Brownout in software engineering is a technique that involves disabling certain features of an application. == Description == Brownout is used to increase the robustness of an application to computing capacity shortage. If too many users are simultaneously accessing an application hosted online, the underlying computing infrastructure may become overloaded, rendering the application unresponsive. Users are likely to abandon the application and switch to competing alternatives, hence incurring long-term revenue loss. To better deal with such a situation, the application can be given brownout capabilities: The application will disable certain features – e.g., an online shop will no longer display recommendations of related products – to avoid overload. Although reducing features generally has a negative impact on the short-term revenue of the application owner, long-term revenue loss can be avoided. The technique is inspired by brownouts in power grids, which consists in reducing the power grid's voltage in case electricity demand exceeds production. Some consumers, such as incandescent light bulbs, will dim – hence originating the term – and draw less power, thus helping match demand with production. Similarly, a brownout application helps match its computing capacity requirements to what is available on the target infrastructure. Brownout complements elasticity. The former can help the application withstand short-term capacity shortage, but does so without changing the capacity available to the application. In contrast, elasticity consists of adding (or removing) capacity to the application, preferably in advance, so as to avoid capacity shortage altogether. The two techniques can be combined; e.g., brownout is triggered when the number of users increases unexpectedly until elasticity can be triggered, the latter usually requiring minutes to show an effect. Brownout is relatively non-intrusive for the developer, for example, it can be implemented as an advice in aspect-oriented programming. However, surrounding components, such as load-balancers, need to be made brownout-aware to distinguish between cases where an application is running normally and cases where the application maintains a low response time by triggering brownout. == Usage in phased deprecation == A related use of the brownout concept in software engineering is the deliberate introduction of temporary outages to a system, API or feature that is being phased out. This is sometimes also called a "scream test" when it is used to discover unknown dependents of a system or API. The intention is to allow detection of downstream consumers of an API or service who may otherwise have missed deprecation announcements or to uncover hidden side-effects of the deprecation that may have been overlooked. The intention is that developers of dependent systems will notice their own system failures caused by the upstream brownout. Such brownouts are typically pre-announced scheduled outages or probabilistic in nature (such as artificially failing a percentage of requests). As a brownout is only a temporary or partial outage, it provides downstream consumers of an API or service time to remove any discovered dependencies on the deprecated API before it is fully retired. For consumers that have already prepared for the deprecation, a brownout provides valuable testing that the final removal of the service won't cause any unexpected problems.
Aggregation (linguistics)
In linguistics, aggregation is a subtask of natural language generation, which involves merging syntactic constituents (such as sentences and phrases) together. Sometimes aggregation can be done at a conceptual level. == Examples == A simple example of syntactic aggregation is merging the two sentences John went to the shop and John bought an apple into the single sentence John went to the shop and bought an apple. Syntactic aggregation can be much more complex than this. For example, aggregation can embed one of the constituents in the other; e.g., we can aggregate John went to the shop and The shop was closed into the sentence John went to the shop, which was closed. From a pragmatic perspective, aggregating sentences together often suggests to the reader that these sentences are related to each other. If this is not the case, the reader may be confused. For example, someone who reads John went to the shop and bought an apple may infer that the apple was bought in the shop; if this is not the case, then these sentences should not be aggregated. == Algorithms and issues == Aggregation algorithms must do two things: Decide when two constituents should be aggregated Decide how two constituents should be aggregated, and create the aggregated structure The first issue, deciding when to aggregate, is poorly understood. Aggegration decisions certainly depend on the semantic relations between the constituents, as mentioned above; they also depend on the genre (e.g., bureaucratic texts tend to be more aggregated than instruction manuals). They probably should depend on rhetorical and discourse structure. The literacy level of the reader is also probably important (poor readers need shorter sentences). But we have no integrated model which brings all these factors together into a single algorithm. With regard to the second issue, there have been some studies of different types of aggregation, and how they should be carried out. Harbusch and Kempen describe several syntactic aggregation strategies. In their terminology, John went to the shop and bought an apple is an example of forward conjunction Reduction Much less is known about conceptual aggregation. Di Eugenio et al. show how conceptual aggregation can be done in an intelligent tutoring system, and demonstrate that performing such aggregation makes the system more effective (and that conceptual aggregation make a bigger impact than syntactic aggregation). == Software == Unfortunately there is not much software available for performing aggregation. However the SimpleNLG system does include limited support for basic aggregation. For example, the following code causes SimpleNLG to print out The man is hungry and buys an apple.
Physics-informed neural networks
In machine learning, physics-informed neural networks (PINNs), also referred to as theory-trained neural networks (TTNs), are a type of universal function approximator that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples. Because they process continuous spatial and time coordinates and output continuous PDE solutions, they can be categorized as neural fields. == Function approximation == Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization. Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. To remedy this, Physics-Informed Neural Networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training datasets that do not necessarily need to be large or complete. An accurate solution of partial differential equations can potentially be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINNs may be used for finding an optimal solution with high fidelity. PINNs can be applied to a wide range of problems in computational science, and are a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a mesh-free alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, a trained PINN network can be used to predict values on simulation grids of different resolutions without needing to be retrained. Additionally, the derivatives used in the partial differential equations can be computed using automatic differentiation (AD), which is assessed to be superior to numerical or symbolic differentiation. == Modeling and computation == A general nonlinear partial differential equation can be written as: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} where u ( t , x ) {\displaystyle u(t,x)} denotes the solution, N [ ⋅ ; λ ] {\displaystyle {\mathcal {N}}[\cdot ;\lambda ]} is a nonlinear operator parameterized by λ {\displaystyle \lambda } , and Ω {\displaystyle \Omega } is a subset of R D {\displaystyle \mathbb {R} ^{D}} . This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems: data-driven solutions of partial differential equations data-driven discovery of partial differential equations === Data-driven solution of partial differential equations === The data-driven solution of PDE computes the hidden state u ( t , x ) {\displaystyle u(t,x)} of the system given boundary data and/or measurements z {\displaystyle z} , and fixed model parameters λ {\displaystyle \lambda } . We solve: u t + N [ u ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u]=0,\quad x\in \Omega ,\quad t\in [0,T]} . by defining the residual f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ] {\displaystyle f:=u_{t}+{\mathcal {N}}[u]} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} is the error between the PINN u ( t , x ) {\displaystyle u(t,x)} and the set of boundary conditions and measured data on the set of points Γ {\displaystyle \Gamma } where the boundary conditions and data are defined. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the PDE during the training process. This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE. === Data-driven discovery of partial differential equations === Given noisy and incomplete measurements z {\displaystyle z} of the state of the system, the data-driven discovery of PDEs results in computing the unknown state u ( t , x ) {\displaystyle u(t,x)} and learning model parameters λ {\displaystyle \lambda } that best describe the observed data: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} By defining f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ; λ ] = 0 {\displaystyle f:=u_{t}+{\mathcal {N}}[u;\lambda ]=0} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network, f ( t , x ) {\displaystyle f(t,x)} results in a PINN. This network can be derived using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} , together with the parameter λ {\displaystyle \lambda } of the differential operator can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} , with u {\displaystyle u} and z {\displaystyle z} state solutions and measurements at sparse location Γ {\displaystyle \Gamma } , respectively. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation. == Extensions and applications == === For piece-wise function approximation === PINNs are unable to approximate PDEs that have strong non-linearity or sharp gradients (such as those that commonly occur in practical fluid flow problems). Piecewise approximation has been an old practic
SPL notation
SPL (Sentence Plan Language) is an abstract notation representing the semantics of a sentence in natural language. In a classical Natural Language Generation (NLG) workflow, an initial text plan (hierarchically or sequentially organized factoids, often modelled in accordance with Rhetorical Structure Theory) is transformed by a sentence planner (generator) component to a sequence of sentence plans modelled in a Sentence Plan Language. A surface generator can be used to transform the SPL notation into natural language sentences. Probably the most widely used SPL language used today (2022) is AMR (Abstract Meaning Representation, see there for further references), but is owes parts of its popularity to its application to NLP problems other than NLG, e.g., machine translation and semantic parsing.
SmarterChild
SmarterChild was a chatbot available on AOL Instant Messenger and Windows Live Messenger (previously MSN Messenger) networks. == History == SmarterChild was an apparently intelligent agent or "bot" developed by ActiveBuddy, Inc., with offices in New York and Sunnyvale. It was widely distributed across global instant messaging networks. SmarterChild became very popular, attracting over 30 million Instant Messenger "buddies" on AIM (AOL), MSN and Yahoo Messenger over the course of its lifetime. Founded in 2000, ActiveBuddy was the brainchild of Robert Hoffer and Timothy Kay, who later brought seasoned advertising executive Peter Levitan on board as CEO. The concept for conversational instant messaging bots came from the founder's vision to add natural language comprehension functionality to the increasingly popular AIM instant messaging application. The original implementation took shape as a demo that Kay programmed in Perl in his Los Altos garage to connect a single buddy name, "ActiveBuddy", to look up stock symbols, and later allow AIM users to play Colossal Cave Adventure, a word-based adventure game, and MIT's Boris Katz Start Question Answering System but quickly grew to include a wide range of database applications the company called 'knowledge domains' including instant access to news, weather, stock information, movie times, yellow pages listings, and detailed sports data, as well as a variety of tools (personal assistant, calculators, translator, etc.). None of the individual domains which the company had named “stocksBuddy”, “sportsBuddy”, etc. ever launched publicly. When Stephen Klein came on board as COO — and eventually CEO — he insisted that all of the disparate test “buddies” be launched together with the company’s highly-developed colloquial chat domain. He suggested using “SmarterChild”, a username coined by Tim Kay which Tim was using to test various things. The bundled domains were launched publicly as SmarterChild (on AIM initially) in June 2001. SmarterChild provided information wrapped in fun and quirky conversation. The company generated no revenue from SmarterChild, but used it as a demonstration of the power of what Klein called “conversational computing”. The company subsequently marketed Automated Service Agents—delivering immediate answers to customer service inquiries—-to large corporations, like Comcast, Cingular, TimeWarner Cable, etc. SmarterChild's popularity spawned targeted marketing-oriented bots for Radiohead, Austin Powers, Intel, Keebler, The Sporting News and others. ActiveBuddy co-founders, Kay and Hoffer, as co-inventors, were issued two controversial U.S. patents in 2002. ActiveBuddy changed its name to Colloquis (briefly Conversagent) and targeted development of consumer-facing enterprise customer service agents, which the company marketed as Automated Service Agents. Microsoft acquired Colloquis in October 2006 and proceeded to de-commission SmarterChild and kill off the Automated Service Agent business as well. Robert Hoffer, ActiveBuddy co-founder, licensed the technology from Microsoft after Microsoft abandoned the Colloquis technology.
Perplexity AI
Perplexity AI, Inc., or simply Perplexity, is an American privately held software company offering a web search engine that processes user queries and synthesizes responses. Perplexity products use large language models and incorporate real-time web search capabilities, providing responses based on current Internet content, citing sources used. Its real-time search engine is called Sonar and is based on Meta's Llama model. A free public version is available, while a paid Pro subscription offers access to more advanced language models and additional features. Perplexity AI, Inc., was founded in August 2022 by Aravind Srinivas, Denis Yarats, Johnny Ho, and Andy Konwinski. As of September 2025, the company was valued at US$20 billion. Perplexity AI has attracted legal scrutiny over allegations of copyright infringement, unauthorized content use, and trademark issues from several major media organizations, including the BBC, Dow Jones, and The New York Times. According to separate analyses by Wired and later Cloudflare, Perplexity uses undisclosed web crawlers with spoofed user-agent strings to scrape the content of websites which prohibit, or explicitly block, web scraping. == History == In August 2022, Perplexity AI, Inc., was founded by Aravind Srinivas, Denis Yarats, Johnny Ho, and Andy Konwinski, engineers with backgrounds in back-end systems, artificial intelligence (AI) and machine learning. It launched its main search engine on December 7, 2022, and has since released a Google Chrome extension and apps for iOS and Android. In February 2023, Perplexity reported two million unique visitors. By April 2024, Perplexity had raised $165 million in funding, valuing the company at over $1 billion. As of June 2025, Perplexity closed a $500 million round of funding that elevated its valuation to $14 billion. Investors in Perplexity AI have included Jeff Bezos, Tobias Lütke, Nat Friedman, Nvidia, and Databricks. Perplexity has also received funding from 1789 Capital, a venture capital firm notable for its association with Donald Trump Jr. During Bloomberg’s Tech Summit 2025, Srinivas shared that the company processed 780 million queries in May 2025, experiencing more than 20% month-over-month growth, processing around 30 million queries daily. In July 2024, Perplexity announced the launch of a new publishers' program to share advertising revenue with partners. On January 18, 2025, the day before the impending U.S. ban on the social media app TikTok, Perplexity submitted a proposal for a merger with TikTok US. On August 12, 2025, Perplexity made a bid to buy Chrome from Google for $34.5 billion. Perplexity stated that the sale could remedy anti-trust litigation against Google, in which a judge was considering compelling the sale of Chrome. In December 2025, Cristiano Ronaldo took an undisclosed stake in Perplexity AI and entered a global brand partnership with the company. === Business Strategy and Finance (2026) === As of early 2026, Perplexity AI reached a valuation of $21.21 billion following its Series E-6 funding round. The company's Annual Recurring Revenue (ARR) grew from $80 million in late 2024 to an estimated $200 million by February 2026. In January 2026, the company entered into a three-year, $750 million commitment with Microsoft Azure to secure the GPU capacity required for its advanced "Deep Research" and "Model Council" features. In February 2026, Perplexity transitioned to a subscription-first model by discontinuing its AI-integrated advertising strategy. Leadership stated the move was intended to preserve user trust in the "answer engine," prioritizing objective results over ad revenue. The company also introduced the "Model Council" feature on February 5, 2026, which allows users to compare outputs from multiple large language models, such as GPT-5.2 and Claude 4.6, simultaneously. To expand its user base, Perplexity began offering a free year of Pro access to students, U.S. Military Veterans, and government employees. == Products and services == === Search engine web portal === Perplexity’s primary offering is an online information retrieval system (search engine) that uses large language models to generate responses to user queries by searching and summarizing web-based content. Perplexity offers a feature known as Perplexity Pages that generates structured summaries and report-like content from user queries by aggregating cited sources. Perplexity is available without charge or registration to Web users, a freemium model. === Perplexity Pro === Perplexity Pro is a subscription tier, a more capable paid "enterprise" service, including stronger security and data protection and additional tools, including the ability to search uploaded documents alongside web content and access to a programmatic application programming interface (API). It allows the user to select between backend models such as GPT-5.4, Claude 4.6 and Gemini 3.1 Pro. The company has also developed its own models, Sonar (based on Llama 3.3) and R1 1776 (based on DeepSeek R1). === Internal Knowledge Search === Internal Knowledge Search enables Pro and Enterprise Pro users to simultaneously search across web content and internal documents. Users can upload and search through Excel, Word, PDF, and other common file formats. Enterprise Pro users can upload and index up to 500 files. === Search API === Perplexity's Search API provides AI developers with programmatic access to the company's search infrastructure. The September 2025 release includes a software development kit, an open-source evaluation framework called search_evals, and documentation detailing the API's design and optimization. === Shopping hub === Perplexity's Shopping Hub is an online shopping platform that provides AI-generated product recommendations, and enables users to purchase products directly through Perplexity's interface. It was launched in November 2024 with backing by Amazon and Nvidia. === Finance === In October 2024, Perplexity AI introduced new finance-related features, including looking up stock prices and company earnings data. The tool provides real-time stock quotes and price tracking, industry peer comparisons and basic financial analysis tools. The platform sources its financial data from Financial Modeling Prep. === Assistant === In January 2025, Perplexity launched the Perplexity Assistant, an AI-powered tool designed to enhance the functionality of its search engine. It can perform tasks across multiple apps, such as hailing a ride or searching for a song, and can maintain context across actions. The assistant is also multi-modal, meaning it can use a phone's camera to provide answers about the user's surroundings or on-screen content. Perplexity has acknowledged that the assistant is still in development and may not always function as expected. For instance, certain features, such as summarizing unread emails or upcoming calendar events, require users to enable a workaround based on notifications. === Comet === In July 2025, Perplexity launched Comet, an AI browser based on Chromium. Initially, access to the browser was limited to users subscribed to the most expensive subscription tier. The browser was later released for free download in October 2025. A key feature is integration of the Perplexity search engine, which can perform a variety of tasks such as generating article summaries, describing an image, conducting research about a topic and composing emails. === Truth Social chatbot === Perplexity has been contracted to produce a chatbot for Donald Trump's social media platform Truth Social. == Leadership == Aravind Srinivas is the CEO and co-founder of Perplexity AI. He previously held research positions at OpenAI, Google DeepMind, and other AI research institutions focusing on machine learning and artificial intelligence. In a March 2026 All-In episode, Srinivas said the incoming AI-related layoffs were "glorious future" to "look forward", as it freed people from jobs they didn't like and gave them opportunities to pursue entrepreneurship. == Controversies == === Copyright and trademark infringement allegations === In June 2024, Forbes publicly criticized Perplexity for using their content. According to Forbes, Perplexity published a story largely copied from a proprietary Forbes article without mentioning or prominently citing Forbes. In response, Srinivas said that the feature had some "rough edges" and accepted feedback but maintained that Perplexity only "aggregates" rather than plagiarizes information. In October 2024, The New York Times sent a cease-and-desist notice to Perplexity to stop accessing and using NYT content, claiming that Perplexity is violating its copyright by scraping data from its website. In June 2024, Dow Jones and New York Post filed a lawsuit against Perplexity, alleging copyright infringement. The lawsuit also alleged that Perplexity harmed their brand by attributing hallucinated quotes, for example on F-16 jets for Ukraine, to artic
Convolutional neural network
A convolutional neural network (CNN) is a type of feedforward neural network that learns features via filter (or kernel) optimization. This type of deep learning network has been applied to process and make predictions from many different types of data including text, images and audio. CNNs are the de-facto standard in deep learning-based approaches to computer vision and image processing, and have only recently been replaced—in some cases—by newer architectures such as the transformer. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by the regularization that comes from using shared weights over fewer connections. For example, for each neuron in the fully-connected layer, 10,000 weights would be required for processing an image sized 100 × 100 pixels. However, applying cascaded convolution (or cross-correlation) kernels, only 25 weights for each convolutional layer are required to process 5x5-sized tiles. Higher-layer features are extracted from wider context windows, compared to lower-layer features. Some applications of CNNs include: image and video recognition, recommender systems, image classification, image segmentation, medical image analysis, natural language processing, brain–computer interfaces, and financial time series. CNNs are also known as shift invariant or space invariant artificial neural networks, based on the shared-weight architecture of the convolution kernels or filters that slide along input features and provide translation-equivariant responses known as feature maps. Counter-intuitively, most convolutional neural networks are not invariant to translation, due to the downsampling operation they apply to the input. Feedforward neural networks are usually fully connected networks, that is, each neuron in one layer is connected to all neurons in the next layer. The "full connectivity" of these networks makes them prone to overfitting data. Typical ways of regularization, or preventing overfitting, include: penalizing parameters during training (such as weight decay) or trimming connectivity (skipped connections, dropout, etc.) Robust datasets also increase the probability that CNNs will learn the generalized principles that characterize a given dataset rather than the biases of a poorly-populated set. Convolutional networks were inspired by biological processes in that the connectivity pattern between neurons resembles the organization of the animal visual cortex. Individual cortical neurons respond to stimuli only in a restricted region of the visual field known as the receptive field. The receptive fields of different neurons partially overlap such that they cover the entire visual field. CNNs use relatively little pre-processing compared to other image classification algorithms. This means that the network learns to optimize the filters (or kernels) through automated learning, whereas in traditional algorithms these filters are hand-engineered. This simplifies and automates the process, enhancing efficiency and scalability overcoming human-intervention bottlenecks. == Architecture == A convolutional neural network consists of an input layer, hidden layers and an output layer. In a convolutional neural network, the hidden layers include one or more layers that perform convolutions. Typically this includes a layer that performs a dot product of the convolution kernel with the layer's input matrix. This product is usually the Frobenius inner product, and its activation function is commonly ReLU. As the convolution kernel slides along the input matrix for the layer, the convolution operation generates a feature map, which in turn contributes to the input of the next layer. This is followed by other layers such as pooling layers, fully connected layers, and normalization layers. Here it should be noted how close a convolutional neural network is to a matched filter. === Convolutional layers === In a CNN, the input is a tensor with shape: (number of inputs) × (input height) × (input width) × (input channels) After passing through a convolutional layer, the image becomes abstracted to a feature map, also called an activation map, with shape: (number of inputs) × (feature map height) × (feature map width) × (feature map channels). Convolutional layers convolve the input and pass its result to the next layer. This is similar to the response of a neuron in the visual cortex to a specific stimulus. Each convolutional neuron processes data only for its receptive field. Although fully connected feedforward neural networks can be used to learn features and classify data, this architecture is generally impractical for larger inputs (e.g., high-resolution images), which would require massive numbers of neurons because each pixel is a relevant input feature. A fully connected layer for an image of size 100 × 100 has 10,000 weights for each neuron in the second layer. Convolution reduces the number of free parameters, allowing the network to be deeper. For example, using a 5 × 5 tiling region, each with the same shared weights, requires only 25 neurons. Using shared weights means there are many fewer parameters, which helps avoid the vanishing gradients and exploding gradients problems seen during backpropagation in earlier neural networks. To speed processing, standard convolutional layers can be replaced by depthwise separable convolutional layers, which are based on a depthwise convolution followed by a pointwise convolution. The depthwise convolution is a spatial convolution applied independently over each channel of the input tensor, while the pointwise convolution is a standard convolution restricted to the use of 1 × 1 {\displaystyle 1\times 1} kernels. === Pooling layers === Convolutional networks may include local and/or global pooling layers along with traditional convolutional layers. Pooling layers reduce the dimensions of data by combining the outputs of neuron clusters at one layer into a single neuron in the next layer. Local pooling combines small clusters, tiling sizes such as 2 × 2 are commonly used. Global pooling acts on all the neurons of the feature map. There are two common types of pooling in popular use: max and average. Max pooling uses the maximum value of each local cluster of neurons in the feature map, while average pooling takes the average value. === Fully connected layers === Fully connected layers connect every neuron in one layer to every neuron in another layer. It is the same as a traditional multilayer perceptron neural network (MLP). Each neuron in the fully connected layer receives input from all the neurons in the previous layer. These inputs are weighted and summed with the corresponding biases, and then passed through an activation function to perform a nonlinear transformation, generating the output. The flattened matrix goes through a fully connected layer to classify the images. === Receptive field === In neural networks, each neuron receives input from some number of locations in the previous layer. In a convolutional layer, each neuron receives input from only a restricted area of the previous layer called the neuron's receptive field. Typically the area is a square (e.g. 5 by 5 neurons). Whereas, in a fully connected layer, the receptive field is the entire previous layer. Thus, in each convolutional layer, each neuron takes input from a larger area in the input than previous layers. This is due to applying the convolution over and over, which takes the value of a pixel into account, as well as its surrounding pixels. When using dilated layers, the number of pixels in the receptive field remains constant, but the field is more sparsely populated as its dimensions grow when combining the effect of several layers. To manipulate the receptive field size as desired, there are some alternatives to the standard convolutional layer. For example, atrous or dilated convolution expands the receptive field size without increasing the number of parameters by interleaving visible and blind regions. Moreover, a single dilated convolutional layer can comprise filters with multiple dilation ratios, thus having a variable receptive field size. === Weights === Each neuron in a neural network computes an output value by applying a specific function to the input values received from the receptive field in the previous layer. The function that is applied to the input values is determined by a vector of weights and a bias (typically real numbers). Learning consists of iteratively adjusting these biases and weights. The vectors of weights and biases are called filters and represent particular features of the input (e.g., a particular shape). A distinguishing feature of CNNs is that many neurons can share the same filter. This reduces the memory footprint because a single bias and a single vector of weights are used across all receptive fields that share that filter, as opposed to each receptive field having its own bias and vector