{{short description|Computational model used in machine learning}} {{about|the computational models used for artificial intelligence||Neural network (disambiguation)}} {{update|date=May 2026}} {{cs1 config|name-list-style=vanc|display-authors=6}} {{Use dmy dates|date=March 2023}} [[File:Colored_neural_network.svg|thumb|A neural network is an interconnected group of nodes, inspired by a simplification of neurons in a brain. Here, each blue/green circular node in the hidden and output layers represents an artificial neuron and each red circular node in the far left layer represents an input data value. An arrow represents a connection from the output of one neuron (or data node) to the input of another.]] {{Machine learning|Artificial neural network}} In machine learning, a '''neural network''' ('''NN''') or '''neural net''', 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 neurons'', which loosely model the neurons 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 synapses 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 as part of the training process.
Groups of neurons are aggregated into layers. Each layer performs a transformation on its inputs. Signals travel from the first layer (the ''input layer'') to the last layer (the ''output layer''), typically passing through multiple intermediate layers (''hidden layers''). A network is typically called a deep neural network if it has at least two hidden layers. Deep neural networks are capable of learning sophisticated hierarchical representations.{{Citation needed|date=March 2026}}<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 |url=https://www.microsoft.com/en-us/research/wp-content/uploads/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf}}</ref>
Training neural networks is a compute-intensive process, accelerated by the use of graphics processing units (GPUs), and large datasets.
Architectural innovations such as convolutional neural networks (CNNs) significantly improved performance in computer vision tasks, while recurrent neural networks (RNNs) enabled modeling of sequential data such as speech and time-series information. Transformer architectures introduced attention mechanisms that allow neural networks to model long-range dependencies in data and have the basis of large language models.<ref name=":63">{{cite journal |last1=Vaswani |first1=Ashish |last2=Shazeer |first2=Noam |last3=Parmar |first3=Niki |title=Attention Is All You Need |journal=Advances in Neural Information Processing Systems |volume=30 |year=2017 |url=https://proceedings.neurips.cc/paper_files/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf|arxiv=1706.03762}}</ref>{{AI-generated inline|date=May 2026|reason=The preceding text may have been generated with AI assistance and should be checked for accuracy and tone. See Talk:Neural network (machine learning)#AI-generated (May 2026)}}
Artificial neural networks are used for a myriad of tasks including chatbots, large-scale text, image, and video generation, and robotics. {{toclimit|3}} {{multiple image | direction = vertical | total_width = 300 | footer = | image1 = Simplified neural network training example.svg | alt1 = | 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 urchins, which are correlated with "nodes" that represent visual features. The starfish match with a ringed texture and a star outline, whereas most sea urchins match with a striped texture and oval shape. However, the instance of a ring textured sea urchin creates a weakly weighted association between them. | image2 = Simplified neural network example.svg | alt2 = | caption2 = Subsequent run of the network on an input image:<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}} {{One source|date=March 2026|talk=Possible one-source reliance in History section|section}}
=== Mathematical foundations === Deep neural networks are based on statistics developed 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 weights. The sum of the products of the weights and the inputs is calculated at each node. The mean squared errors between these calculated outputs and the given target values are minimized by adjusting to the weights. This technique is the method of least squares or linear regression. It was used to find a rough linear fit to a set of points by Legendre (1805) and Gauss (1795) for the prediction of planetary movement.<ref name="gauss17952">{{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 |bibcode=1981AnSta...945451S |doi-access=free }}</ref><ref name="DLhistory2">{{Citation |last=Schmidhuber |first=Juergen |title=Annotated History of Modern AI and Deep Learning |date=2025-12-29 |url=http://arxiv.org/abs/2212.11279 |access-date=2026-03-27 |arxiv=2212.11279 }}</ref><ref name="stigler2">{{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>
=== Perceptrons === Computers are based on John von Neumann's model. They excecute explicit lists of instructions with access to memory to record their changing state. Neural networks instead originated from efforts to model information processing in biological systems via connectionism. Unlike the von Neumann model, connectionist computing does not separate memory and processing.{{Citation needed|date=March 2026}}
Warren McCulloch and Walter Pitts<ref name="WM2">{{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 |bibcode=1943BMaB....5..115M |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> (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 one branch focused on biological processes and another focused on artificial intelligence. McCulloch and Pitts also developed mathematical models of artificial neurons capable of representing logical functions.<ref name="WM2" />
In the late 1940s, 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 neural plasticity that became known as Hebbian learning. It was used in many early neural network experiments, such as Rosenblatt's perceptron and the Hopfield network. Farley and 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 networks computational machines were created by 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 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 19752">{{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="Olazaran2">{{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="joseph19602">{{Cite thesis |last=Joseph |first=R. D. |title=Contributions to perceptron theory |date=1961 |degree=PhD |publisher=Cornell University |url=https://www.proquest.com/openview/f305deb1a2b1624b59a0321efb5b810d/1?pq-origsite=gscholar&cbl=18750&diss=y}}</ref>{{Verify source|date=March 2026}} mentioned an earlier perceptron-like device by B. G. Farley and W. A. Clark of the MIT Lincoln Laboratory;<ref name="DLhistory2" /> however, according to Joseph, "they dropped the subject."<ref name="joseph19602" />{{Verify source|date=March 2026}}
The first perceptrons did not have adaptive hidden units. However, Joseph (1960)<ref name="joseph19602" /> discussed multilayer that did. Rosenblatt (1962)<ref name="rosenblatt19622">{{cite book |last=Rosenblatt |first=Frank |author-link=Frank Rosenblatt |title=Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms |publisher=Spartan Books |year=1962|url=https://books.google.com/books?id=7FhRAAAAMAAJ}}</ref>{{rp|section 16}} cited and adopted these ideas, crediting work by H. D. Block and B. W. Knight. However, these early efforts did not lead to a working learning algorithm for hidden units, i.e., deep learning.{{Citation needed|date=March 2026}}
The perceptron raised public excitement in 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=":082">{{Cite book |author=Russel |author2=Norvig, Peter |url=https://people.engr.tamu.edu/guni/csce625/slides/AI.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 |first=Stuart}}</ref>
=== Historical foundations and the Dartmouth proposal === Artificial neural networks were identified as a promising direction for artificial intelligence research in the 1955 proposal for the Dartmouth Summer Research Project on Artificial Intelligence.<ref name=":72">{{cite web |title=A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence |last1=McCarthy |first1=John |last2=Minsky |first2=Marvin |last3=Rochester |first3=Nathaniel |last4=Shannon |first4=Claude |year=1955 |url=http://www-formal.stanford.edu/jmc/history/dartmouth/dartmouth.html |website=Stanford University |access-date=2026-03-15 }}</ref> Neural network models initially faced major limitations. Hardware constraints limited network size and training efficiency, while theoretical understanding of learning algorithms remained incomplete. Many models used single-layer perceptrons, which were restricted to solving linearly separable problems. These limitations were highlighted in the book ''Perceptrons'' by Marvin Minsky and Seymour Papert, which deflated interest during the late 1960s and 1970s.<ref>{{cite book |last1=Minsky |first1=Marvin |last2=Papert |first2=Seymour |title=Perceptrons: An Introduction to Computational Geometry |publisher=MIT Press |year=1969 |url=https://direct.mit.edu/books/monograph/3132/PerceptronsAn-Introduction-to-Computational|isbn=978-0-262-34393-0|doi=10.7551/mitpress/11301.001.0001}}</ref>{{AI-generated inline|date=May 2026|reason=The preceding text may have been generated with AI assistance and should be checked for accuracy and tone. See Talk:Neural network (machine learning)#AI-generated (May 2026)}}
=== 1960s and 1970s=== Fundamental research was conducted on NNs 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 Valentin Lapa in the Soviet Union (1965). They regarded it as a form of polynomial regression,<ref name="ivak19652">{{cite book|first1=Alexey G. |last1=Ivakhnenko |first2=Valentin Grigorʹevich |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> generalizing 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 |bibcode=1970Autom...6..207I |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="ivak19712">{{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> training layer by layer via regression analysis. Superfluous hidden units were pruned using a separate validation set. The activation functions of the nodes were Kolmogorov-Gabor polynomials, the first deep networks with multiplicative units or "gates".<ref name="DLhistory2" />
The first deep learning multilayer perceptron (MLP) trained by stochastic gradient descent<ref name="robbins19512">{{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="Amari19672">{{cite journal |last1=Amari |first1=Shun'ichi |author-link=Shun'ichi Amari|title=A theory of adaptive pattern classifiers|journal= IEEE Transactions |date=1967 |volume=EC |issue=16 |pages=279–307|url=https://people.idsia.ch/~juergen/amari1967.pdf}}</ref> In computer experiments conducted by Amari's student S. Saito, a five layer MLP with two modifiable layers learned internal representations to classify non-linearily separable pattern classes.<ref name="DLhistory2" /> Subsequent developments in hardware and hyperparameter tuning made end-to-end stochastic gradient descent the dominant technique for reducing loss (error).{{Citation needed|date=March 2026}}
In 1969, Kunihiko Fukushima introduced the ReLU (rectified linear unit) activation function.<ref name="DLhistory2" /><ref name="Fukushima19692">{{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="sonoda172">{{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> RelLU is the most common activation function.<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 after Minsky and Papert (1969),<ref name=":1322">{{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 perceptrons were incapable of processing the exclusive-or circuit. However, this insight was irrelevant for the deep networks of Ivakhnenko (1965) and Amari (1967).
In 1976, transfer learning was introduced.<ref>{{Cite journal |last=Bozinovski |first=Stevo |date=2020-09-15 |title=Reminder of the First Paper on Transfer Learning in Neural Networks, 1976 |url=https://www.informatica.si/index.php/informatica/article/view/2828 |journal=Informatica |volume=44 |issue=3 |doi=10.31449/inf.v44i3.2828 |issn=1854-3871}}</ref>
=== Backpropagation === Interest in neural networks revived during the 1980s because of the novel backpropagation algorithm, which allowed multi-layer neural networks to be trained efficiently by propagating error gradients backward (from output back to input) through network layers.<ref>{{cite journal |last1=Rumelhart |first1=David |last2=Hinton |first2=Geoffrey |last3=Williams |first3=Ronald |title=Learning representations by back-propagating errors |journal=Nature |year=1986 |volume=323 |issue=6088 |pages=533–536 |doi=10.1038/323533a0 |bibcode=1986Natur.323..533R }}</ref>{{AI-generated inline|date=March 2026}} Backpropagation is an efficient application of the chain rule derived by Gottfried Wilhelm Leibniz in 1673<ref name="leibniz167622">{{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 introduced in 1962 by Rosenblatt,<ref name="rosenblatt19622" /> but he did not describe how to implement this. Henry J. Kelley developed a precursor of backpropagation in 1960 in the context of control theory.<ref name="kelley196022">{{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="lin197032">{{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="lin197632">{{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="DLhistory2" /> G.M. Ostrovski et al. republished it in 1971.<ref name="ostrowski19712">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="backprop2">{{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> Paul Werbos applied backpropagation to neural networks in 1982<ref name="werbos19822">{{Citation |last=Werbos |first=Paul J. |title=Applications of advances in nonlinear sensitivity analysis |date=1982 |work=System Modeling and Optimization |volume=38 |pages=762–770 |editor-last=Drenick |editor-first=R. F. |series=Lecture Notes in Control and Information Sciences |url=https://werbos.com/Neural/SensitivityIFIPSeptember1981.pdf |access-date=2026-03-27 |archive-url=https://web.archive.org/web/20160414055503/http://werbos.com/Neural/SensitivityIFIPSeptember1981.pdf |archive-date=2016-04-14 |place=Berlin/Heidelberg |publisher=Springer-Verlag |language=en |doi=10.1007/bfb0006203 |isbn=978-3-540-11691-2 |editor2-last=Kozin |editor2-first=F.}}</ref><ref name=":12">{{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="werbos19742">{{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 describe the algorithm).<ref name="backprop2" /> In 1986, David E. Rumelhart, et al., popularized 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 === Deep learning architectures for convolutional neural networks (CNNs) with convolutional layers and downsampling layers and weight replication began with the neocognitron introduced by Kunihiko Fukushima in 1979.<ref name="FUKU19792">{{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="FUKU19802">{{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="SCHIDHUB42">{{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 |bibcode=2015NN.....61...85S |s2cid=11715509}}</ref> Fukushima's CNN architecture 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 CNNs to phoneme recognition. It used convolutions, weight sharing, and backpropagation.<ref name="Waibel19872">{{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="speechsignal2">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 recognizing individual letters.<ref name="wz19882">{{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 recognizing handwritten ZIP codes on mail. Training required 3 days.<ref name="LECUN19892">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="wz19902">{{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 hand-written digits, was applied by banks to recognize numbers on checks digitized in 32×32 pixel images.<ref name="lecun982">{{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="Qian19882">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="Bohr19882">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 alignments.<ref name="Rost19932">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 source of RNN was statistical mechanics. In 1972, Shun'ichi Amari proposed to modify the weights of an Ising model by Hebbian learning rule as a model of associative memory, adding in 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="Hopfield198222">{{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, Cajal observed "recurrent semicircles" in the 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> 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="WM2" />
In 1982 Crossbar Adaptive Array, a recurrent neural network with an array architecture (rather than a multilayer perceptron architecture),<ref name="CAA19822">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=2">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 to computing actions (decisions), it computed internal state evaluations (emotions) of consequence situations. Eliminating the external supervisor, it introduced the self-learning method in neural networks.
In cognitive psychology, the ''American Psychologist'' journal in the early 1980s debated the relation between cognition and emotion. Social psychologist Robert Zajonc in 1980 stated that emotion is computed first and is independent from cognition, while Richard 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>{{Cite journal |last=Lazarus |first=Richard S. |date=November 1982 |title=Thoughts on the relations between emotion and cognition. |url=https://doi.apa.org/doi/10.1037/0003-066X.37.9.1019 |journal=American Psychologist |language=en |volume=37 |issue=9 |pages=1019–1024 |doi=10.1037/0003-066X.37.9.1019 |bibcode=1982AmPsy..37.1019L |issn=1935-990X}}</ref><ref>{{Cite journal |last=Bozinovski |first=Stevo |date=2014 |title=Modeling Mechanisms of Cognition-emotion Interaction in Artificial Neural Networks, since 1981 |url=https://linkinghub.elsevier.com/retrieve/pii/S1877050914015567 |journal=Procedia Computer Science |language=en |volume=41 |pages=255–263 |doi=10.1016/j.procs.2014.11.111 |archive-url=https://web.archive.org/web/20190323204838/https://core.ac.uk/download/pdf/81973924.pdf |archive-date=23 March 2019}}</ref> It was an example of a debate where an RNN contributed to an issue and also addressed cognitive psychology.{{Importance inline|date=March 2026}}
The Jordan network (1986) and the Elman network (1990) applied RNN to study cognitive psychology.{{Citation needed|date=March 2026}}
In the 1980s, backpropagation did not work well for deep RNNs. In 1991, Jürgen Schmidhuber proposed the "neural sequence chunker" or "neural history compressor"<ref name="chunker19912">{{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="schmidhuber19922">{{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 self-supervised pre-training (the "P" in ChatGPT) and neural knowledge distillation.<ref name="DLhistory2" /> In 1993, a neural history compressor system solved a "Very Deep Learning" task that required more than 1000 layers in an RNN unfolded in time.<ref name="schmidhuber199322">{{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 identified and analyzed the vanishing gradient problem<ref name="HOCH19912">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><ref name="HOCH20012">{{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 connections to solve it. He and Schmidhuber introduced long short-term memory (LSTM), which set accuracy records in multiple application domains.<ref>{{Cite Q|Q98967430}}</ref><ref name="lstm22">{{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 the ultimate version of LSTM, which required the forget gate, and was introduced in 1999.<ref name="lstm19992">{{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.{{Citation needed|date=March 2026}}
During 1985–1995, inspired by statistical mechanics, several architectures and methods were developed by Terry Sejnowski, Peter Dayan, Geoffrey Hinton, and others, 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="nc952">{{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=":133">{{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.{{Citation needed|date=March 2026}}
=== Modern deep learning === Between 2009 and 2012, NNs began winning prizes in image recognition contests, approaching human performance on various tasks, initially in pattern recognition and handwriting recognition.<ref>{{Cite web |first=Juergen |last=Schmidhuber |title=How bio-inspired deep learning keeps winning competitions « the Kurzweil Library |url=https://www.thekurzweillibrary.com/how-bio-inspired-deep-learning-keeps-winning-competitions |access-date=2026-05-19 |language=en-US|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><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|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, ''DanNet,<ref name=":322">{{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 |bibcode=2010NeCom..22.3207C |s2cid=1918673}}</ref>''<ref name=":622">{{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> a CNN by Dan Ciresan, Ueli Meier, Jonathan Masci, Luca Maria Gambardella, and Schmidhuber achieved for the first time superhuman performance in a visual pattern recognition contest, outperforming traditional methods by a factor of 3.<ref name="SCHIDHUB42" /> It then won more contests.<ref name=":822">{{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="ciresan2013miccai2">{{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 }}</ref> They also showed how max-pooling CNNs on GPU improved performance.<ref name=":92">{{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 Hinton<ref name="krizhevsky201222">{{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="VGG2">{{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="szegedy2">{{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, Ng and Dean created a network that learned to recognize higher-level concepts, such as cats, from training only on unlabeled images.<ref name="ng20122">{{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 GPUs and distributed computing allowed the use of larger networks, particularly in image and visual recognition problems, which became known as "deep learning".<ref name=":42">{{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>
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="SAB12">{{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>
Generative adversarial networks (GANs) (Ian Goodfellow et al., 2014)<ref name="GANnips2">{{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 from 2014–2018. GAN was originally published in 1991 by 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="curiosity19912">{{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="gancurpm20202">{{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 was achieved by Nvidia's StyleGAN (2018)<ref name="SyncedReview2018222">{{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="progressiveGAN2017222">{{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 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 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 thereafter, with systems such as DALL·E 2 (2022) and Stable Diffusion (2022).
In 2014, the state of the art was training "[a] 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 accuracy,<ref name="prelu22">{{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="resnet22">{{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, training very deep networks advanced with the highway network, published in May,<ref name="highway201532">{{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.<ref name="resnet201532">{{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.{{Clarify|date=March 2026}}
=== Transformers === {{Main|Transformer (deep learning architecture)#History}} During the 2010s, the seq2seq model was developed, and attention mechanisms were added. This led to the modern transformer architecture in 2017 in "Attention Is All You Need".<ref name=":63" /> It requires computation time that is quadratic in the size of the context window. Schmidhuber's fast weight controller (1992)<ref name="transform199222">{{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="fastlinear202022">{{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="schlag202122">{{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="DLhistory2" /> Transformers have increasingly become the model of choice for natural language processing.<ref name="wolf20202">{{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 models such as GPT, Gemini, Grok, DeepSeek, and Qwen use this architecture.
==Elements== {{Confusing|section|date=April 2017}}{{Further|Mathematics of artificial neural networks}} right|thumb|Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals NNs began as an attempt to replicate the architecture of the brain in the digital realm. NNs immediately showed promise for handling non-linear relationships, but encountered obstacles. Models soon reoriented to applying mathematical insights to improve empirical results, at the expense of biological fidelity.
=== Neuron === alt=Artificial neuron structure|thumb|306x306px|Structure of an artificial neuron NNs are composed of digital neurons, conceptually derived from biological neurons. Each neuron has one or more numerical inputs and produces a single numerical output.<ref name="Abbod20072">{{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> An input (e.g., an image) is typically parceled across the set of input neurons (each getting a piece of the image). Each neuron's inputs, weighted by the weights of the connections from each of those inputs are summed. A bias term is added to this sum.<ref name="DAWSON19982">{{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> bias) The result is then passed through a nonlinear activation function to produce the neuron's output. Small enough outputs may be zeroed out (ignored).<ref name=":52">{{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><ref>{{Cite journal |last1=Kufel |first1=Jakub |last2=Bargieł-Łączek |first2=Katarzyna |last3=Kocot |first3=Szymon |last4=Koźlik |first4=Maciej |last5=Bartnikowska |first5=Wiktoria |last6=Janik |first6=Michał |last7=Czogalik |first7=Łukasz |last8=Dudek |first8=Piotr |last9=Magiera |first9=Mikołaj |last10=Lis |first10=Anna |last11=Paszkiewicz |first11=Iga |last12=Nawrat |first12=Zbigniew |last13=Cebula |first13=Maciej |last14=Gruszczyńska |first14=Katarzyna |date=2023-08-03 |title=What Is Machine Learning, Artificial Neural Networks and Deep Learning?-Examples of Practical Applications in Medicine |journal=Diagnostics (Basel, Switzerland) |volume=13 |issue=15 |page=2582 |doi=10.3390/diagnostics13152582 |doi-access=free|issn=2075-4418 |pmc=10417718 |pmid=37568945}}</ref><ref>{{Cite journal |last=Sazlı |first=Murat H. |date=2006-01-01 |title=A brief review of feed-forward neural networks |url=https://dergipark.org.tr/en/pub/aupse/article/890416 |journal=Communications Faculty of Sciences University of Ankara Series A2-A3 Physical Sciences and Engineering |language=en |volume=50 |issue=1 |pages=11–17 |doi=10.1501/commua1-2_0000000026 |bibcode=2006CFSAU..50...11M |issn=1303-6009}}</ref>
=== Network === NNs connect neurons to each other, using the output of some neurons as the input of others, akin to biological axon-synapse-dendrite connections.<ref name="Winston2">{{cite book |title=Artificial intelligence |publisher=Addison-Wesley Pub. Co |isbn=0-201-53377-4 |edition=3rd|year=1992 }}</ref> The network forms a directed, weighted graph. The weights apply to both the graph's the nodes and edges.<ref name="Zell1994ch5.22">{{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>
Nodes are arranged in layers, with the bottom layers directly addressing raw data, while the top layers reveal the final results. Intermediate layers gradually increase the level of abstraction, so what began as for example, pixels in an image, gradually resolves into things such as object boundaries, and then into real-world objects such as letters and faces.<ref name="Zell1994ch5.22" /><ref name=":52" /> Single layer and unlayered networks are also used.{{Citation needed|date=March 2026}}thumb|Structure of a feedforward neural network with four layers Multiple connection patterns have been used. Traditionally, they are ''fully connected'', with every neuron in one layer connecting to every neuron in the next layer.<ref>{{Cite journal |last1=Elizondo |first1=D. |last2=Fiesler |first2=E. |date=October 1997 |title=A Survey of Partially Connected Neural Networks |url=https://www.worldscientific.com/doi/abs/10.1142/s0129065797000513 |journal=International Journal of Neural Systems |volume=08 |issue=5n06 |pages=535–558 |doi=10.1142/S0129065797000513 |pmid=10065835 |issn=0129-0657}}</ref> However, in convolutional neural networks, some layers are ''convolutional,'' meaning each neuron in one layer is connected to a subset of neurons in the previous layer, such as those representing one section of an image.<ref name="flexible2">{{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><ref>{{Citation |last=Herberg |first=Evelyn |title=Lecture Notes: Neural Network Architectures |date=2023-04-18 |url=http://arxiv.org/abs/2304.05133 |access-date=2026-03-26 |arxiv=2304.05133 }}</ref>{{Clarify|reason=Explain how the neurons are "spatially grouped"|date=March 2026}}
In most neural networks,{{Citation needed|date=March 2026}} the outputs of the neurons in one layer are connected only to neurons in the immediately following layer (directed acyclic graph), meaning information only flows forward from one layer to the next. These are known as feedforward networks.<ref name="Zell1994p732">{{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> In contrast, networks that allow connections between neurons in the same or previous layers are known as 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>
== Learning == {{No footnotes|date=August 2019|section}}{{See also|Mathematical optimization|Estimation theory|Machine learning}} Training/learning involves adjusting the weights of the network to improve the accuracy of the result. NNs typically require vast numbers of sample inputs (far more than biological brains) to achieve a given level of function. This is done by minimizing the observed errors among sample observations. Training takes place before a network is deployed, and (unlike brains) does not continue thereafter. Instead, the network may be retrained from scratch as more sample data becomes available.
Empirical risk minimization adjusts node and link weights to minimize the difference (empirical risk), between the predicted output and the known values in the training samples.<ref name=":23">{{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> A defined loss function measures the degree of error.<ref name=":42" /> Backpropagation spreads the error (adjusts the weights) from the output nodes across the network to the input nodes.<ref name=":23" /> The intent is to allow the network to process data that is not included in the training samples.
As long as the value of the loss function (its cost) continues to decline, the network is continuing to improve. The function typically produces a statistic whose value is only approximate. When the cost is low, the difference between the output (''e.g.'' almost certainly a cat) and the correct answer (a cat) is small. Most learning models can be viewed as a straightforward application of optimization theory and statistical estimation.<ref name="Zell1994ch5.22" /><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 typically ends when additional observations do not usefully reduce the cost. The cost typically approaches, but does not reach, 0. If no feasible amount of sample data yields a low cost, training is labeled a failure.
=== Loss function === {{Expand section|Talk more about how loss functions are described, what types exist for different problems|date=March 2026}} While it is possible to assess a loss function ad hoc, typically it must exhibit desirable properties such as convexity, differentiability, and robustness.<ref>{{Cite journal |last=Terven |first=Juan |last2=Cordova-Esparza |first2=Diana-Margarita |last3=Romero-González |first3=Julio-Alejandro |last4=Ramírez-Pedraza |first4=Alfonso |last5=Chávez-Urbiola |first5=E. A. |date=2025-04-11 |title=A comprehensive survey of loss functions and metrics in deep learning |url=https://link.springer.com/10.1007/s10462-025-11198-7 |journal=Artificial Intelligence Review |language=en |volume=58 |issue=7 |doi=10.1007/s10462-025-11198-7 |issn=1573-7462}}</ref> In a probabilistic model, the model's posterior probability can be used as an inverse cost (higher values are better).{{citation needed|date=October 2024}}
=== Backpropagation === {{Main|Backpropagation}} Backpropagation is a method used to adjust the connection weights to compensate for errors found during learning. The error amount is essentially divided among the connections. Technically, backpropagation calculates the gradient (the derivative) of the loss 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 machines'',<ref>{{cite journal|last1=Huang|first1=Guang-Bin|last2=Zhu |first2=Qin-Yu|last3=Siew|first3=Chee-Kheong|year=2006|title=Extreme learning machine: theory and applications|journal=Neurocomputing|volume=70|issue=1 |pages=489–501|doi=10.1016/j.neucom.2005.12.126 |citeseerx=10.1.1.217.3692|s2cid=116858 }}</ref> "no-prop" networks,<ref>{{cite journal|year=2013|title=The no-prop algorithm: A new learning algorithm for multilayer neural networks |journal=Neural Networks|volume=37 |pages=182–188|doi=10.1016/j.neunet.2012.09.020|pmid=23140797|last1=Widrow|first1=Bernard|display-authors=etal}}</ref> training without backtracking,<ref>{{cite arXiv|eprint=1507.07680|first1=Yann |last1=Ollivier|first2=Guillaume|last2=Charpiat|title=Training recurrent networks without backtracking |year=2015|class=cs.NE}}</ref> "weightless" networks,<ref name="RBMTRAIN2">{{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> and non-connectionist neural networks.{{citation needed|date=June 2022}}
=== Hyperparameters === {{Main|Hyperparameter (machine learning)}} A hyperparameter is a parameter defining any configurable part of the network and 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, sample batch size, number of nodes and layers, etc .<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=978-0-262-03561-3 |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 hyperparameter choices, and thus may be adjusted during training (typically between training runs), 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 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 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 accelerating learning. In order to avoid oscillations such as connection weights that cycle between high and low values, 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}}
===Learning paradigms=== {{Unreferenced section|date=May 2026}} Machine learning has involved a variety of approaches to training models, including 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> 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> and self-supervised learning.
==== Supervised learning ==== Supervised learning pairs inputs and desired outputs. The learning task is to produce the desired output for each input. In this case, the cost is related to eliminating incorrect outputs.<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 include pattern recognition (classification) and regression (function approximation). Supervised learning is 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.
====Unsupervised learning==== In unsupervised learning, input data is given along with the cost function for the data <math>\textstyle x</math> and the output, without an "answer sheet". The cost function is dependent on the task (the model domain) and reflects ''a priori'' assumptions (implicit model properties, its parameters and the observed variables). For example, the model <math>\textstyle f(x) = a</math> treats <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 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 (in both examples, those quantities are maximized). Unsupervised learning is typically applied to estimation problems; applications include clustering, the estimation of statistical distributions, compression, and filtering.
==== Self-supervised learning ==== {{Excerpt|Self-supervised learning|only=paragraphs}}
====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 (the game) after each one. The goal is to win the game (get the highest score). The cost is the inverse of the score. In reinforcement learning, the aim is to weight the network to increase the score. After each action the game generates an observation and an instantaneous cost, according to its rules. The rules and the long-term cost can only 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 (policies) 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.
NNs serve as the learning component.<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 NNs (giving 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 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 NNs' ability to mitigate cost even when reducing the discretization grid density for numerically approximating control tasks.
====Self-learning==== Self-learning was introduced in 1982 along with a ''crossbar adaptive array'' (CAA) NN that could teach itself.<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 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, its behavioral environment, and its genetic environment, from which 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 NN topologies and weights using evolutionary computation. It is competitive with 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> Neuroevolution 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 Sherrington–Kirkpatrick models are a type of neural network built by introducing random variations into the network, either by giving neurons stochastic transfer functions,<ref name="Ackley19852">{{cite journal |last1=Ackley |first1=David H. |last2=Hinton |first2=Geoffrey E. |last3=Sejnowski |first3=Terrence J. |title=A Learning Algorithm for Boltzmann Machines |journal=Cognitive Science |date=1985 |volume=9 |issue=1 |pages=147–169 |doi=10.1207/s15516709cog0901_7}}</ref> or by giving them stochastic weights. This makes them useful tools for optimization problems, since the random fluctuations help the network escape from 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 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, 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> integrates topology with deep neural networks to address high-order data. Initially rooted in algebraic topology, TDL evolved into a versatile framework incorporating tools from mathematical disciplines such as differential topology and geometric topology.
===Other=== In a Bayesian framework, a distribution over the set of allowed models is chosen to minimize 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, 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="Qin12">{{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="Qin22">{{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 ==== {{Unreferenced section|date=May 2026}} Learning can be either stochastic or batch. Stochastic learning creates a weight adjustment for each sample. 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 == {{Excerpt|Types of artificial neural networks}}
== Network design == The choice of model depends on the data and the application. Models that work well with textual data are typically not the best choice for image data, etc. An important element is which training/learning the model uses.<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>
Neural architecture search (NAS) uses machine learning to automate NN design. NAS has yielded networks that compare well with hand-designed systems. The basic 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> Efforts include 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 library provides functions to help with building a deep network from scratch.
Hyperparameters are design choices (they are not learned).<ref name="abs1502.021272">{{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> ==Theoretical properties== ===Computational power=== The multilayer perceptron is a universal function approximator, as proven by the universal approximation theorem. However, the proof does not specify the number of neurons required, the network topology, the weights or the learning parameters.
A recurrent architecture with 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 linear connections. Further, the use of irrational values for weights results in a machine with 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 is 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. Information capacity and the VC dimension are two metrics. 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=ACM| pages=2709–2710| doi=10.1145/3343031.3350545| isbn=978-1-4503-6889-6| s2cid=204837170}}</ref> that derive from considering a neuron as an electrical element.
The information capacity captures the functions that the network can model given any data as input. The VC dimension uses the principles of measure theory and finds the maximum capacity under optimal circumstances, given input data in a specific form. 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><ref name="auto2">{{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>
===Convergence=== Models may not consistently converge on a single solution, because the system may get stuck in a local minima. Alternatively, the optimization method used might not guarantee to converge should it begin far from any local minima. Thirdly, for sufficiently large data or parameters, some methods are impractically slow/expensive. Training may also cross some saddle point that may then prevent access to the solution.
When the width of a network approaches infinity, it is well described by its first order Taylor expansion throughout training, and so inherits the convergence behavior of 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> When parameter numbers are small, NNs often fit target functions from low to high frequencies. This behavior is referred to as spectral bias, or the frequency principle.<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 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 of the behavior of some well-studied iterative numerical schemes such as the Jacobi method. Deeper neural networks are 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=== {{Unreferenced section|date=May 2026}} Applications that must generalize well to unseen examples, must avoid over-training. This arises in convoluted or over-specified systems when the network capacity is much larger than needed.
Two approaches address over-training. Cross-validation and similar techniques can check for over-training and select appropriate hyperparameters to minimize generalization error. ''Regularization'' in a probabilistic (Bayesian) framework can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize 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. right|thumb|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 adopting 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.
The softmax activation function is:
: <math>y_i=\frac{e^{x_i}}{\sum_{j=1}^c e^{x_j}}</math>
== Applications == Neural networks support a broad range of applications in image processing, speech recognition, natural language processing, finance, and medicine.{{cn|date=May 2026}} Because of their ability to model and reproduce nonlinear processes, 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, 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="SAB12" /> 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, 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="TurekNeuralNet2">{{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, 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 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 markets) * Quantum chemistry<ref name="Balabin_20092">{{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="Ciaramella2">{{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> * Disaster response<ref>{{cite journal |last1=Maity |first1=Abhishek |title=CrisisSense: Transforming Social Signals into Real-Time Disaster Awareness via Deep Neural Intelligence |journal=2026 IEEE Madhya Pradesh Section Conference (MPCON) |date=March 2026 |pages=1501–1506 |doi=10.1109/MPCON69668.2026.11508516 |url=https://ieeexplore.ieee.org/document/11508516}}</ref>
NNs have been used to diagnose 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 data.<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>
NNs 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 settling in building foundations.<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> They are used to mitigate flooding by 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> NNs have 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> NNs have been employed in cybersecurity, with the objective to discriminate between legitimate and malicious activities. 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 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.
NNs have been proposed as a tool to solve partial differential equations 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 systems.<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> In brain research NNs have studied short-term behavior of 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.
NNs show promise in profiling a user's interests from photos<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> and discovering new stable materials by efficiently predicting the total energy of crystals.<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>
=== Image processing === NNs are employed in computer vision tasks such as image classification, object and facial recognition, and image segmentation. They have been applied to automated surveillance medical imaging for diagnosis.<ref name=":072">{{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>
=== Speech recognition === NNs are used for speaker identification, speech-to-text, and text-to-speech conversion. NNs have conquered large vocabulary continuous speech recognition, outperforming traditional techniques.<ref name=":072" /><ref name=":152">{{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.<ref>{{Cite journal |last1=Natarajan |first1=Sureshkumar |last2=Rahman Al-Haddad |first2=Syed Abdul |last3=Ahmad |first3=Faisul Arif |last4=Kamil |first4=Raja |last5=Hassan |first5=Mohd Khair |last6=Azrad |first6=Syaril |last7=Macleans |first7=June Francis |last8=Abdulhussain |first8=Sadiq H. |last9=Mahmmod |first9=Basheera M. |last10=Saparkhojayev |first10=Nurbek |last11=Dauitbayeva |first11=Aigul |date=2025-07-01 |title=Deep neural networks for speech enhancement and speech recognition: A systematic review |url=https://www.sciencedirect.com/science/article/pii/S2090447925001467 |journal=Ain Shams Engineering Journal |volume=16 |issue=7 |article-number=103405 |doi=10.1016/j.asej.2025.103405 |issn=2090-4479}}</ref><ref>{{Cite journal |last1=Lv |first1=Zhihan |last2=Poiesi |first2=Fabio |last3=Dong |first3=Qi |last4=Lloret |first4=Jaime |last5=Song |first5=Houbing |date=2022-11-11 |title=Deep Learning for Intelligent Human–Computer Interaction |journal=Applied Sciences |language=en |volume=12 |issue=22 |article-number=11457 |doi=10.3390/app122211457 |doi-access=free|issn=2076-3417}}</ref>
=== Natural language processing === In natural language processing, NNs are used for tasks such as text classification, sentiment analysis, machine translation, to answer free-form questions, act as chatbots, and to summarize and analyze texts.<ref name=":072" /><ref name=":152" /> 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>{{Vague|reason=What sort of implications?|date=March 2026}}
=== Control systems === NNs are used to model dynamic systems for tasks such as system identification, control design, autonomous vehicles, and optimization.<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>{{Vague|reason=How are they important?|date=March 2026}}
=== Finance === {{excerpt|Applications of artificial intelligence#Trading and investment}}
=== Medicine === NNs analyze 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=":152" /> In drug discovery, NNs speed up the identification of potential drug candidates and predict their efficacy and safety, significantly reducing development time and costs.<ref name=":072" /> Additionally, their application in personalized medicine and healthcare data analysis allows tailored therapies and efficient patient care management.<ref name=":152" />
=== Cybersecurity === Neural networks are widely applied in cybersecurity for anomaly detection, malware classification, and intrusion detection. By learning patterns of normal system or network behavior, NNs can identify deviations that indicate malicious activity.<ref>{{cite journal |last1=Buczak |first1=Anna |last2=Guven |first2=Erhan |title=A Survey of Data Mining and Machine Learning Methods for Cyber Security Intrusion Detection |journal=IEEE Communications Surveys & Tutorials |year=2016 |volume=18 |issue=2 |pages=1153–1176 |doi=10.1109/COMST.2015.2494502 }}</ref>
=== Content creation === Transformers are used for content creation across numerous industries.<ref name=":092">{{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> They can analyze samples and produce outputs that match the style of an artist or musician. For instance, DALL-E trained on 650 million pairs of images and texts and can create artworks based on user text.<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> Companies such as AIVA and Jukedeck have used transformers to create original music.<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> NNs have been used to create personalized advertisements.<ref name=":092" /> Film production companies have used NNs to analyze the financial success of a film.<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> NNs have found uses in video game creation.<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>
==Issues== ===Training === NNs require millions of training samples to become functional. As of 2026, training a commercial LLM (GPT, Grok, Gemini) typically required hundreds of thousands of computers, and cost 10s of millions of dollars.<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><ref name="Qin12" />
===Theory=== A central claim{{citation needed|date=January 2023}} of NNs is that they embody new and powerful general principles for processing information. However, at root they use statistical long-developed methods.
===Hardware=== Neural networks require enormous computing resources for training.<ref name=":02">{{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 requires only 20 watts of power,<ref>{{Cite web |title=Power of a Human Brain - The Physics Factbook |url=https://hypertextbook.com/facts/2001/JacquelineLing.shtml |access-date=2026-05-19 |website=hypertextbook.com}}</ref> training commercial transformers requires (2026) data centers with hundreds of megawatts.<ref>{{Cite web |title=2024 United States Data Center Energy Usage Report {{!}} Energy Technologies Area |url=https://eta.lbl.gov/publications/2024-lbnl-data-center-energy-usage-report |access-date=2026-05-19 |website=eta.lbl.gov |language=en}}</ref>
From 1991 to 2015, computing power, especially as delivered by GPGPUs (on GPUs), increased around a million-fold, enabling the use of the backpropagation algorithm.<ref name="SCHIDHUB42" />
Neuromorphic engineering or a physical neural network constructed non-von-Neumann chips to directly implement neural networks in circuitry. For example, Alphabet introduced custom chips (Tensor Processing Unit).<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>
=== Concept drift === The statistical properties of input data may change over time, a phenomenon known as concept drift or 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 levels observed during training.
Several strategies have been developed to monitor neural networks for drift and degradation:
* '''Error-based monitoring''': comparing current predictions against ground-truth labels. This approach directly quantifies predictive performance but may be impractical when labels are delayed or impractical 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>
=== Dataset bias === Neural networks are dependent on the quality of their training data; low quality data can lead the model to product poor results.<ref name=":0102">{{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=":172">{{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> Biased data produces biased results, requiring trainers to detect and correct them. For example, data that underrepresents some demographic groups may prevent cause errors, e.g., in facial recognition and law enforcement.<ref name=":172" /><ref name=":222">{{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> In 2018, Amazon scrapped a recruiting tool because the model favored men over women for jobs in software engineering due to the larger number of male workers in the field.<ref name=":222" /> The system penalized resumes with the word "woman" or the name of a women's college. One corrective is to add synthetic data to offset the bias.<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>
=== Lack of interpretability === NNs are "black box" systems that have complicated understanding of their decision-making processes. NNs are further vulnerable to adversarial examples ("poison"), that can cause incorrect predictions.
These concerns have led to increased research in explainable artificial intelligence (XAI), robust machine learning, and hybrid AI approaches that combine neural learning with symbolic reasoning.{{Citation needed|date=March 2026}} Advocates of hybrid models also 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>
Progress has included local vs. non-local learning and shallow vs. deep architectures.<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>{{Clarify|date=March 2026}}
Analyzing human thought (a biological neural network) has also proven difficult.
==Gallery== <gallery widths="220"> File:Single_layer_ann.svg|alt=A single-layer feedforward artificial neural network. Arrows originating from <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> x <nowiki> </nowiki> <nowiki> </nowiki> 2 <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki><nowiki> {\displaystyle x_{2}} </nowiki><nowiki> </nowiki> <nowiki> </nowiki>are omitted for clarity. There are p inputs to this network and q outputs. In this system, the value of the qth output, <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> y <nowiki> </nowiki> <nowiki> </nowiki> q <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki><nowiki> {\displaystyle y_{q}} </nowiki><nowiki> </nowiki> , is calculated as <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> y <nowiki> </nowiki> <nowiki> </nowiki> q <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> = <nowiki> </nowiki> K <nowiki> </nowiki> ⋅ <nowiki> </nowiki> <nowiki> </nowiki> ( <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> ∑ <nowiki> </nowiki> <nowiki> </nowiki> i <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> ( <nowiki> </nowiki> <nowiki> </nowiki> x <nowiki> </nowiki> <nowiki> </nowiki> i <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> w <nowiki> </nowiki> <nowiki> </nowiki> i <nowiki> </nowiki> q <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> ) <nowiki> </nowiki> − <nowiki> </nowiki> <nowiki> </nowiki> b <nowiki> </nowiki> <nowiki> </nowiki> q <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> ) <nowiki> </nowiki> <nowiki> </nowiki> . <nowiki> </nowiki> <nowiki> </nowiki> <nowiki> </nowiki> {\displaystyle y_{q}=K\cdot \left(\sum _{i}(x_{i}w_{iq})-b_{q}\right).}|A single-layer feedforward 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 ''q''th output, <math>y_q</math>, is calculated as <math>y_q = K\cdot\left(\sum_i(x_i w_{iq})-b_q\right).</math> File:Two_layer_ann.svg|A two-layer feedforward neural network File:Artificial_neural_network.svg|An neural network File:Ann_dependency_(graph).svg|An NN dependency graph File:Single-layer_feedforward_artificial_neural_network.png|A single-layer feedforward 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 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>
== 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}}
==Notes== <references group="note"></references>
==References== <references ></references>
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==External links== {{Spoken Wikipedia|date=27 November 2011|En-Neural_network.ogg}}
* [http://www.dkriesel.com/en/science/neural_networks A Brief Introduction to Neural Networks (D. Kriesel)] – Illustrated, bilingual manuscript about 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 NN] * [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 |last=Sanderson |first=Grant |date=5 October 2017 |title=But what ''is'' a Neural Network? |url=https://www.youtube.com/watch?v=aircAruvnKk&list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211107/aircAruvnKk |archive-date=7 November 2021 |work=3Blue1Brown |via=YouTube}}{{cbignore}} * {{Cite journal |last1=Hakim |first1=MD Azizul |last2=Alam |first2=Mohammad Ifazul |year=2025 |title=Biologically inspired neural network layer with homeostatic regulation and adaptive repair mechanisms |journal=Scientific Reports |publisher=Nature Publishing Group |volume=15 |issue=1 |page=33903 |bibcode=2025NatSR..1533903H |doi=10.1038/s41598-025-09114-8 |issn=2045-2322 |pmc=12484884 |pmid=41028030}} {{Navboxes|list={{Artificial intelligence navbox}} {{Complex systems topics}} {{Control theory}} {{Neuroscience}} {{Self-driving cars and enabling technologies}}}} {{Authority control}} Category:Computational statistics Category:Artificial neural networks Category:Classification algorithms Category:Computational neuroscience Category:Market research Category:Mathematical psychology Category:Mathematical and quantitative methods (economics) Category:Bioinspiration