=4 |pages=1175–1183 |doi=10.1109/18.605580 |citeseerx=10.1.1.411.7782 }}</ref>{{Failed verification|date=May 2023}}

===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 [[David J. C. MacKay|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=[[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 M. Cover|Thomas Cover]].<ref>{{cite journal|last=Cover|first=Thomas|author-link=Thomas M. Cover|year=1965|publisher=[[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|bibcode=1965ITECm..14..326C |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 book| last=Gerald | first=Friedland| title=Proceedings of the 27th ACM International Conference on Multimedia| chapter=Reproducibility and Experimental Design for Machine Learning on Audio and Multimedia Data| author-link=Gerald Friedland|year=2019|publisher=[[Association for Computing Machinery|ACM]]| 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 electrical element. The information capacity captures the functions modelable by the network given any data as input. The second notion, is the [[VC dimension]]. VC Dimension uses the principles of [[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/}}</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 [[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 [[Convergent series|convergence]] behavior of [[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 fit 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 |title=Neural Information Processing |date=2019 |veditors=Gedeon T, Wong K, Lee M |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 [[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|title=Deep Frequency Principle Towards Understanding Why Deeper Learning is Faster |date=18 May 2021|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 [[Overfitting|over-training]]. This arises in convoluted or over-specified systems when the network capacity significantly exceeds the needed free parameters. 

Two approaches address over-training. The first is to use [[cross-validation (statistics)|cross-validation]] and similar techniques to check for the presence of over-training and to select hyperparameters to minimize the generalization error. The second is to use some form of ''[[regularization (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 [[mean squared error]] (MSE) cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the [[confidence interval]] of network output, assuming a [[normal distribution]]. A confidence analysis made this way is statistically valid as long as the output [[probability distribution]] stays the same and the network is not modified.

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

The softmax activation function is:

:<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 many training samples for real-world operation.<ref>{{cite journal |last1=Parisi |first1=German I. |last2=Kemker |first2=Ronald |last3=Part |first3=Jose L. |last4=Kanan |first4=Christopher |last5=Wermter |first5=Stefan |date=1 May 2019 |title=Continual lifelong learning with neural networks: A review |journal=Neural Networks |volume=113 |pages=54–71 |doi=10.1016/j.neunet.2019.01.012 |pmid=30780045 |issn=0893-6080|doi-access=free |arxiv=1802.07569 }}</ref>
Any learning machine needs sufficient representative examples in order to capture the underlying structure that allows it to generalize to new cases. 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 [[cerebellar model articulation controller|CMAC]].<ref name="Qin1"/>
Dean Pomerleau uses a neural network to train a robotic vehicle to drive on multiple types of roads (single lane, multi-lane, dirt, etc.), and a large amount of his research is devoted to extrapolating multiple training scenarios from a single training experience, and 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).<ref>Dean Pomerleau, "Knowledge-based Training of Artificial Neural Networks for Autonomous Robot Driving"</ref>

===Theory===

A central claim{{citation needed|date=January 2023}} of ANNs is that they embody new and powerful general principles for processing information. These principles are [[ill-defined]]. This allows simple statistical association (the basic function of artificial neural networks) to be described as learning or recognition. In 1997, [[Alexander Dewdney]], a former ''[[Scientific American]]'' columnist, commented that as a result, artificial neural networks have a 
{{quote|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|page=82}}</ref>}} One response to Dewdney is that neural networks have been successfully used to 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 [[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 }}</ref>
}}

Although it is true that analyzing what has been learned by an artificial neural network is difficult, it is much easier to do so than to analyze what has been learned by a biological neural network. Moreover, recent emphasis on the [[Explainable artificial intelligence|explainability]] of AI has contributed towards the development of methods, notably those based on [[Attention (machine learning)|attention]] mechanisms, for visualizing and explaining learned neural networks. Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering generic principles that allow a learning machine to be successful. For example, Bengio and LeCun (2007) wrote an article regarding local vs non-local learning, as well as shallow vs deep architecture.<ref>{{cite web|url=http://www.iro.umontreal.ca/~lisa/publications2/index.php/publications/show/4|title=Scaling Learning Algorithms towards {AI} – LISA – Publications – Aigaion 2.0|website=iro.umontreal.ca }}</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] {{Webarchive|url=https://web.archive.org/web/20240519082645/https://www.amazon.com/Natural-Artificial-Intelligence-Introduction-Computational/dp/0985875720 |date=19 May 2024 }}," 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 [[Graph (discrete mathematics)|graph]] of neurons, simulating even a simplified neuron on [[von Neumann architecture]] may consume vast amounts of [[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 [[Central processing unit|CPU]] power and time.{{citation needed|date=October 2024}}

Some argue 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 [[General-purpose computing on graphics processing units (software)|GPGPUs]] (on [[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="SCHIDHUB4"/> The use of accelerators such as [[Field-programmable gate array|FPGA]]s and GPUs can reduce training times from months to days.{{r|:0}}<ref>{{Cite web |title=The Bitter Lesson |url=http://www.incompleteideas.net/IncIdeas/BitterLesson.html |access-date=7 August 2024 |website=incompleteideas.net|first=Rich|last=Sutton|authorlink=Rich Sutton|date=March 13, 2019}}</ref>

[[Neuromorphic engineering]] or a [[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 [[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 [[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><ref>Sun and Bookman, 1990</ref>

=== Dataset bias ===
Neural networks are dependent on the quality of the data they are trained on, thus low quality data with imbalanced representativeness can lead to the model learning and perpetuating societal biases.<ref name=":010">{{Cite journal |last1=Norori |first1=Natalia |last2=Hu |first2=Qiyang |last3=Aellen |first3=Florence Marcelle |last4=Faraci |first4=Francesca Dalia |last5=Tzovara |first5=Athina |date=October 2021 |title=Addressing bias in big data and AI for health care: A call for open science |journal=Patterns |language=en |volume=2 |issue=10 |article-number=100347 |doi=10.1016/j.patter.2021.100347|doi-access=free |pmid=34693373 |pmc=8515002 }}</ref><ref name=":17">{{Cite journal |last=Carina |first=Wang |date=27 October 2022 |title=Failing at Face Value: The Effect of Biased Facial Recognition Technology on Racial Discrimination in Criminal Justice |journal=Scientific and Social Research |volume=4 |issue=10 |pages=29–40 |doi=10.26689/ssr.v4i10.4402 |issn=2661-4332|doi-access=free }}</ref> These inherited biases become especially critical when the ANNs are integrated into real-world scenarios where the training data may be imbalanced due to the scarcity of data for a specific race, gender or other attribute.<ref name=":010" /> This imbalance can result in the model having inadequate representation and understanding of underrepresented groups, leading to discriminatory outcomes that exacerbate societal inequalities, especially in applications like [[Facial recognition system|facial recognition]], hiring processes, and [[law enforcement]].<ref name=":17" /><ref name=":22">{{Cite journal |last=Chang |first=Xinyu |date=13 September 2023 |title=Gender Bias in Hiring: An Analysis of the Impact of Amazon's Recruiting Algorithm |url=https://aemps.ewapublishing.org/article.html?pk=e5b93601b03d453c855d54d3153875ba |journal=Advances in Economics, Management and Political Sciences |volume=23 |issue=1 |pages=134–140 |doi=10.54254/2754-1169/23/20230367 |issn=2754-1169 |doi-access=free |access-date=9 December 2023 |archive-date=9 December 2023 |archive-url=https://web.archive.org/web/20231209135207/https://aemps.ewapublishing.org/article.html?pk=e5b93601b03d453c855d54d3153875ba |url-status=live }}</ref> For example, in 2018, [[Amazon (company)|Amazon]] had to scrap a recruiting tool because the model favored men over women for jobs in software engineering due to the higher number of male workers in the field.<ref name=":22" /> The program would penalize any resume with the word "woman" or the name of any women's college. However, the use of [[synthetic data]] can help reduce dataset bias and increase representation in datasets.<ref>{{Cite book |last1=Kortylewski |first1=Adam |last2=Egger |first2=Bernhard |last3=Schneider |first3=Andreas |last4=Gerig |first4=Thomas |last5=Morel-Forster |first5=Andreas |last6=Vetter |first6=Thomas |chapter=Analyzing and Reducing the Damage of Dataset Bias to Face Recognition with Synthetic Data |date=June 2019 |title=2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) |pages=2261–2268 |publisher=IEEE |doi=10.1109/cvprw.2019.00279 |isbn=978-1-7281-2506-0 |s2cid=198183828 |url=https://edoc.unibas.ch/75257/1/20200128164027_5e3055eb775f1.pdf |access-date=30 December 2023 |archive-date=19 May 2024 |archive-url=https://web.archive.org/web/20240519082642/https://edoc.unibas.ch/75257/1/20200128164027_5e3055eb775f1.pdf |url-status=live }}</ref>

==Gallery==
<gallery widths="220">
File:Single layer ann.svg|A single-layer feedforward artificial neural network. Arrows originating from <math>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>y_q</math>, is calculated as <math>y_q = K*(\sum_i(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 nodes and 2 outputs. Given position state and direction, it 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 nodes and 2 outputs. Given position state, direction and other environment values, it outputs thruster based control values.
File:Cmac.jpg|Parallel pipeline structure of CMAC neural network. This learning algorithm can converge in one step. 
</gallery>

== Recent advancements and future directions ==
Artificial neural networks (ANNs) have undergone significant advancements, particularly in their ability to model complex systems, handle large data sets, and adapt to various types of applications. Their evolution over the past few decades has been marked by a broad range of applications in fields such as image processing, speech recognition, natural language processing, finance, and medicine.{{citation needed|date=October 2024}}

=== Image processing ===
In the realm of image processing, ANNs are employed in tasks such as image classification, object recognition, and image segmentation. For instance, deep convolutional neural networks (CNNs) have been important in handwritten digit recognition, achieving state-of-the-art performance.<ref name=":07">{{Cite journal |last=Huang |first=Yanbo |date=2009 |title=Advances in Artificial Neural Networks – Methodological Development and Application |journal=Algorithms |language=en |volume=2 |issue=3 |pages=973–1007 |doi=10.3390/algor2030973 |issn=1999-4893 |doi-access=free }}</ref> This demonstrates the ability of ANNs to effectively process and interpret complex visual information, leading to advancements in fields ranging from automated surveillance to medical imaging.<ref name=":07"/>

=== Speech recognition ===
By modeling speech signals, ANNs are used for tasks like speaker identification and speech-to-text conversion. Deep neural network architectures have introduced significant improvements in large vocabulary continuous speech recognition, outperforming traditional techniques.<ref name=":07"/><ref name=":15">{{Cite journal |last1=Kariri |first1=Elham |last2=Louati |first2=Hassen |last3=Louati |first3=Ali |last4=Masmoudi |first4=Fatma |date=2023 |title=Exploring the Advancements and Future Research Directions of Artificial Neural Networks: A Text Mining Approach |journal=Applied Sciences |language=en |volume=13 |issue=5 |page=3186 |doi=10.3390/app13053186 |issn=2076-3417 |doi-access=free }}</ref> These advancements have enabled the development of more accurate and efficient voice-activated systems, enhancing user interfaces in technology products.{{citation needed|date=October 2024}}

=== Natural language processing ===
In natural language processing, ANNs are used for tasks such as text classification, sentiment analysis, and machine translation. They have enabled the development of models that can accurately translate between languages, understand the context and sentiment in textual data, and categorize text based on content.<ref name=":07"/><ref name=":15"/> This has implications for automated customer service, content moderation, and language understanding technologies.<ref>{{Cite web |last=Staff |first=Coursera |date=2024-09-26 |title=What Is an Artificial Neural Network, and Why Does It Matter for AI? |url=https://www.coursera.org/articles/artificial-neural-network |access-date=2025-09-07 |website=Coursera |language=en}}</ref>

=== Control systems ===
In the domain of control systems, ANNs are used to model dynamic systems for tasks such as system identification, control design, and optimization. For instance, deep feedforward neural networks are important in system identification and control applications.<ref>{{Cite journal |last1=Pillonetto |first1=Gianluigi |last2=Aravkin |first2=Aleksandr |last3=Gedon |first3=Daniel |last4=Ljung |first4=Lennart |last5=Ribeiro |first5=Antônio H. |last6=Schön |first6=Thomas B. |date=2025-01-01 |title=Deep networks for system identification: A survey |url=https://www.sciencedirect.com/science/article/pii/S0005109824004011 |journal=Automatica |volume=171 |article-number=111907 |doi=10.1016/j.automatica.2024.111907 |issn=0005-1098}}</ref>

=== Finance ===
{{more|Applications of artificial intelligence#Trading and investment}}
ANNs are used for [[quantitative investing|stock market prediction]] and [[credit scoring]]: 
*In investing, ANNs can process vast amounts of financial data, recognize complex patterns, and forecast stock market trends, aiding investors and risk managers in making informed decisions.<ref name=":07"/> 
*In credit scoring, ANNs offer data-driven, personalized assessments of creditworthiness, improving the accuracy of default predictions and automating the lending process.<ref name=":15"/> 
ANNs require high-quality data and careful tuning, and their "black-box" nature can pose challenges in interpretation. Nevertheless, ongoing advancements suggest that ANNs continue to play a role in finance, offering valuable insights and enhancing [[financial risk management|risk management strategies]].{{citation needed|date=October 2024}}

=== Medicine ===
ANNs are able to process and analyze vast medical datasets. They enhance diagnostic accuracy, especially by interpreting complex [[medical imaging]] for early disease detection, and by predicting patient outcomes for personalized treatment planning.<ref name=":15"/> In [[drug discovery]], ANNs speed up the identification of potential drug candidates and predict their efficacy and safety, significantly reducing development time and costs.<ref name=":07"/> Additionally, their application in personalized medicine and healthcare data analysis allows tailored therapies and efficient patient care management.<ref name=":15" /> Ongoing research is aimed at addressing remaining challenges such as data privacy and model interpretability, as well as expanding the scope of ANN applications in medicine.{{citation needed|date=October 2024}}

=== Content creation ===
ANNs such as generative adversarial networks (GAN) and [[Transformer (machine learning model)|transformers]] are used for [[content creation]] across numerous industries.<ref name=":09">{{Cite journal |last1=Fui-Hoon Nah |first1=Fiona |last2=Zheng |first2=Ruilin |last3=Cai |first3=Jingyuan |last4=Siau |first4=Keng |last5=Chen |first5=Langtao |date=3 July 2023 |title=Generative AI and ChatGPT: Applications, challenges, and AI-human collaboration |journal=Journal of Information Technology Case and Application Research |language=en |volume=25 |issue=3 |pages=277–304 |doi=10.1080/15228053.2023.2233814 |issn=1522-8053|doi-access=free }}</ref> This is because deep learning models are able to learn the style of an artist or musician from huge datasets and generate completely new artworks and music compositions. For instance, [[DALL-E]] is a deep neural network trained on 650 million pairs of images and texts across the internet that can create artworks based on text entered by the user.<ref>{{Cite web |title=DALL-E 2's Failures Are the Most Interesting Thing About It – IEEE Spectrum |url=https://spectrum.ieee.org/openai-dall-e-2 |access-date=9 December 2023 |website=[[IEEE]] |language=en |archive-date=15 July 2022 |archive-url=https://web.archive.org/web/20220715204154/https://spectrum.ieee.org/openai-dall-e-2 |url-status=live }}</ref> In the field of music, transformers are used to create original music for commercials and documentaries through companies such as [[AIVA]] and [[Jukedeck]].<ref>{{Cite journal |last=Briot |first=Jean-Pierre |date=January 2021 |title=From artificial neural networks to deep learning for music generation: history, concepts and trends |journal=Neural Computing and Applications |language=en |volume=33 |issue=1 |pages=39–65 |doi=10.1007/s00521-020-05399-0 |issn=0941-0643|doi-access=free }}</ref> In the marketing industry, generative models are used to create personalized advertisements for consumers.<ref name=":09" /> Additionally, major film companies are partnering with technology companies to analyze the financial success of a film, such as the partnership between Warner Bros and technology company Cinelytic established in 2020.<ref>{{Cite journal |last=Chow |first=Pei-Sze |date=6 July 2020 |title=Ghost in the (Hollywood) machine: Emergent applications of artificial intelligence in the film industry |journal=NECSUS_European Journal of Media Studies |doi=10.25969/MEDIAREP/14307 |issn=2213-0217}}</ref> Furthermore, neural networks have found uses in video game creation, where Non Player Characters (NPCs) can make decisions based on all the characters currently in the game.<ref>{{Cite book |last1=Yu |first1=Xinrui |last2=He |first2=Suoju |last3=Gao |first3=Yuan |last4=Yang |first4=Jiajian |last5=Sha |first5=Lingdao |last6=Zhang |first6=Yidan |last7=Ai |first7=Zhaobo |chapter=Dynamic difficulty adjustment of game AI for video game Dead-End |date=June 2010 |pages=583–587 |title=The 3rd International Conference on Information Sciences and Interaction Sciences |publisher=IEEE |doi=10.1109/icicis.2010.5534761|isbn=978-1-4244-7384-7 |s2cid=17555595 }}</ref>

== See also ==
{{cols|colwidth=18em}}
* [[ADALINE]]
* [[Autoencoder]]
* [[Bio-inspired computing]]
* [[Blue Brain Project]]
* [[Catastrophic interference]]
* [[Cognitive architecture]]
* [[Connectionist expert system]]
* [[Connectomics]]
* [[Deep image prior]]
* [[Digital morphogenesis]]
* [[Efficiently updatable neural network]]
* [[Evolutionary algorithm]]
* [[Family of curves]]
* [[Genetic algorithm]]
* [[Hyperdimensional computing]]
* [[In situ adaptive tabulation]]
* [[Large width limits of neural networks]]
* [[List of machine learning concepts]]
* [[Memristor]]
* [[Mind uploading]]
* [[Neural gas]]
* [[Neural network software]]
* [[Optical neural network]]
* [[Parallel distributed processing]]
* [[Philosophy of artificial intelligence]]
* [[Predictive analytics]]
* [[Quantum neural network]]
* [[Support vector machine]]
* [[Spiking neural network]]
* [[Stochastic parrot]]
* [[Tensor product network]]
* [[Topological deep learning]]
{{colend}}

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

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* {{Cite book |title=Cognitive science: integrative synchronization mechanisms in cognitive neuroarchitectures of the modern connectionism |last=Maurer |first=Harald |year=2021 |publisher=CRC Press |isbn=978-1-351-04352-6 |doi=10.1201/9781351043526 |s2cid=242963768}}
* {{cite book |url={{google books |plainurl=y |id=m12UR8QmLqoC}} |title=Pattern Recognition and Neural Networks |last=Ripley |first=Brian D. |author-link=Brian D. Ripley |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-71770-0}}
* {{cite journal |last1=Siegelmann |first1=H.T. |first2=Eduardo D. |last2=Sontag |s2cid=2456483 |year=1994 |title=Analog computation via neural networks |url=https://archive.org/details/sim_theoretical-computer-science_1994-09-12_131_2/page/330 |journal=Theoretical Computer Science |volume=131 |issue=2 |pages=331–360 |doi=10.1016/0304-3975(94)90178-3|doi-access=free }} 
* {{Cite book |title=Neural networks for statistical modeling |last1=Smith |first1=Murray |date=1993 |publisher=Van Nostrand Reinhold |isbn=978-0-442-01310-3 |oclc=27145760}}
* {{Cite book |title=Advanced methods in neural computing |last=Wasserman |first=Philip D. |year=1993 |publisher=Van Nostrand Reinhold |isbn=978-0-442-00461-3 |oclc=27429729}}
* {{cite book |last1=Wilson |first1=Halsey |title=Artificial intelligence |date=2018 |publisher=Grey House Publishing |isbn=978-1-68217-867-6}}
{{colend}}

==External links==
{{Spoken Wikipedia|En-Neural_network.ogg|date=27 November 2011}}
* [http://www.dkriesel.com/en/science/neural_networks 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.
*[http://www.msm.cam.ac.uk/phase-trans/abstracts/neural.review.html Review of Neural Networks in Materials Science] {{Webarchive|url=https://web.archive.org/web/20150607101310/http://www.msm.cam.ac.uk/phase-trans/abstracts/neural.review.html |date=7 June 2015 }}
*[https://web.archive.org/web/20090318133122/http://www.gc.ssr.upm.es/inves/neural/ann1/anntutorial.html Artificial Neural Networks Tutorial in three languages (Univ. Politécnica de Madrid)]
*[https://web.archive.org/web/20091216110504/http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/cs11/report.html Another introduction to ANN]
*[https://www.youtube.com/watch?v=AyzOUbkUf3M Next Generation of Neural Networks] {{Webarchive|url=https://web.archive.org/web/20110124234328/http://www.youtube.com/watch?v=AyzOUbkUf3M |date=24 January 2011 }} – Google Tech Talks
*[http://www.msm.cam.ac.uk/phase-trans/2009/performance.html Performance of Neural Networks]
*[http://www.msm.cam.ac.uk/phase-trans/2009/review_Bhadeshia_SADM.pdf Neural Networks and Information] {{Webarchive|url=https://web.archive.org/web/20090709153828/http://www.msm.cam.ac.uk/phase-trans/2009/review_Bhadeshia_SADM.pdf |date=9 July 2009 }}
*{{cite web |first=Grant |last=Sanderson |title=But what ''is'' a Neural Network? |work=[[3Blue1Brown]] |date=5 October 2017 |url=https://www.youtube.com/watch?v=aircAruvnKk&list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi | archive-url=https://ghostarchive.org/varchive/youtube/20211107/aircAruvnKk| archive-date=7 November 2021 | url-status=live|via=[[YouTube]] }}{{cbignore}}

* {{Cite journal | last1 = Hakim | first1 = MD Azizul | last2 = Alam | first2 = Mohammad Ifazul | title = Biologically inspired neural network layer with homeostatic regulation and adaptive repair mechanisms | journal = Scientific Reports | volume = 15 | issue = 1 | pages = 33903 | year = 2025 | publisher = Nature Publishing Group | doi = 10.1038/s41598-025-09114-8 | pmid = 41028030 | pmc = 12484884 | bibcode = 2025NatSR..1533903H | url = https://doi.org/10.1038/s41598-025-09114-8 | issn = 2045-2322}}




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