AI Code Programming

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  Spatial embedding

  Spatial embedding is one of feature learning techniques used in spatial analysis where points, lines, polygons or other spatial data types. representing geographic locations are mapped to vectors of real numbers. Conceptually it involves a mathematical embedding from a space with many dimensions per geographic object to a continuous vector space with a much lower dimension. Such embedding methods allow complex spatial data to be used in neural networks and have been shown to improve performance in spatial analysis tasks == Embedded data types == Geographic data can take many forms: text, images, graphs, trajectories, polygons. Depending on the task, there may be a need to combine multimodal data from different sources. The next section describes examples of different types of data and their uses. === Text === Geolocated posts on social media can be used to acquire a library of documents bound to a given place that can be later transformed to embedded vectors using word embedding techniques. === Image === Satellites and aircraft collect digital spatial data acquired from remotely sensed images which can be used in machine learning. They are sometimes hard to analyse using basic image analysis methods and convolutional neural networks can be used to acquire an embedding of images bound to a given geographical object or a region. === Point === A single point of interest (POI) can be assigned multiple features that can be used in machine learning. These could be demographic, transportation, meteorological, or economic data, for example. When embedding single points, it is common to consider the entire set of available points as nodes in a graph. === Line / multiline === Among other things, motion trajectories are represented as lines (multilines). Individual trajectories are embedded taking into account travel time, distances and also features of points visited along the way. Embedding of trajectories allows to improve performance of such tasks as clustering and also categorization. === Polygon === The geographic areas analyzed in machine learning are defined by both administrative boundaries and top-down division into grids of regular shapes such as rectangles, for example. Both types are represented as polygons and, like points, can be assigned different demographic, transportation, or economic features. A polygon can also have features related to the size of the area or shape it represents. === Graph === An example domain where graph representation is used is the street layout in a city, where vertices can be intersections and edges can be roads. The vertices can also be destination points like public transport stops or important points in the city, and the edges represent the flow between them. Embedding graphs or single vertices allows to improve accuracy of analysis methods in which the treated geographical domain can be represented as a network. == Usage == POI recommendation - generating personalized point of interest recommendations based on user preferences. Next/future location prediction - prediction of the next location a person will go to based on their historical trajectory. Zone functions classification - based on different mobility of people or POI distribution a function of a given area in a city can be predicted. Crime prediction - estimation of crime rate in different regions of a city. Local event detection - studying spatio-temporal changes in embeddings can provide valuable information in detection of local event occurring in specific location. Regional mobility popularity prediction - analysis of mobility can show patterns in popularity of different regions in a city. Shape matching - finding a similar shape of given polygon, for example finding building with the same shape as input building. Travel time estimation - predicting estimated travel time given current traffic conditions and special occurring events. Time estimation for on-demand food delivery - estimation of delivery time when placing an order through the website. == Temporal aspect == Some of the data analyzed has a timestamp associated with it. In some cases of data analysis this information is omitted and in others it is used to divide the set into groups. The most common division is the separation of weekdays from weekends or division into hours of the day. This is particularly important in the analysis of mobility data, because the characteristics of mobility during the week and at different times of the day are very different from each other. Another area in which time division into, for example, individual months can be used is in the analysis of tourism of a given region. In order to take such a split into account, embedding methods treat the time stamp specifically or separate versions of the model are developed for different subgroups of the analyzed set.

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  Tucker decomposition

  In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal. It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data. For a 3rd-order tensor T ∈ F n 1 × n 2 × n 3 {\displaystyle T\in F^{n_{1}\times n_{2}\times n_{3}}} , where F {\displaystyle F} is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , Tucker Decomposition can be denoted as follows, T = T × 1 U ( 1 ) × 2 U ( 2 ) × 3 U ( 3 ) {\displaystyle T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}} where T ∈ F d 1 × d 2 × d 3 {\displaystyle {\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}} is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of T {\displaystyle T} , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor T {\displaystyle {\mathcal {T}}} respectively. U ( 1 ) , U ( 2 ) , U ( 3 ) {\displaystyle U^{(1)},U^{(2)},U^{(3)}} are unitary matrices in F d 1 × n 1 , F d 2 × n 2 , F d 3 × n 3 {\displaystyle F^{d_{1}\times n_{1}},F^{d_{2}\times n_{2}},F^{d_{3}\times n_{3}}} respectively. The k-mode product (k = 1, 2, 3) of T {\displaystyle {\mathcal {T}}} by U ( k ) {\displaystyle U^{(k)}} is denoted as T × U ( k ) {\displaystyle {\mathcal {T}}\times U^{(k)}} with entries as ( T × 1 U ( 1 ) ) ( i 1 , j 2 , j 3 ) = ∑ j 1 = 1 d 1 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) ( T × 2 U ( 2 ) ) ( j 1 , i 2 , j 3 ) = ∑ j 2 = 1 d 2 T ( j 1 , j 2 , j 3 ) U ( 2 ) ( j 2 , i 2 ) ( T × 3 U ( 3 ) ) ( j 1 , j 2 , i 3 ) = ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle {\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(i_{1},j_{2},j_{3})&=\sum _{j_{1}=1}^{d_{1}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(j_{1},i_{2},j_{3})&=\sum _{j_{2}=1}^{d_{2}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(2)}(j_{2},i_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(j_{1},j_{2},i_{3})&=\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(3)}(j_{3},i_{3})\end{aligned}}} Altogether, the decomposition may also be written more directly as T ( i 1 , i 2 , i 3 ) = ∑ j 1 = 1 d 1 ∑ j 2 = 1 d 2 ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) U ( 2 ) ( j 2 , i 2 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle T(i_{1},i_{2},i_{3})=\sum _{j_{1}=1}^{d_{1}}\sum _{j_{2}=1}^{d_{2}}\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})U^{(2)}(j_{2},i_{2})U^{(3)}(j_{3},i_{3})} Taking d i = n i {\displaystyle d_{i}=n_{i}} for all i {\displaystyle i} is always sufficient to represent T {\displaystyle T} exactly, but often T {\displaystyle T} can be compressed or efficiently approximately by choosing d i < n i {\displaystyle d_{i} Read more →

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  Multilayer perceptron

  In deep learning, a multilayer perceptron (MLP) is a kind of modern feedforward neural network consisting of fully connected neurons with nonlinear activation functions, organized in layers, notable for being able to distinguish data that is not linearly separable. Modern neural networks are trained using backpropagation and are colloquially referred to as "vanilla" networks. MLPs grew out of an effort to improve on single-layer perceptrons, which could only be applied to linearly separable data. A perceptron traditionally used a Heaviside step function as its nonlinear activation function. However, the backpropagation algorithm requires that modern MLPs use continuous activation functions such as sigmoid or ReLU. Multilayer perceptrons form the basis of deep learning, and are applicable across a vast set of diverse domains. == Timeline == In 1943, Warren McCulloch and Walter Pitts proposed the binary artificial neuron as a logical model of biological neural networks. In 1958, Frank Rosenblatt proposed the multilayered perceptron model, consisting of an input layer, a hidden layer with randomized weights that did not learn, and an output layer with learnable connections. In 1962, Rosenblatt published many variants and experiments on perceptrons in his book Principles of Neurodynamics, including up to 2 trainable layers by "back-propagating errors". However, it was not the backpropagation algorithm, and he did not have a general method for training multiple layers. In 1965, Alexey Grigorevich Ivakhnenko and Valentin Lapa published Group Method of Data Handling. It was one of the first deep learning methods, used to train an eight-layer neural net in 1971. In 1967, Shun'ichi Amari reported the first multilayered neural network trained by stochastic gradient descent, was able to classify non-linearily separable pattern classes. Amari's student Saito conducted the computer experiments, using a five-layered feedforward network with two learning layers. Backpropagation was independently developed multiple times in early 1970s. The earliest published instance was Seppo Linnainmaa's master thesis (1970). Paul Werbos developed it independently in 1971, but had difficulty publishing it until 1982. In 1986, David E. Rumelhart et al. popularized backpropagation. In 2003, interest in backpropagation networks returned due to the successes of deep learning being applied to language modelling by Yoshua Bengio with co-authors. In 2021, a very simple NN architecture combining two deep MLPs with skip connections and layer normalizations was designed and called MLP-Mixer; its realizations featuring 19 to 431 millions of parameters were shown to be comparable to vision transformers of similar size on ImageNet and similar image classification tasks. == Mathematical foundations == === Activation function === If a multilayer perceptron has a linear activation function in all neurons, that is, a linear function that maps the weighted inputs to the output of each neuron, then linear algebra shows that any number of layers can be reduced to a two-layer input-output model. In MLPs some neurons use a nonlinear activation function that was developed to model the frequency of action potentials, or firing, of biological neurons. The two historically common activation functions are both sigmoids, and are described by y ( v i ) = tanh ⁡ ( v i ) and y ( v i ) = ( 1 + e − v i ) − 1 {\displaystyle y(v_{i})=\tanh(v_{i})~~{\textrm {and}}~~y(v_{i})=(1+e^{-v_{i}})^{-1}} . The first is a hyperbolic tangent that ranges from −1 to 1, while the other is the logistic function, which is similar in shape but ranges from 0 to 1. Here y i {\displaystyle y_{i}} is the output of the i {\displaystyle i} th node (neuron) and v i {\displaystyle v_{i}} is the weighted sum of the input connections. Alternative activation functions have been proposed, including the rectifier and softplus functions. More specialized activation functions include radial basis functions (used in radial basis networks, another class of supervised neural network models). In recent developments of deep learning the rectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numerical problems related to the sigmoids. === Layers === The MLP consists of three or more layers (an input and an output layer with one or more hidden layers) of nonlinearly-activating nodes. Since MLPs are fully connected, each node in one layer connects with a certain weight w i j {\displaystyle w_{ij}} to every node in the following layer. === Learning === Learning occurs in the perceptron by changing connection weights after each piece of data is processed, based on the amount of error in the output compared to the expected result. This is an example of supervised learning, and is carried out through backpropagation, a generalization of the least mean squares algorithm in the linear perceptron. We can represent the degree of error in an output node j {\displaystyle j} in the n {\displaystyle n} th data point (training example) by e j ( n ) = d j ( n ) − y j ( n ) {\displaystyle e_{j}(n)=d_{j}(n)-y_{j}(n)} , where d j ( n ) {\displaystyle d_{j}(n)} is the desired target value for n {\displaystyle n} th data point at node j {\displaystyle j} , and y j ( n ) {\displaystyle y_{j}(n)} is the value produced by the perceptron at node j {\displaystyle j} when the n {\displaystyle n} th data point is given as an input. The node weights can then be adjusted based on corrections that minimize the error in the entire output for the n {\displaystyle n} th data point, given by E ( n ) = 1 2 ∑ output node j e j 2 ( n ) {\displaystyle {\mathcal {E}}(n)={\frac {1}{2}}\sum _{{\text{output node }}j}e_{j}^{2}(n)} . Using gradient descent, the change in each weight w i j {\displaystyle w_{ij}} is Δ w j i ( n ) = − η ∂ E ( n ) ∂ v j ( n ) y i ( n ) {\displaystyle \Delta w_{ji}(n)=-\eta {\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}y_{i}(n)} where y i ( n ) {\displaystyle y_{i}(n)} is the output of the previous neuron i {\displaystyle i} , and η {\displaystyle \eta } is the learning rate, which is selected to ensure that the weights quickly converge to a response, without oscillations. In the previous expression, ∂ E ( n ) ∂ v j ( n ) {\displaystyle {\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}} denotes the partial derivate of the error E ( n ) {\displaystyle {\mathcal {E}}(n)} according to the weighted sum v j ( n ) {\displaystyle v_{j}(n)} of the input connections of neuron i {\displaystyle i} . The derivative to be calculated depends on the induced local field v j {\displaystyle v_{j}} , which itself varies. It is easy to prove that for an output node this derivative can be simplified to − ∂ E ( n ) ∂ v j ( n ) = e j ( n ) ϕ ′ ( v j ( n ) ) {\displaystyle -{\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}=e_{j}(n)\phi ^{\prime }(v_{j}(n))} where ϕ ′ {\displaystyle \phi ^{\prime }} is the derivative of the activation function described above, which itself does not vary. The analysis is more difficult for the change in weights to a hidden node, but it can be shown that the relevant derivative is − ∂ E ( n ) ∂ v j ( n ) = ϕ ′ ( v j ( n ) ) ∑ k − ∂ E ( n ) ∂ v k ( n ) w k j ( n ) {\displaystyle -{\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}=\phi ^{\prime }(v_{j}(n))\sum _{k}-{\frac {\partial {\mathcal {E}}(n)}{\partial v_{k}(n)}}w_{kj}(n)} . This depends on the change in weights of the k {\displaystyle k} th nodes, which represent the output layer. So to change the hidden layer weights, the output layer weights change according to the derivative of the activation function, and so this algorithm represents a backpropagation of the activation function.

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  Evolutionary algorithm

  Evolutionary algorithms (EA) reproduce essential elements of biological evolution in a computer algorithm in order to solve "difficult" problems, at least approximately, for which no exact or satisfactory solution methods are known. They are metaheuristics and population-based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, recombination and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolution (microevolutionary processes) and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor. In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems; therefore, there may be no direct link between algorithm complexity and problem complexity. == Generic definition == The following is an example of a generic evolutionary algorithm: Randomly generate the initial population of individuals, the first generation. Evaluate the fitness of each individual in the population. Check, if the goal is reached and the algorithm can be terminated. Select individuals as parents, preferably of higher fitness. Produce offspring with optional crossover (mimicking reproduction). Apply mutation operations on the offspring. Select individuals preferably of lower fitness for replacement with new individuals (mimicking natural selection). Return to 2 == Types == Similar techniques differ in genetic representation and other implementation details, and the nature of the particular applied problem. Genetic algorithm – This is the most popular type of EA. One seeks the solution of a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved), by applying operators such as recombination and mutation (sometimes one, sometimes both). This type of EA is often used in optimization problems. Genetic programming – Here the solutions are in the form of computer programs, and their fitness is determined by their ability to solve a computational problem. There are many variants of Genetic Programming: Cartesian genetic programming Gene expression programming Grammatical evolution Linear genetic programming Multi expression programming Evolutionary programming – Similar to evolution strategy, but with a deterministic selection of all parents. Evolution strategy (ES) – Works with vectors of real numbers as representations of solutions, and typically uses self-adaptive mutation rates. The method is mainly used for numerical optimization, although there are also variants for combinatorial tasks. CMA-ES Natural evolution strategy Differential evolution – Based on vector differences and is therefore primarily suited for numerical optimization problems. Coevolutionary algorithm – Similar to genetic algorithms and evolution strategies, but the created solutions are compared on the basis of their outcomes from interactions with other solutions. Solutions can either compete or cooperate during the search process. Coevolutionary algorithms are often used in scenarios where the fitness landscape is dynamic, complex, or involves competitive interactions. Neuroevolution – Similar to genetic programming but the genomes represent artificial neural networks by describing structure and connection weights. The genome encoding can be direct or indirect. Learning classifier system – Here the solution is a set of classifiers (rules or conditions). A Michigan-LCS evolves at the level of individual classifiers whereas a Pittsburgh-LCS uses populations of classifier-sets. Initially, classifiers were only binary, but now include real, neural net, or S-expression types. Fitness is typically determined with either a strength or accuracy based reinforcement learning or supervised learning approach. Quality–Diversity algorithms – QD algorithms simultaneously aim for high-quality and diverse solutions. Unlike traditional optimization algorithms that solely focus on finding the best solution to a problem, QD algorithms explore a wide variety of solutions across a problem space and keep those that are not just high performing, but also diverse and unique. == Theoretical background == The following theoretical principles apply to all or almost all EAs. === No free lunch theorem === The no free lunch theorem of optimization states that all optimization strategies are equally effective when the set of all optimization problems is considered. Under the same condition, no evolutionary algorithm is fundamentally better than another. This can only be the case if the set of all problems is restricted. This is exactly what is inevitably done in practice. Therefore, to improve an EA, it must exploit problem knowledge in some form (e.g. by choosing a certain mutation strength or a problem-adapted coding). Thus, if two EAs are compared, this constraint is implied. In addition, an EA can use problem specific knowledge by, for example, not randomly generating the entire start population, but creating some individuals through heuristics or other procedures. Another possibility to tailor an EA to a given problem domain is to involve suitable heuristics, local search procedures or other problem-related procedures in the process of generating the offspring. This form of extension of an EA is also known as a memetic algorithm. Both extensions play a major role in practical applications, as they can speed up the search process and make it more robust. === Convergence === For EAs in which, in addition to the offspring, at least the best individual of the parent generation is used to form the subsequent generation (so-called elitist EAs), there is a general proof of convergence under the condition that an optimum exists. Without loss of generality, a maximum search is assumed for the proof: From the property of elitist offspring acceptance and the existence of the optimum it follows that per generation k {\displaystyle k} an improvement of the fitness F {\displaystyle F} of the respective best individual x ′ {\displaystyle x'} will occur with a probability P > 0 {\displaystyle P>0} . Thus: F ( x 1 ′ ) ≤ F ( x 2 ′ ) ≤ F ( x 3 ′ ) ≤ ⋯ ≤ F ( x k ′ ) ≤ ⋯ {\displaystyle F(x'_{1})\leq F(x'_{2})\leq F(x'_{3})\leq \cdots \leq F(x'_{k})\leq \cdots } I.e., the fitness values represent a monotonically non-decreasing sequence, which is bounded due to the existence of the optimum. From this follows the convergence of the sequence against the optimum. Since the proof makes no statement about the speed of convergence, it is of little help in practical applications of EAs. But it does justify the recommendation to use elitist EAs. However, when using the usual panmictic population model, elitist EAs tend to converge prematurely more than non-elitist ones. In a panmictic population model, mate selection (see step 4 of the generic definition) is such that every individual in the entire population is eligible as a mate. In non-panmictic populations, selection is suitably restricted, so that the dispersal speed of better individuals is reduced compared to panmictic ones. Thus, the general risk of premature convergence of elitist EAs can be significantly reduced by suitable population models that restrict mate selection. === Virtual alphabets === With the theory of virtual alphabets, David E. Goldberg showed in 1990 that by using a representation with real numbers, an EA that uses classical recombination operators (e.g. uniform or n-point crossover) cannot reach certain areas of the search space, in contrast to a coding with binary numbers. This results in the recommendation for EAs with real representation to use arithmetic operators for recombination (e.g. arithmetic mean or intermediate recombination). With suitable operators, real-valued representations are more effective than binary ones, contrary to earlier opinion. == Comparison to other concepts == === Biological processes === A possible limitation of many evolutionary algorithms is their lack of a clear genotype–phenotype distinction. In nature, the fertilized egg cell undergoes a complex process known as embryogenesis to become a mature p

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  Belief–desire–intention model

  For popular psychology, the belief–desire–intention (BDI) model of human practical reasoning was developed by Michael Bratman as a way of explaining future-directed intention. BDI is fundamentally reliant on folk psychology (the 'theory theory'), which is the notion that our mental models of the world are theories. It was used as a basis for developing the belief–desire–intention software model. == Applications == BDI was part of the inspiration behind the BDI software architecture, which Bratman was also involved in developing. Here, the notion of intention was seen as a way of limiting time spent on deliberating about what to do, by eliminating choices inconsistent with current intentions. BDI has also aroused some interest in psychology. BDI formed the basis for a computational model of childlike reasoning CRIBB.

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  Spatial Analysis of Principal Components

  Spatial Principal Component Analysis (sPCA) is a multivariate statistical technique that complements the traditional Principal Component Analysis (PCA) by incorporating spatial information into the analysis of genetic variation. While traditional PCA can be used to find spatial patterns, it focuses on reducing data dimensionality by identifying uncorrelated principal components that capture maximum variance, thus often lacking power to identify non-trivial spatial genetic patterns. By accounting for spatial autocorrelation, sPCA is able to uncover spatial patterns in the data and find the spatial structure of datasets where observations are either geographically or topologically linked. This statistical power improvement allows the investigation of cryptic spatial patterns of genetic variability otherwise overlooked. sPCA has been applied in various fields, including geography, ecology and genetics. == History == sPCA was introduced in 2008 by Thibaut Jombart, Sébastien Devillard, Anne-Béatrice Dufour, and D. Pontier as a spatially explicit method to investigate the spatial pattern of genetic variation among individuals or populations. In 2017, Valeria Montano and Thibaut Jombart published an alternative non-parametric test to evaluate the significance of global and local spatial genetic patterns with improved statistical power. == Details == sPCA modifies the PCA framework by integrating spatial weights, typically in the form of connectivity matrices or spatial adjacency graphs. It identifies principal components (PCs) that maximize both genentic variance and spatial autocorreation, as measured by Moran's I. These weights represent relationships between observations based on geographic distance or other spatial criteria. The method decomposes variance into two components: Global structures, correspond to positive autocorrelation, that is, reflect broad-scale spatial patterns where similar values cluster over large regions. Local structures, correspond to negative autocorrelation, that is, capture fine-scale spatial variations or localized patterns. The core of sPCA relies on the eigenanalysis of a spatially weighted covariance or correlation matrix. The spatial weight matrix can be constructed using techniques such as Delaunay triangulation, nearest-neighbor graphs, or distance-based criteria. Applications of sPCA should be used only as an explorative tool. == Applications == sPCA has been widely used in many fields, including: Ecology: To find spatial patterns in species distributions and environmental gradients. Genetics: Population structure and gene flow analysis while allowing for spatial autocorrelation considerations. Biogeography: To identify historical dispersal routes, and barriers to gene flow, providing insights into species distribution patterns and evolutionary history. == Software/Source Code == sPCA implementations are available in R in adegenet and ntbox . These tools facilitate the application of sPCA by providing functions for constructing spatial weight matrices, performing eigenanalysis, and obtaining spatial principal components in an easy-to-read form.

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  Taguchi loss function

  The Taguchi loss function is graphical depiction of loss developed by the Japanese business statistician Genichi Taguchi to describe a phenomenon affecting the value of products produced by a company. Praised by Dr. W. Edwards Deming (the business guru of the 1980s American quality movement), it made clear the concept that quality does not suddenly plummet when, for instance, a machinist exceeds a rigid blueprint tolerance. Instead 'loss' in value progressively increases as variation increases from the intended condition. This was considered a breakthrough in describing quality, and helped fuel the continuous improvement movement. The concept of Taguchi's quality loss function was in contrast with the American concept of quality, popularly known as goal post philosophy, the concept given by American quality guru Phil Crosby. Goal post philosophy emphasizes that if a product feature doesn't meet the designed specifications it is termed as a product of poor quality (rejected), irrespective of amount of deviation from the target value (mean value of tolerance zone). This concept has similarity with the concept of scoring a 'goal' in the game of football or hockey, because a goal is counted 'one' irrespective of the location of strike of the ball in the 'goal post', whether it is in the center or towards the corner. This means that if the product dimension goes out of the tolerance limit the quality of the product drops suddenly. Through his concept of the quality loss function, Taguchi explained that from the customer's point of view this drop of quality is not sudden. The customer experiences a loss of quality the moment product specification deviates from the 'target value'. This 'loss' is depicted by a quality loss function and it follows a parabolic curve mathematically given by L = k(y–m)2, where m is the theoretical 'target value' or 'mean value' and y is the actual size of the product, k is a constant and L is the loss. This means that if the difference between 'actual size' and 'target value' i.e. (y–m) is large, loss would be more, irrespective of tolerance specifications. In Taguchi's view tolerance specifications are given by engineers and not by customers; what the customer experiences is 'loss'. This equation is true for a single product; if 'loss' is to be calculated for multiple products the loss function is given by L = k[S2 + ( y ¯ {\displaystyle {\bar {y}}} – m)2], where S2 is the 'variance of product size' and y ¯ {\displaystyle {\bar {y}}} is the average product size. == Overview == The Taguchi loss function is important for a number of reasons—primarily, to help engineers better understand the importance of designing for variation.

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  Mean squared prediction error

  In statistics the mean squared prediction error (MSPE), also known as mean squared error of the predictions, of a smoothing, curve fitting, or regression procedure is the expected value of the squared prediction errors (PE), the square difference between the fitted values implied by the predictive function g ^ {\displaystyle {\widehat {g}}} and the values of the (unobservable) true value g. It is an inverse measure of the explanatory power of g ^ , {\displaystyle {\widehat {g}},} and can be used in the process of cross-validation of an estimated model. Knowledge of g would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated. == Formulation == If the smoothing or fitting procedure has projection matrix (i.e., hat matrix) L, which maps the observed values vector y {\displaystyle y} to predicted values vector y ^ = L y , {\displaystyle {\hat {y}}=Ly,} then PE and MSPE are formulated as: P E i = g ( x i ) − g ^ ( x i ) , {\displaystyle \operatorname {PE_{i}} =g(x_{i})-{\widehat {g}}(x_{i}),} MSPE = E ⁡ [ PE i 2 ] = ∑ i = 1 n PE i 2 ⁡ / n . {\displaystyle \operatorname {MSPE} =\operatorname {E} \left[\operatorname {PE} _{i}^{2}\right]=\sum _{i=1}^{n}\operatorname {PE} _{i}^{2}/n.} The MSPE can be decomposed into two terms: the squared bias (mean error) of the fitted values and the variance of the fitted values: MSPE = ME 2 + VAR , {\displaystyle \operatorname {MSPE} =\operatorname {ME} ^{2}+\operatorname {VAR} ,} ME = E ⁡ [ g ^ ( x i ) − g ( x i ) ] {\displaystyle \operatorname {ME} =\operatorname {E} \left[{\widehat {g}}(x_{i})-g(x_{i})\right]} VAR = E ⁡ [ ( g ^ ( x i ) − E ⁡ [ g ( x i ) ] ) 2 ] . {\displaystyle \operatorname {VAR} =\operatorname {E} \left[\left({\widehat {g}}(x_{i})-\operatorname {E} \left[{g}(x_{i})\right]\right)^{2}\right].} The quantity SSPE=nMSPE is called sum squared prediction error. The root mean squared prediction error is the square root of MSPE: RMSPE=√MSPE. == Computation of MSPE over out-of-sample data == The mean squared prediction error can be computed exactly in two contexts. First, with a data sample of length n, the data analyst may run the regression over only q of the data points (with q < n), holding back the other n – q data points with the specific purpose of using them to compute the estimated model’s MSPE out of sample (i.e., not using data that were used in the model estimation process). Since the regression process is tailored to the q in-sample points, normally the in-sample MSPE will be smaller than the out-of-sample one computed over the n – q held-back points. If the increase in the MSPE out of sample compared to in sample is relatively slight, that results in the model being viewed favorably. And if two models are to be compared, the one with the lower MSPE over the n – q out-of-sample data points is viewed more favorably, regardless of the models’ relative in-sample performances. The out-of-sample MSPE in this context is exact for the out-of-sample data points that it was computed over, but is merely an estimate of the model’s MSPE for the mostly unobserved population from which the data were drawn. Second, as time goes on more data may become available to the data analyst, and then the MSPE can be computed over these new data. == Estimation of MSPE over the population == When the model has been estimated over all available data with none held back, the MSPE of the model over the entire population of mostly unobserved data can be estimated as follows. For the model y i = g ( x i ) + σ ε i {\displaystyle y_{i}=g(x_{i})+\sigma \varepsilon _{i}} where ε i ∼ N ( 0 , 1 ) {\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)} , one may write n ⋅ MSPE ⁡ ( L ) = g T ( I − L ) T ( I − L ) g + σ 2 tr ⁡ [ L T L ] . {\displaystyle n\cdot \operatorname {MSPE} (L)=g^{\text{T}}(I-L)^{\text{T}}(I-L)g+\sigma ^{2}\operatorname {tr} \left[L^{\text{T}}L\right].} Using in-sample data values, the first term on the right side is equivalent to ∑ i = 1 n ( E ⁡ [ g ( x i ) − g ^ ( x i ) ] ) 2 = E ⁡ [ ∑ i = 1 n ( y i − g ^ ( x i ) ) 2 ] − σ 2 tr ⁡ [ ( I − L ) T ( I − L ) ] . {\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[g(x_{i})-{\widehat {g}}(x_{i})\right]\right)^{2}=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\operatorname {tr} \left[\left(I-L\right)^{T}\left(I-L\right)\right].} Thus, n ⋅ MSPE ⁡ ( L ) = E ⁡ [ ∑ i = 1 n ( y i − g ^ ( x i ) ) 2 ] − σ 2 ( n − tr ⁡ [ L ] ) . {\displaystyle n\cdot \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-\operatorname {tr} \left[L\right]\right).} If σ 2 {\displaystyle \sigma ^{2}} is known or well-estimated by σ ^ 2 {\displaystyle {\widehat {\sigma }}^{2}} , it becomes possible to estimate MSPE by n ⋅ M S P E ^ ⁡ ( L ) = ∑ i = 1 n ( y i − g ^ ( x i ) ) 2 − σ ^ 2 ( n − tr ⁡ [ L ] ) . {\displaystyle n\cdot \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-\operatorname {tr} \left[L\right]\right).} Colin Mallows advocated this method in the construction of his model selection statistic Cp, which is a normalized version of the estimated MSPE: C p = ∑ i = 1 n ( y i − g ^ ( x i ) ) 2 σ ^ 2 − n + 2 p . {\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2p.} where p the number of estimated parameters p and σ ^ 2 {\displaystyle {\widehat {\sigma }}^{2}} is computed from the version of the model that includes all possible regressors. That concludes this proof.

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  Web Intents

  Web Intents was an experimental framework for web-based inter-application communication and service discovery. Web Intents consists of a discovery mechanism and a very light-weight RPC system between web applications, modelled after the Intents system in Android. In the context of the framework an Intent equals an action to be performed by a provider. Web Intents allow two web applications to communicate with each other, without either of them having to actually know what the other one is. == Support == === Client === Google Chrome versions 18 to 23 natively supported Web Intents. This support was disabled in version 24, citing the existence of a "number of areas for development in both the API and specific user experience in Chrome". There is a JavaScript shim with support for IE 8, IE 9, Opera, Safari, Firefox 3+ and Chrome 3+. === Server === There are some Web Intents proxy pages that make available some real services that don't yet support intents. AddThis supports Web Intents by their sharing tools regardless of browser support. == History == Paul Kinlan of Google announced the Web Intents project in December 2010. He soon released a prototype API to GitHub. In August 2011 Google announced that Chrome would support Web Intents. Google and Mozilla have started co-operating to unify Web Intents and Mozilla's Web Activities (which tries to solve the same problem) into one proposal. In November 2012, Greg Billock of Google announced that experimental support of Web Intents had been removed from Chrome.

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  Andrej Mrvar

  Andrej Mrvar is a Slovenian computer scientist and a professor at the University of Ljubljana's Faculty of Social Sciences. He is known for his work in network analysis, graph drawing, decision making, virtual reality, timing and data processing of sports competitions. == Education and career == He is well known for his work on Pajek, a free software for analysis and visualization of large networks. Mrvar began work on Pajek in 1996 with Vladimir Batagelj. His book Exploratory Social Network Analysis with Pajek, coauthored with Wouter de Nooy and Vladimir Batagelj, is his most cited work. It was published by Cambridge University Press in three editions (first 2005, second 2011, and third 2018). The book was translated into Japanese (2009) and Chinese (first edition 2012, second 2014). With Anuška Ferligoj, he was a founding co-editor-in-chief of the Metodološki zvezki - Advances in Methodology and Statistics journal. == Awards and honors == Vidmar Award (Faculty of Electrical and Computer Engineering, University of Ljubljana): 1988, 1990 First prizes for contributions (with Vladimir Batagelj) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. Award of University of Ljubljana for contributions in education and research (Svečana listina Univerze v Ljubljani za pomembne dosežke na področju vzgojnoizobraževalnega in znanstvenoraziskovalega dela): 2001 The INSNA's William D. Richards Software award for work on Pajek (with Vladimir Batagelj): 2013 Award of Faculty of Social Sciences, University of Ljubljana for scientific excellence (Priznanje za znanstveno odličnost): 2013 == Selected publications == Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press (First Edition: 2005, Second Edition: 2011, Third Edition: 2018 ). Japanese Translation (2010). Chinese Translation (First Edition: 2012, Second Edition: 2014) Andrej Mrvar and Vladimir Batagelj, Analysis and visualization of large networks with program package Pajek. Complex Adaptive Systems Modeling, 4:6. SpringerOpen, 2016 Vladimir Batagelj and Andrej Mrvar, Some Analyses of Erdős Collaboration Graph, Social Networks, 22, 173–186, 2000 Vladimir Batagelj and Andrej Mrvar, A Subquadratic Triad Census Algorithm for Large Sparse Networks with Small Maximum Degree. Social Networks, 23, 237–243, 2001 Patrick Doreian and Andrej Mrvar, A Partitioning Approach to Structural Balance, Social Networks, 18, 149–168, 1996 Patrick Doreian and Andrej Mrvar, Partitioning Signed Social Networks, Social Networks, 31, 1–11, 2009 Andrej Mrvar and Patrick Doreian, Partitioning Signed Two-Mode Networks, Journal of Mathematical Sociology, 33, 196–221, 2009 Patrick Doreian and Andrej Mrvar, The international reach of the Koch brothers network. In: Antonyuk, A. and Basov, N. (Eds.): Networks in the Global World V. NetGloW 2020. Lecture Notes in Networks and Systems, 181, 225–235. Springer, 2021 Patrick Doreian and Andrej Mrvar, Delineating Changes in the Fundamental Structure of Signed Networks, Frontiers in Physics, 294, 1–11, 2021 Patrick Doreian and Andrej Mrvar, Hubs and Authorities in the Koch Brothers Network. Social Networks, Social Networks, 64, 148–157, 2021 Patrick Doreian and Andrej Mrvar, Public issues, policy proposals, social movements, and the interests of the Koch Brothers network of allies, Quality and Quantity, 56, 305–322, 2022 Douglas R. White, Vladimir Batagelj, Andrej Mrvar, Analyzing Large Kinship and Marriage Networks with Pgraph and Pajek. Social Science Computer Review, 17, 245–274, 1999 Ion Georgiou, Ronald Concer, Andrej Mrvar, A Systemic Approach to Sociometric Group Research: Advancing The Work of Leslie Day Zeleny, 1939–1947, Social Networks, 63, 174–200, 2020

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  Silhouette (clustering)

  Silhouette is a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette value ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. A clustering with an average silhouette width of over 0.7 is considered to be "strong", a value over 0.5 "reasonable", and over 0.25 "weak". However, with an increasing dimensionality of the data, it becomes difficult to achieve such high values because of the curse of dimensionality, as the distances become more similar. The silhouette score is specialized for measuring cluster quality when the clusters are convex-shaped, and may not perform well if the data clusters have irregular shapes or are of varying sizes. The silhouette value can be calculated with any distance metric, such as Euclidean distance or Manhattan distance. == Definition == Assume the data have been clustered via any technique, such as k-medoids or k-means, into k {\displaystyle k} clusters. For data point i ∈ C i {\displaystyle i\in C_{i}} (data point i {\displaystyle i} in the cluster C i {\displaystyle C_{i}} ), calculate a ( i ) {\displaystyle a(i)} , the average distance that i {\displaystyle i} is from all other points in that cluster: a ( i ) = 1 | C i | − 1 ∑ j ∈ C i , i ≠ j d ( i , j ) {\displaystyle a(i)={\frac {1}{|C_{i}|-1}}\sum _{j\in C_{i},i\neq j}d(i,j)} where | C i | {\displaystyle |C_{i}|} is the number of points belonging to cluster C i {\displaystyle C_{i}} , and d ( i , j ) {\displaystyle d(i,j)} is the distance between data points i {\displaystyle i} and j {\displaystyle j} in the cluster C i {\displaystyle C_{i}} (we divide by | C i | − 1 {\displaystyle |C_{i}|-1} because the distance d ( i , i ) {\displaystyle d(i,i)} is not included in the sum). a ( i ) {\displaystyle a(i)} can be interpreted as a measure of how well i {\displaystyle i} is assigned to its cluster (the smaller the value, the better the assignment). We then define the mean dissimilarity of point i {\displaystyle i} to some cluster C j {\displaystyle C_{j}} as the mean of the distance from i {\displaystyle i} to all points in C j {\displaystyle C_{j}} (where C j ≠ C i {\displaystyle C_{j}\neq C_{i}} ). For each data point i ∈ C i {\displaystyle i\in C_{i}} , we now define b ( i ) {\displaystyle b(i)} as the average distance between i {\displaystyle i} and the points in the closest cluster (hence: "min") that i {\displaystyle i} does not belong to: b ( i ) = min j ≠ i 1 | C j | ∑ l ∈ C j d ( i , l ) {\displaystyle b(i)=\min _{j\neq i}{\frac {1}{|C_{j}|}}\sum _{l\in C_{j}}d(i,l)} The cluster with the smallest mean dissimilarity is said to be the "neighboring cluster" of i {\displaystyle i} because it is the next best fit cluster for point i {\displaystyle i} . We now define a silhouette (value) of one data point i {\displaystyle i} s ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } {\displaystyle s(i)={\frac {b(i)-a(i)}{\max\{a(i),b(i)\}}}} , if | C i | > 1 {\displaystyle |C_{i}|>1} and s ( i ) = 0 {\displaystyle s(i)=0} , if | C i | = 1 {\displaystyle |C_{i}|=1} , which can also be written as s ( i ) = { 1 − a ( i ) b ( i ) , if a ( i ) < b ( i ) 0 , if a ( i ) = b ( i ) b ( i ) a ( i ) − 1 , if a ( i ) > b ( i ) {\displaystyle s(i)={\begin{cases}1-{\frac {a(i)}{b(i)}},&{\mbox{ if }}a(i)b(i)\\\end{cases}}} From the above definition, s ( i ) {\displaystyle s(i)} is bounded to the interval [ − 1 , 1 ] {\displaystyle [-1,1]} , i.e. − 1 ≤ s ( i ) ≤ 1. {\displaystyle -1\leq s(i)\leq 1.} Note that a ( i ) {\displaystyle a(i)} is not clearly defined for clusters with size = 1, in which case we set s ( i ) = 0 {\displaystyle s(i)=0} . This choice is arbitrary, but neutral in the sense that it is at the midpoint of the bounds, -1 and 1. For s ( i ) {\displaystyle s(i)} to be close to 1 we require a ( i ) ≪ b ( i ) {\displaystyle a(i)\ll b(i)} . As a ( i ) {\displaystyle a(i)} is a measure of how dissimilar i {\displaystyle i} is to its own cluster, a small value means it is well matched. Furthermore, a large b ( i ) {\displaystyle b(i)} implies that i {\displaystyle i} is badly matched to its neighbouring cluster. Thus an s ( i ) {\displaystyle s(i)} close to 1 means that the data is appropriately clustered. If s ( i ) {\displaystyle s(i)} is close to -1, then by the same logic we see that i {\displaystyle i} would be more appropriate if it was clustered in its neighbouring cluster. An s ( i ) {\displaystyle s(i)} near zero means that the datum is on the border of two natural clusters. The mean s ( i ) {\displaystyle s(i)} over all points of a cluster is a measure of how tightly grouped all the points in the cluster are. Thus the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset is a measure of how appropriately the data have been clustered. If there are too many or too few clusters, as may occur when a poor choice of k {\displaystyle k} is used in the clustering algorithm (e.g., k-means), some of the clusters will typically display much narrower silhouettes than the rest. Thus silhouette plots and means may be used to determine the natural number of clusters within a dataset. One can also increase the likelihood of the silhouette being maximized at the correct number of clusters by re-scaling the data using feature weights that are cluster specific. Kaufman et al. introduced the term silhouette coefficient for the maximum value of the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset, i.e., S C = max k s ~ ( k ) , {\displaystyle SC=\max _{k}{\tilde {s}}\left(k\right),} where s ~ ( k ) {\displaystyle {\tilde {s}}\left(k\right)} represents the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset for a specific number of clusters k {\displaystyle k} . The silhouette coefficient describes the best possible clustering possible for a given number of clusters, as measured by the highest average silhouette score for all points in the dataset. == Simplified and medoid silhouette == Computing the silhouette coefficient needs all O ( N 2 ) {\displaystyle {\mathcal {O}}(N^{2})} pairwise distances, making this evaluation much more costly than clustering with k-means. For a clustering with centers μ C I {\displaystyle \mu _{C_{I}}} for each cluster C I {\displaystyle C_{I}} , we can use the following simplified Silhouette for each point i ∈ C I {\displaystyle i\in C_{I}} instead, which can be computed using only O ( N k ) {\displaystyle {\mathcal {O}}(Nk)} distances: a ′ ( i ) = d ( i , μ C I ) {\displaystyle a'(i)=d(i,\mu _{C_{I}})} and b ′ ( i ) = min C J ≠ C I d ( i , μ C J ) {\displaystyle b'(i)=\min _{C_{J}\neq C_{I}}d(i,\mu _{C_{J}})} , which has the additional benefit that a ′ ( i ) {\displaystyle a'(i)} is always defined, then define accordingly the simplified silhouette and simplified silhouette coefficient s ′ ( i ) = b ′ ( i ) − a ′ ( i ) max { a ′ ( i ) , b ′ ( i ) } {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{\max\{a'(i),b'(i)\}}}} S C ′ = max k 1 N ∑ i s ′ ( i ) {\displaystyle SC'=\max _{k}{\frac {1}{N}}\sum _{i}s'\left(i\right)} . If the cluster centers are medoids (as in k-medoids clustering) instead of arithmetic means (as in k-means clustering), this is also called the medoid-based silhouette or medoid silhouette. If every object is assigned to the nearest medoid (as in k-medoids clustering), we know that a ′ ( i ) ≤ b ′ ( i ) {\displaystyle a'(i)\leq b'(i)} , and hence s ′ ( i ) = b ′ ( i ) − a ′ ( i ) b ′ ( i ) = 1 − a ′ ( i ) b ′ ( i ) {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{b'(i)}}=1-{\frac {a'(i)}{b'(i)}}} . == Silhouette clustering == Instead of using the average silhouette to evaluate a clustering obtained from, e.g., k-medoids or k-means, we can try to directly find a solution that maximizes the Silhouette. We do not have a closed form solution to maximize this, but it will usually be best to assign points to the nearest cluster as done by these methods. Van der Laan et al. proposed to adapt the standard algorithm for k-medoids, PAM, for this purpose and call this algorithm PAMSIL: Choose initial medoids by using PAM Compute the average silhouette of this initial solution For each pair of a medoid m and a non-medoid x swap m and x compute the average silhouette of the resulting solution remember the best swap un-swap m and x for the next iteration Perform the best swap and return to

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  Variational autoencoder

  In machine learning, a variational autoencoder (VAE) is an artificial neural network architecture introduced by Diederik P. Kingma and Max Welling in 2013. It is part of the families of probabilistic graphical models and variational Bayesian methods. In addition to being seen as an autoencoder neural network architecture, variational autoencoders can also be studied within the mathematical formulation of variational Bayesian methods, connecting a neural encoder network to its decoder through a probabilistic latent space (for example, as a multivariate Gaussian distribution) that corresponds to the parameters of a variational distribution. Thus, the encoder maps each point (such as an image) from a large complex dataset into a distribution within the latent space, rather than to a single point in that space. The decoder has the opposite function, which is to map from the latent space to the input space, again according to a distribution (although in practice, noise is rarely added during the decoding stage). By mapping a point to a distribution instead of a single point, the network can avoid overfitting the training data. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately. Although this type of model was initially designed for unsupervised learning, its effectiveness has been proven for semi-supervised learning and supervised learning. == Overview of architecture and operation == A variational autoencoder is a generative model with a prior and noise distribution respectively. Usually such models are trained using the expectation-maximization meta-algorithm (e.g. probabilistic PCA, (spike & slab) sparse coding). Such a scheme optimizes a lower bound of the data likelihood, which is usually computationally intractable, and in doing so requires the discovery of q-distributions, or variational posteriors. These q-distributions are normally parameterized for each individual data point in a separate optimization process. However, variational autoencoders use a neural network as an amortized approach to jointly optimize across data points. In that way, the same parameters are reused for multiple data points, which can result in massive memory savings. The first neural network takes as input the data points themselves, and outputs parameters for the variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder. The decoder is the second neural network of this model. It is a function that maps from the latent space to the input space, e.g. as the means of the noise distribution. It is possible to use another neural network that maps to the variance, however this can be omitted for simplicity. In such a case, the variance can be optimized with gradient descent. To optimize this model, one needs to know two terms: the "reconstruction error", and the Kullback–Leibler divergence (KL-D). Both terms are derived from the free energy expression of the probabilistic model, and therefore differ depending on the noise distribution and the assumed prior of the data, here referred to as p-distribution. For example, a standard VAE task such as IMAGENET is typically assumed to have a gaussianly distributed noise; however, tasks such as binarized MNIST require a Bernoulli noise. The KL-D from the free energy expression maximizes the probability mass of the q-distribution that overlaps with the p-distribution, which unfortunately can result in mode-seeking behaviour. The "reconstruction" term is the remainder of the free energy expression, and requires a sampling approximation to compute its expectation value. More recent approaches replace Kullback–Leibler divergence (KL-D) with various statistical distances, see "Statistical distance VAE variants" below. == Formulation == From the point of view of probabilistic modeling, one wants to maximize the likelihood of the data x {\displaystyle x} by their chosen parameterized probability distribution p θ ( x ) = p ( x | θ ) {\displaystyle p_{\theta }(x)=p(x|\theta )} . This distribution is usually chosen to be a Gaussian N ( x | μ , σ ) {\displaystyle N(x|\mu ,\sigma )} which is parameterized by μ {\displaystyle \mu } and σ {\displaystyle \sigma } respectively, and as a member of the exponential family it is easy to work with as a noise distribution. Simple distributions are easy enough to maximize, however distributions where a prior is assumed over the latents z {\displaystyle z} results in intractable integrals. Let us find p θ ( x ) {\displaystyle p_{\theta }(x)} via marginalizing over z {\displaystyle z} . p θ ( x ) = ∫ z p θ ( x , z ) d z , {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x,z})\,dz,} where p θ ( x , z ) {\displaystyle p_{\theta }({x,z})} represents the joint distribution under p θ {\displaystyle p_{\theta }} of the observable data x {\displaystyle x} and its latent representation or encoding z {\displaystyle z} . According to the chain rule, the equation can be rewritten as p θ ( x ) = ∫ z p θ ( x | z ) p θ ( z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz} In the vanilla variational autoencoder, z {\displaystyle z} is usually taken to be a finite-dimensional vector of real numbers, and p θ ( x | z ) {\displaystyle p_{\theta }({x|z})} to be a Gaussian distribution. Then p θ ( x ) {\displaystyle p_{\theta }(x)} is a mixture of Gaussian distributions. It is now possible to define the set of the relationships between the input data and its latent representation as Prior p θ ( z ) {\displaystyle p_{\theta }(z)} Likelihood p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} Posterior p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} Unfortunately, the computation of p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} is expensive and in most cases intractable. To speed up the calculus to make it feasible, it is necessary to introduce a further function to approximate the posterior distribution as q ϕ ( z | x ) ≈ p θ ( z | x ) {\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})} with ϕ {\displaystyle \phi } defined as the set of real values that parametrize q {\displaystyle q} . This is sometimes called amortized inference, since by "investing" in finding a good q ϕ {\displaystyle q_{\phi }} , one can later infer z {\displaystyle z} from x {\displaystyle x} quickly without doing any integrals. In this way, the problem is to find a good probabilistic autoencoder, in which the conditional likelihood distribution p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is computed by the probabilistic decoder, and the approximated posterior distribution q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is computed by the probabilistic encoder. Parametrize the encoder as E ϕ {\displaystyle E_{\phi }} , and the decoder as D θ {\displaystyle D_{\theta }} . == Evidence lower bound (ELBO) == Like many deep learning approaches that use gradient-based optimization, VAEs require a differentiable loss function to update the network weights through backpropagation. For variational autoencoders, the idea is to jointly optimize the generative model parameters θ {\displaystyle \theta } to reduce the reconstruction error between the input and the output, and ϕ {\displaystyle \phi } to make q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} as close as possible to p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . As reconstruction loss, mean squared error and cross entropy are often used. The Kullback–Leibler divergence D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) {\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))} can be used as a loss function to squeeze q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} under p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . This divergence loss expands to D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( z | x ) ] = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x ) p θ ( x , z ) ] = ln ⁡ p θ ( x ) + E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x , z ) ] . {\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right].\end{aligned}}} Now, define the evidence lower bound (ELBO): L θ , ϕ ( x ) := E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ p θ ( x , z ) q ϕ ( z | x ) ] = ln ⁡ p θ ( x ) − D K L ( q ϕ ( ⋅ | x ) ∥ p θ ( ⋅ | x ) ) {\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))} Maximizing the ELBO θ ∗ , ϕ ∗ = argmax θ , ϕ L θ , ϕ ( x ) {\dis

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  Lexical choice

  Lexical choice is the subtask of Natural language generation that involves choosing the content words (nouns, non-auxiliary verbs, adjectives, and adverbs) in a generated text. Function words (determiners, for example) are usually chosen during realisation. == Examples == The simplest type of lexical choice involves mapping a domain concept (perhaps represented in an ontology) to a word. For example, the concept Finger might be mapped to the word finger. A more complex situation is when a domain concept is expressed using different words in different situations. For example, the domain concept Value-Change can be expressed in many ways: The temperature rose: the verb rose is used for a Value-Change in temperature which increases the value. The temperature fell: the verb fell is used for a Value-Change in temperature which decreases the value. The rain got heavier: the phrase got heavier is used for a Value-Change in precipitation amount when the precipitation is rain. Sometimes words can communicate additional contextual information, for example: The temperature plummeted: the verb plummeted is used for a Value-Change in temperature which decreases the value, when the change is rapid and large. Contextual information is especially significant for vague terms such as tall. For example, a 2m tall man is tall, but a 2m tall horse is small. == Linguistic perspective == Lexical choice modules must be informed by linguistic knowledge of how the system's input data maps onto words. This is a question of semantics, but it is also influenced by syntactic factors (such as collocation effects) and pragmatic factors (such as context). Hence NLG systems need linguistic models of how meaning is mapped to words in the target domain (genre) of the NLG system. Genre tends to be very important; for example the verb veer has a very specific meaning in weather forecasts (wind direction is changing in a clockwise direction) which it does not have in general English, and a weather-forecast generator must be aware of this genre-specific meaning. In some cases there are major differences in how different people use the same word; for example, some people use by evening to mean 6PM and others use it to mean midnight. Psycholinguists have shown that when people speak to each other, they agree on a common interpretation via lexical alignment; this is not something which NLG systems can yet do. Ultimately, lexical choice must deal with the fundamental issue of how language relates to the non-linguistic world. For example, a system which chose colour terms such as red to describe objects in a digital image would need to know which RGB pixel values could generally be described as red; how this was influenced by visual (lighting, other objects in the scene) and linguistic (other objects being discussed) context; what pragmatic connotations were associated with red (for example, when an apple is called red, it is assumed to be ripe as well as have the colour red); and so forth. == Algorithms and models == A number of algorithms and models have been developed for lexical choice in the research community, for example Edmonds developed a model for choosing between near-synonyms (words with similar core meanings but different connotations). However such algorithms and models have not been widely used in applied NLG systems; such systems have instead often used quite simple computational models, and invested development effort in linguistic analysis instead of algorithm development.

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  Rectified linear unit

  In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU ⁡ ( x ) = x + = max ( 0 , x ) = x + | x | 2 = { x if x > 0 , 0 x ≤ 0 {\displaystyle \operatorname {ReLU} (x)=x^{+}=\max(0,x)={\frac {x+|x|}{2}}={\begin{cases}x&{\text{if }}x>0,\\0&x\leq 0\end{cases}}} where x {\displaystyle x} is the input to a neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognition using deep neural nets and computational neuroscience. == History == The ReLU was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used ReLU in the context of visual feature extraction in hierarchical neural networks. In 1998, Gregory Woodbury demonstrated that the rectified linear function could account for a broad range of emergent properties in the visual cortex. His work showed that a single unified model could drive the joint development of refined retinotopic maps, ocular dominance columns, and orientation selectivity. By utilizing the rectifier's "cutoff" property, Woodbury achieved a close quantitative fit to biological data, matching the spatial periodicities and topographic refinement patterns observed in macaque and cat cortical maps. Furthermore, he extended this framework to adult plasticity, accurately replicating the spatial and temporal dynamics of lesion-induced cortical reorganization. This research established that the rectified linear response was a necessary mechanism for the stable self-organisation and maintenance of complex, multi-feature neural maps. In 2000, Hahnloser et al. argued that ReLU approximates the biological relationship between neural firing rates and input current, in addition to enabling recurrent neural network dynamics to stabilise under weaker criteria. Prior to 2010, most activation functions used were the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more numerically efficient counterpart, the hyperbolic tangent. Around 2010, the use of ReLU became common again. Jarrett et al. (2009) noted that rectification by either absolute or ReLU (which they called "positive part") was critical for object recognition in convolutional neural networks (CNNs), specifically because it allows average pooling without neighboring filter outputs cancelling each other out. They hypothesized that the use of sigmoid or tanh was responsible for poor performance in previous CNNs. Nair and Hinton (2010) made a theoretical argument that the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by considering a single binary neuron in a Boltzmann machine that takes x {\displaystyle x} as input, and produces 1 as output with probability σ ( x ) = 1 1 + e − x {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}} . They then considered extending its range of output by making infinitely many copies of it X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } , that all take the same input, offset by an amount 0.5 , 1.5 , 2.5 , … {\displaystyle 0.5,1.5,2.5,\dots } , then their outputs are added together as ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} . They then demonstrated that ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} is approximately equal to N ( log ⁡ ( 1 + e x ) , σ ( x ) ) {\displaystyle {\mathcal {N}}(\log(1+e^{x}),\sigma (x))} , which is also approximately equal to ReLU ⁡ ( N ( x , σ ( x ) ) ) {\displaystyle \operatorname {ReLU} ({\mathcal {N}}(x,\sigma (x)))} , where N {\displaystyle {\mathcal {N}}} stands for the gaussian distribution. They also argued for another reason for using ReLU: that it allows "intensity equivariance" in image recognition. That is, multiplying input image by a constant k {\displaystyle k} multiplies the output also. In contrast, this is false for other activation functions like sigmoid or tanh. They found that ReLU activation allowed good empirical performance in restricted Boltzmann machines. Glorot et al (2011) argued that ReLU has the following advantages over sigmoid or tanh: ReLU is more similar to biological neurons' responses in their main operating regime. ReLU avoids vanishing gradients. ReLU is cheaper to compute. ReLU creates sparse representation naturally, because many hidden units output exactly zero for a given input. They also found empirically that deep networks trained with ReLU can achieve strong performance without unsupervised pre-training, especially on large, purely supervised tasks. In 2017, the rectified linear function became a central component of the transformer architecture introduced in the Vaswani et al paper "Attention Is All You Need". Within every transformer layer, ReLU is utilized in the position-wise feed-forward networks (FFN), defined by Equation 2 of their paper: FFN ⁡ ( x ) = max ( 0 , x W 1 + b 1 ) W 2 + b 2 {\displaystyle \operatorname {FFN} (x)=\max(0,xW_{1}+b_{1})W_{2}+b_{2}} This equation is foundational to the model's capacity; while the attention mechanism determines the relationships between tokens, the ReLU-based FFN performs the majority of the numerical computation and houses the bulk of the model's parameters. The efficiency and scalability of this rectified framework triggered a global technological revolution, enabling the development of Large Language Models that have had a profound economic impact. The industrial response to this architecture—including the massive expansion of AI-specific hardware and the birth of the generative AI sector—has positioned the Transformer as a cornerstone of 21st-century infrastructure. During the post 2017 period of rapid AI advancement, the rectified linear unit function has been key to achieving increased model performance and scaling due to the fact that it zeros out responses that are immaterial for a given stimuli, preventing them from accumulating in massive scale models. It is the complete silencing of the parts of the model found to be stimuli-irrelevant during learning that allows for scaling. As the stimuli-irrelevant proportion of the model becomes more massive, these highly numerous connections within the model would inevitably accumulate during scaling no matter how small each individual response is. Therefore, the rectified linear unit function, with its absolute zeroing property, enabled the scaling to hundred billion parameter models and beyond. Early Transformer scaling giants like GPT-3 (2020) and Falcon-180B (2023) relied on the rectified linear unit function explicitly, while successors such as GPT-4 (2023) and Llama 3 (2024) utilized smoother variants like GELU or SwiGLU. These variants were used to improve training stability while fundamentally preserving the rectified principle of zeroing low responses. At the centre of modern artificial intelligence ReLU and its variants maintain absolute zero response across the bulk of the model at any one time, while maintaining approximately linear reponses for stimuli-relevant connections enabling high performance on each specific cognitive task. This feature of activation sparsity has been critical for massive scaling and performance gains of AI models right up to the present day. == Advantages == Advantages of ReLU include: Sparse activation: for example, in a randomly initialized network, only about 50% of hidden units are activated (i.e. have a non-zero output). Better gradient propagation: fewer vanishing gradient problems compared to sigmoidal activation functions that saturate in both directions. Efficiency: only requires comparison and addition. Scale-invariant (homogeneous, or "intensity equivariance"): max ( 0 , a x ) = a max ( 0 , x ) for a ≥ 0 {\displaystyle \max(0,ax)=a\max(0,x){\text{ for }}a\geq 0} . == Potential problems == Possible downsides can include: Non-differentiability at zero (however, it is differentiable anywhere else, and the value of the derivative at zero can be chosen to be 0 or 1 arbitrarily). Not zero-centered: ReLU outputs are always non-negative. This can make it harder for the network to learn during backpropagation, because gradient updates tend to push weights in one direction (positive or negative). Batch normalization can help address this. ReLU is unbounded. Redundancy of the parametrization: Because ReLU is scale-invariant, the network computes the exact same function by scaling the weights and biases in front of a ReLU activation by k {\displaystyle k} , and the weights after by 1 / k {\displaystyle 1/k} . Dying ReLU: ReLU neurons can sometimes be pushed into states

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  Types of artificial neural networks

  Types of neural networks (NN) include a family of techniques. The simplest types have static components, including number of units, number of layers, unit weights and topology. Dynamic NNs evolve via learning. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. The main types are: Transformers: these use attention to analyze every token in the input stream against every other token in the stream. That technique has enabled neural networks to reach the general public via chatbots, code generators and many other forms. Convolutional neural networks (CNN): a FNN that uses kernels and regularization to evade problems in prior generations of NNs. They are typically used to analyze visual and other two-dimensional data. Generative adversarial networks set networks (of varying structure) against each other, each trying to push the other(s) to produce better results such as winning a game or to deceive the opponent about the authenticity of an input. == Feedforward == In feedforward neural networks the information moves from the input to output directly in every layer. There can be hidden layers with or without cycles/loops to sequence inputs. Feedforward networks can be constructed with various types of units, such as binary McCulloch–Pitts neurons, the simplest of which is the perceptron. Continuous neurons, frequently with sigmoidal activation, are used in the context of backpropagation. == Group method of data handling == The Group Method of Data Handling (GMDH) features fully automatic structural and parametric model optimization. The node activation functions are Kolmogorov–Gabor polynomials that permit additions and multiplications. It uses a deep multilayer perceptron with eight layers. It is a supervised learning network that grows layer by layer, where each layer is trained by regression analysis. Useless items are detected using a validation set, and pruned through regularization. The size and depth of the resulting network depends on the task. == Autoencoder == An autoencoder, autoassociator or Diabolo network is similar to the multilayer perceptron (MLP) – with an input layer, an output layer and one or more hidden layers connecting them. However, the output layer has the same number of units as the input layer. Its purpose is to reconstruct its own inputs (instead of emitting a target value). Therefore, autoencoders are unsupervised learning models. An autoencoder is used for unsupervised learning of efficient codings, typically for the purpose of dimensionality reduction and for learning generative models of data. == Probabilistic == A probabilistic neural network (PNN) is a four-layer feedforward neural network. The layers are Input, hidden pattern, hidden summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each class, the class probability of a new input is estimated and Bayes’ rule is employed to allocate it to the class with the highest posterior probability. It was derived from the Bayesian network and a statistical algorithm called Kernel Fisher discriminant analysis. It is used for classification and pattern recognition. == Time delay == A time delay neural network (TDNN) is a feedforward architecture for sequential data that recognizes features independent of sequence position. In order to achieve time-shift invariance, delays are added to the input so that multiple data points (points in time) are analyzed together. It usually forms part of a larger pattern recognition system. It has been implemented using a perceptron network whose connection weights were trained with back propagation (supervised learning). == Convolutional == A convolutional neural network (CNN, or ConvNet or shift invariant or space invariant) is a class of deep network, composed of one or more convolutional layers with fully connected layers (matching those in typical ANNs) on top. It uses tied weights and pooling layers. In particular, max-pooling. It is often structured via Fukushima's convolutional architecture. They are variations of multilayer perceptrons that use minimal preprocessing. This architecture allows CNNs to take advantage of the 2D structure of input data. Its unit connectivity pattern is inspired by the organization of the visual cortex. Units respond to stimuli in a restricted region of space known as the receptive field. Receptive fields partially overlap, over-covering the entire visual field. Unit response can be approximated mathematically by a convolution operation. CNNs are suitable for processing visual and other two-dimensional data. They have shown superior results in both image and speech applications. They can be trained with standard backpropagation. CNNs are easier to train than other regular, deep, feed-forward neural networks and have many fewer parameters to estimate. Capsule Neural Networks (CapsNet) add structures called capsules to a CNN and reuse output from several capsules to form more stable (with respect to various perturbations) representations. Examples of applications in computer vision include DeepDream and robot navigation. They have wide applications in image and video recognition, recommender systems and natural language processing. == Deep stacking network == A deep stacking network (DSN) (deep convex network) is based on a hierarchy of blocks of simplified neural network modules. It was introduced in 2011 by Deng and Yu. It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization. Each DSN block is a simple module that is easy to train by itself in a supervised fashion without backpropagation for the entire blocks. Each block consists of a simplified multi-layer perceptron (MLP) with a single hidden layer. The hidden layer h has logistic sigmoidal units, and the output layer has linear units. Connections between these layers are represented by weight matrix U; input-to-hidden-layer connections have weight matrix W. Target vectors t form the columns of matrix T, and the input data vectors x form the columns of matrix X. The matrix of hidden units is H = σ ( W T X ) {\displaystyle {\boldsymbol {H}}=\sigma ({\boldsymbol {W}}^{T}{\boldsymbol {X}})} . Modules are trained in order, so lower-layer weights W are known at each stage. The function performs the element-wise logistic sigmoid operation. Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block. Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks. Then learning the upper-layer weight matrix U given other weights in the network can be formulated as a convex optimization problem: min U T f = ‖ U T H − T ‖ F 2 , {\displaystyle \min _{U^{T}}f=\|{\boldsymbol {U}}^{T}{\boldsymbol {H}}-{\boldsymbol {T}}\|_{F}^{2},} which has a closed-form solution. Unlike other deep architectures, such as DBNs, the goal is not to discover the transformed feature representation. The structure of the hierarchy of this kind of architecture makes parallel learning straightforward, as a batch-mode optimization problem. In purely discriminative tasks, DSNs outperform conventional DBNs. === Tensor deep stacking networks === This architecture is a DSN extension. It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer. TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor. While parallelization and scalability are not considered seriously in conventional DNNs, all learning for DSNs and TDSNs is done in batch mode, to allow parallelization. Parallelization allows scaling the design to larger (deeper) architectures and data sets. The basic architecture is suitable for diverse tasks such as classification and regression. == Physics-informed == Such a neural network is designed for the numerical solution of mathematical equations, such as differential, integral, delay, fractional and others. As input parameters, PINN accepts variables (spatial, temporal, and others), transmits them through the network block. At the output, it produces an approximate solution and substitutes it into the mathematical model, considering the initial and boundary conditions. If the solution does not satisfy the required accuracy, one uses the backpropagation and rectify the solution. Besides PINN, other architectures have been developed to produce surrogate models for scientific comput

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