Color quantization

In computer graphics, color quantization or color image quantization is quantization applied to color spaces; it is a process that reduces the number of distinct colors used in an image, usually with the intention that the new image should be as visually similar as possible to the original image. Computer algorithms to perform color quantization on bitmaps have been studied since the 1970s. Color quantization is critical for displaying images with many colors on devices that can only display a limited number of colors, usually due to memory limitations, and enables efficient compression of certain types of images. The name "color quantization" is primarily used in computer graphics research literature; in applications, terms such as optimized palette generation, optimal palette generation, or decreasing color depth are used. Some of these are misleading, as the palettes generated by standard algorithms are not necessarily the best possible. == Algorithms == Most standard techniques treat color quantization as a problem of clustering points in three-dimensional space, where the points represent colors found in the original image and the three axes represent the three color channels. Almost any three-dimensional clustering algorithm can be applied to color quantization, and vice versa. After the clusters are located, typically the points in each cluster are averaged to obtain the representative color that all colors in that cluster are mapped to. The three color channels are usually red, green, and blue, but another popular choice is the Lab color space, in which Euclidean distance is more consistent with perceptual difference. The most popular algorithm by far for color quantization, invented by Paul Heckbert in 1979, is the median cut algorithm. Many variations on this scheme are in use. Before this time, most color quantization was done using the population algorithm or population method, which essentially constructs a histogram of equal-sized ranges and assigns colors to the ranges containing the most points. A more modern popular method is clustering using octrees, first conceived by Gervautz and Purgathofer and improved by Xerox PARC researcher Dan Bloomberg. If the palette is fixed, as is often the case in real-time color quantization systems such as those used in operating systems, color quantization is usually done using the "straight-line distance" or "nearest color" algorithm, which simply takes each color in the original image and finds the closest palette entry, where distance is determined by the distance between the two corresponding points in three-dimensional space. In other words, if the colors are ( r 1 , g 1 , b 1 ) {\displaystyle (r_{1},g_{1},b_{1})} and ( r 2 , g 2 , b 2 ) {\displaystyle (r_{2},g_{2},b_{2})} , we want to minimize the Euclidean distance: ( r 1 − r 2 ) 2 + ( g 1 − g 2 ) 2 + ( b 1 − b 2 ) 2 . {\displaystyle {\sqrt {(r_{1}-r_{2})^{2}+(g_{1}-g_{2})^{2}+(b_{1}-b_{2})^{2}}}.} This effectively decomposes the color cube into a Voronoi diagram, where the palette entries are the points and a cell contains all colors mapping to a single palette entry. There are efficient algorithms from computational geometry for computing Voronoi diagrams and determining which region a given point falls in; in practice, indexed palettes are so small that these are usually overkill. Color quantization is frequently combined with dithering, which can eliminate unpleasant artifacts such as banding that appear when quantizing smooth gradients and give the appearance of a larger number of colors. Some modern schemes for color quantization attempt to combine palette selection with dithering in one stage, rather than perform them independently. A number of other much less frequently used methods have been invented that use entirely different approaches. The Local K-means algorithm, conceived by Oleg Verevka in 1995, is designed for use in windowing systems where a core set of "reserved colors" is fixed for use by the system and many images with different color schemes might be displayed simultaneously. It is a post-clustering scheme that makes an initial guess at the palette and then iteratively refines it. In the early days of color quantization, the k-means clustering algorithm was deemed unsuitable because of its high computational requirements and sensitivity to initialization. In 2011, M. Emre Celebi reinvestigated the performance of k-means as a color quantizer. He demonstrated that an efficient implementation of k-means outperforms a large number of color quantization methods. The high-quality but slow NeuQuant algorithm reduces images to 256 colors by training a Kohonen neural network "which self-organises through learning to match the distribution of colours in an input image. Taking the position in RGB-space of each neuron gives a high-quality colour map in which adjacent colours are similar." It is particularly advantageous for images with gradients. Finally, one of the newer methods is spatial color quantization, conceived by Puzicha, Held, Ketterer, Buhmann, and Fellner of the University of Bonn, which combines dithering with palette generation and a simplified model of human perception to produce visually impressive results even for very small numbers of colors. It does not treat palette selection strictly as a clustering problem, in that the colors of nearby pixels in the original image also affect the color of a pixel. See sample images. == History and applications == In the early days of PCs, it was common for video adapters to support only 2, 4, 16, or (eventually) 256 colors due to video memory limitations; they preferred to dedicate the video memory to having more pixels (higher resolution) rather than more colors. Color quantization helped to justify this tradeoff by making it possible to display many high color images in 16- and 256-color modes with limited visual degradation. Many operating systems automatically perform quantization and dithering when viewing high color images in a 256 color video mode, which was important when video devices limited to 256 color modes were dominant. Modern computers can now display millions of colors at once, far more than can be distinguished by the human eye, limiting this application primarily to mobile devices and legacy hardware. Nowadays, color quantization is mainly used in GIF and PNG images. GIF, for a long time the most popular lossless and animated bitmap format on the World Wide Web, only supports up to 256 colors, necessitating quantization for many images. Some early web browsers constrained images to use a specific palette known as the web colors, leading to severe degradation in quality compared to optimized palettes. PNG images support 24-bit color, but can often be made much smaller in filesize without much visual degradation by application of color quantization, since PNG files use fewer bits per pixel for palettized images. The infinite number of colors available through the lens of a camera is impossible to display on a computer screen; thus converting any photograph to a digital representation necessarily involves some quantization. Practically speaking, 24-bit color is sufficiently rich to represent almost all colors perceivable by humans with sufficiently small error as to be visually identical (if presented faithfully), within the available color space. However, the digitization of color, either in a camera detector or on a screen, necessarily limits the available color space. Consequently there are many colors that may be impossible to reproduce, regardless of how many bits are used to represent the color. For example, it is impossible in typical RGB color spaces (common on computer monitors) to reproduce the full range of green colors that the human eye is capable of perceiving. With the few colors available on early computers, different quantization algorithms produced very different-looking output images. As a result, a lot of time was spent on writing sophisticated algorithms to be more lifelike. === Quantization for image compression === Many image file formats support indexed color. A whole-image palette typically selects 256 "representative" colors for the entire image, where each pixel references any one of the colors in the palette, as in the GIF and PNG file formats. A block palette typically selects 2 or 4 colors for each block of 4x4 pixels, used in BTC, CCC, S2TC, and S3TC. === Editor support === Many bitmap graphics editors contain built-in support for color quantization, and will automatically perform it when converting an image with many colors to an image format with fewer colors. Most of these implementations allow the user to set exactly the number of desired colors. Examples of such support include: Photoshop's Mode→Indexed Color function supplies a number of quantization algorithms ranging from the fixed Windows system and Web palettes to the proprietary Local and Global algorithms for generating palettes suited to a particu

2018 Google data breach

The 2018 Google data breach was a major data privacy scandal in which the Google+ API exposed the private data of over five hundred thousand users. Google+ managers first noticed harvesting of personal data in March 2018, during a review following the Facebook–Cambridge Analytica data scandal. The bug, despite having been fixed immediately, exposed the private data of approximately 500,000 Google+ users to the public. Google did not reveal the leak to the network's users. In November 2018, another data breach occurred following an update to the Google+ API. Although Google found no evidence of failure, approximately 52.5 million personal profiles were potentially exposed. In August 2019, Google declared a shutdown of Google+ due to low use and technological challenges. == Overview of Google+ == Google+ was launched in June 2011 as an invite-only social network, but was opened for public access later in the year. It was managed by Vic Gundotra. Similar to Facebook, Google+ also included key features Circles, Hangouts and Sparks. Circles let users personalize their social groups by sorting friends into different categories. Once allowed into a Circle, users could regulate information in their individual spaces. Hangouts included video chatting and instant messaging between users. Sparks allowed Google to track users' past searches to find news and content related to their interests. Google+ was linked to other Google services, such as YouTube, Google Drive and Gmail, giving it access to roughly 2 billion user accounts. However, less than 400 million consumers actively used Google+, with 90% of those users using it for less than five seconds. == The breaches == In March 2018, Google developers found a data breach within the Google+ People API in which external apps acquired access to Profile fields that were not marked as public. According to The Wall Street Journal, Google didn’t disclose the breach when it was first discovered in March to avoid regulatory scrutiny and reputational damage. 500,000 Google+ accounts were included in the breach, which allowed 438 external apps unauthorized access to private users' names, emails, addresses, occupations, genders and ages. This information was available between 2015 and 2018. Google found no evidence of any user's personal information being misused, nor that any third-party app developers were aware of the leak. In November 2018, a software update created another data breach within the Google+ API. The bug impacted 52.5 million users, where, similarly to the March breach, unauthorized apps were able to access Google+ profiles, including users' names, email addresses, occupations and ages. Apps could not access financial information, national identification, numbers, or passwords. Blog posts, messages and phone numbers also remained inaccessible if marked as private. Unlike the previous breach, access was only available for six days before Google+ learned of the breach. Once more, Google+ found no evidence of data being misused by third-party developers. == Responses == In October 2018, the Wall Street Journal published an article outlining the initial breach and Google's decision to not disclose it to users. At the time, there was no federal law that required Google to inform their consumers of data breaches. Google+ originally did not disclose the breach out of fears of being compared to Facebook's recent data leak and subsequent loss of consumer confidence. In response to the Wall Street Journal article, Google announced the shutdown of Google+ in August 2019. After the second data leak, the date was moved to April 2019. In response to the data breach, enterprise consumers were notified of the bug's impact and given instructions on how to save, download and delete their data prior to the Google+ shut down. Google's Privacy and Data Protection Office found no misuse of user data. Prior to the Google+ shutdown, Google set a 10-month period in which users could download and migrate their data. After the 10-month period, user content was deleted. On 4 February 2019, consumers were no longer able to create new Google+ profiles. Google shut down Google+ APIs on 7 March 2019 to ensure that developers did not continue to rely on the APIs prior to the Google+ shutdown. Google is the principal entity of its parent company, Alphabet Inc. After the data breach, Alphabet Inc. share prices fell by 1% to $1,157.06 on 9 October 2018 after an earlier drop of $1,135.40 that morning, the lowest price since 5 July 2018. After the publication of The Wall Street Journal article, share prices dropped as low as 2.1% in two days on 10 October 2018. Share prices steadily increased from this point and met the 8 October 2018 share price on 5 February 2019. Google planned to rebuild Google+ as a corporate enterprise network. Google Play will now assess which apps can ask for permission to access the user's SMS data. Only the default app for telephone distribution is able to make requests. Prior to the data breaches, apps were able to request access to all of a consumer's data simultaneously. Now, each app must request permission for each aspect of a consumer's profile.

DryvIQ

DryvIQ is a software application that enables businesses to migrate on-site system files and associated data across storage and content management platforms, as well as create synchronized hybrid storage systems. == History == Before it was DryvIQ, the software SkySync was released in 2013 by Ann Arbor, Michigan based company, Portal Architects, Inc. The company created SkySync, a back-end, administrative application designed to transfer content across storage platforms, after abandoning 18 months of development on a desktop application called SkyBrary in 2011. Between 2014 and 2015, Portal Architects established partnerships with the following companies: Autodesk, Box, Dropbox, Egnyte, EMC, Google, Syncplicity, Huddle, IBM, Microsoft, OpenText, Oracle, Citrix ShareFile, Hightail and Internet2. SkySync (currently DryvIQ) was named a "Cool Vendor in Content Management" by Gartner in 2015. In 2022, SkySync changed its name to DryvIQ, which is now what the company is currently known as. == Overview == DryvIQ is a software application that syncs, migrates or backs up files including their associated properties, metadata, versions, user accounts and permissions across on-premises and Cloud-based storage platforms. The software deploys on a server, virtual machine or within Microsoft Azure, Amazon Web Services or other cloud computing services.

Application-release automation

Application-release automation (ARA) refers to the process of packaging and deploying an application or update of an application from development, across various environments, and ultimately to production. ARA solutions must combine the capabilities of deployment automation, environment management and modeling, and release coordination. == Relationship with DevOps == ARA tools help cultivate DevOps best practices by providing a combination of automation, environment modeling and workflow-management capabilities. These practices help teams deliver software rapidly, reliably and responsibly. ARA tools achieve a key DevOps goal of implementing continuous delivery with a large quantity of releases quickly. == Relationship with deployment == ARA is more than just software-deployment automation – it deploys applications using structured release-automation techniques that allow for an increase in visibility for the whole team. It combines workload automation and release-management tools as they relate to release packages, as well as movement through different environments within the DevOps pipeline. ARA tools help regulate deployments, how environments are created and deployed, and how and when releases are deployed. == ARA Solutions == All ARA solutions must include capabilities in automation, environment modeling, and release coordination. Additionally, the solution must provide this functionality without reliance on other tools.

Celia (virtual assistant)

Celia is an artificially intelligent virtual assistant developed by Huawei for their latest HarmonyOS and Android-based EMUI smartphones that lack Google Services and a Google Assistant. The assistant can perform day-to-day tasks, which include making a phone call, setting a reminder and checking the weather. It was unveiled on 7 April 2020 and got publicly released on 27 April 2020 via an OTA update solely to selected devices that can update their software to EMUI 10.1. Huawei had initially referred to the new assistant in late 2019 by having announced that there would be an English version of their already 2018 Chinese speaker assistant—Xiaoyi—to be released into the European markets. Due to the on-going China–United States trade war, the company's newly released smartphones were left without any Google services, including the loss of Google Assistant. This subsequently led to the development and release of Celia. AI technology is integrated into the software of Celia, which allows it to translate text using a phones camera and to identify everyday objects — similar to that of Google Lens. == Features == Celia has many features that are similar to that of its rivals: the Google Assistant and Siri. It can be triggered by the words, 'Hey Celia' or be summoned by pressing and holding down on the power button. The default search engine for Celia is Bing, but this can be changed in settings. Celia can make calls, check the agenda, send a message, show the weather, set alarms and control home appliances. The assistant also has the ability to integrate itself with the stock apps of the EMUI software and toggle with the device's settings, such as by turning on the flashlight and playing multimedia content, but with the users command. With the AI that is installed in Celia, it can identify food, everyday objects and translate text using the phones camera. In China, Chinese Xiaoyi packs with an LLM model called PanGu-Σ 3.0 AI on HarmonyOS 4.0 major upgrade improvements from Celia, making the assistant smarter and more advanced compared to when it was launched in 2020 on EMUI handsets in China and internationally, surpassing Apple and Google by the being the first in the AI industry, with a dedicated AI system framework of APIs on the latest operating system that evolves to a complete large dedicated AI software stack called Harmony Intelligence of Pangu Embedded variant model and MindSpore AI framework with Neural Network Runtime on OpenHarmony-based HarmonyOS NEXT base system to replace the dual framework system with a single frame HarmonyOS 5.0 version by Q4 2024, first introduced on June 21, 2024, in Developer Beta 1 preview release at HDC 2024. == Availability by country and language == Currently, Celia is available only in German, English, French and Spanish, and has been released in Germany, the UK, France, Spain, Chile, Mexico and Colombia. Huawei has said, that there will be more regions and languages to come. == Compatible devices == Celia only became available with the EMUI 10.1 update that was released in April, which means that a limited number of devices are compatible with it. More devices will be added to the list throughout the coming months as Celia's availability increases. The current list is shown below: === Huawei P series === Huawei P50 (Pro) Huawei P40 (Lite, Pro & Pro+) Huawei P30 (Pro) === Huawei Mate series === Huawei Mate 40 Huawei Mate 30 (Lite, Pro & RS Porche Design) Huawei MatePad Pro Huawei Mate 20 (Pro, 20X 4G, 20X 5G and RS Porche Design) Huawei Mate X & Xs === Huawei Nova series === Huawei Nova 6 (Nova 6 5G & Nova 6 SE) Huawei Nova 5 (Nova 5 Pro, Nova 5i Pro & Nova 5Z) Huawei Nova Y60 === Huawei Enjoy series === Huawei Enjoy 10S == Issues == Technology news website Engadget has noted that when saying, 'Hey Celia', out aloud in the presence of an iPhone, Siri will respond along with Celia; this is apparently because 'Celia' sounds similar to 'Siri'.

Digital image correlation and tracking

Digital image correlation and tracking is an optical method that employs tracking and image registration techniques for accurate 2D and 3D measurements of changes in 2D images or 3D volumes. This method is often used to measure full-field displacement and strains, and it is widely applied in many areas of science and engineering. Compared to strain gauges and extensometers, digital image correlation methods provide finer details about deformation, due to the ability to provide both local and average data. == Overview == Digital image correlation (DIC) techniques have been increasing in popularity, especially in micro- and nano-scale mechanical testing applications due to their relative ease of implementation and use. Advances in computer technology and digital cameras have been the enabling technologies for this method and while white-light optics has been the predominant approach, DIC can be and has been extended to almost any imaging technology. The concept of using cross-correlation to measure shifts in datasets has been known for a long time, and it has been applied to digital images since at least the early 1970s. The present-day applications are almost innumerable, including image analysis, image compression, velocimetry, and strain estimation. Much early work in DIC in the field of mechanics was led by researchers at the University of South Carolina in the early 1980s and has been optimized and improved in recent years. Commonly, DIC relies on finding the maximum of the correlation array between pixel intensity array subsets on two or more corresponding images, which gives the integer translational shift between them. It is also possible to estimate shifts to a finer resolution than the resolution of the original images, which is often called "sub-pixel" registration because the measured shift is smaller than an integer pixel unit. For sub-pixel interpolation of the shift, other methods do not simply maximize the correlation coefficient. An iterative approach can also be used to maximize the interpolated correlation coefficient by using non-linear optimization techniques. The non-linear optimization approach tends to be conceptually simpler and can handle large deformations more accurately, but as with most nonlinear optimization techniques, it is slower. The two-dimensional discrete cross correlation r i j {\displaystyle r_{ij}} can be defined in several ways, one possibility being: r i j = ∑ m ∑ n [ f ( m + i , n + j ) − f ¯ ] [ g ( m , n ) − g ¯ ] ∑ m ∑ n [ f ( m , n ) − f ¯ ] 2 ∑ m ∑ n [ g ( m , n ) − g ¯ ] 2 . {\displaystyle r_{ij}={\frac {\sum _{m}\sum _{n}[f(m+i,n+j)-{\bar {f}}][g(m,n)-{\bar {g}}]}{\sqrt {\sum _{m}\sum _{n}{[f(m,n)-{\bar {f}}]^{2}}\sum _{m}\sum _{n}{[g(m,n)-{\bar {g}}]^{2}}}}}.} Here f(m, n) is the pixel intensity or the gray-scale value at a point (m, n) in the original image, g(m, n) is the gray-scale value at a point (m, n) in the translated image, f ¯ {\displaystyle {\bar {f}}} and g ¯ {\displaystyle {\bar {g}}} are mean values of the intensity matrices f and g respectively. However, in practical applications, the correlation array is usually computed using Fourier-transform methods, since the fast Fourier transform is a much faster method than directly computing the correlation. F = F { f } , G = F { g } . {\displaystyle \mathbf {F} ={\mathcal {F}}\{f\},\quad \mathbf {G} ={\mathcal {F}}\{g\}.} Then taking the complex conjugate of the second result and multiplying the Fourier transforms together elementwise, we obtain the Fourier transform of the correlogram, R {\displaystyle \ R} : R = F ∘ G ∗ , {\displaystyle R=\mathbf {F} \circ \mathbf {G} ^{},} where ∘ {\displaystyle \circ } is the Hadamard product (entry-wise product). It is also fairly common to normalize the magnitudes to unity at this point, which results in a variation called phase correlation. Then the cross-correlation is obtained by applying the inverse Fourier transform: r = F − 1 { R } . {\displaystyle \ r={\mathcal {F}}^{-1}\{R\}.} At this point, the coordinates of the maximum of r i j {\displaystyle r_{ij}} give the integer shift: ( Δ x , Δ y ) = arg ⁡ max ( i , j ) { r } . {\displaystyle (\Delta x,\Delta y)=\arg \max _{(i,j)}\{r\}.} == Deformation mapping == For deformation mapping, the mapping function that relates the images can be derived from comparing a set of subwindow pairs over the whole images. (Figure 1). The coordinates or grid points (xi, yj) and (xi, yj) are related by the translations that occur between the two images. If the deformation is small and perpendicular to the optical axis of the camera, then the relation between (xi, yj) and (xi, yj) can be approximated by a 2D affine transformation such as: x ∗ = x + u + ∂ u ∂ x Δ x + ∂ u ∂ y Δ y , {\displaystyle x^{}=x+u+{\frac {\partial u}{\partial x}}\Delta x+{\frac {\partial u}{\partial y}}\Delta y,} y ∗ = y + v + ∂ v ∂ x Δ x + ∂ v ∂ y Δ y . {\displaystyle y^{}=y+v+{\frac {\partial v}{\partial x}}\Delta x+{\frac {\partial v}{\partial y}}\Delta y.} Here u and v are translations of the center of the sub-image in the X and Y directions respectively. The distances from the center of the sub-image to the point (x, y) are denoted by Δ x {\displaystyle \Delta x} and Δ y {\displaystyle \Delta y} . Thus, the correlation coefficient rij is a function of displacement components (u, v) and displacement gradients ∂ u ∂ x , ∂ u ∂ y , ∂ v ∂ x , ∂ v ∂ y . {\displaystyle {\frac {\partial u}{\partial x}},{\frac {\partial u}{\partial y}},{\frac {\partial v}{\partial x}},{\frac {\partial v}{\partial y}}.} DIC has proven to be very effective at mapping deformation in macroscopic mechanical testing, where the application of specular markers (e.g. paint, toner powder) or surface finishes from machining and polishing provide the needed contrast to correlate images well. However, these methods for applying surface contrast do not extend to the application of free-standing thin films for several reasons. First, vapor deposition at normal temperatures on semiconductor grade substrates results in mirror-finish quality films with RMS roughnesses that are typically on the order of several nanometers. No subsequent polishing or finishing steps are required, and unless electron imaging techniques are employed that can resolve microstructural features, the films do not possess enough useful surface contrast to adequately correlate images. Typically this challenge can be circumvented by applying paint that results in a random speckle pattern on the surface, although the large and turbulent forces resulting from either spraying or applying paint to the surface of a free-standing thin film are too high and would break the specimens. In addition, the sizes of individual paint particles are on the order of μms, while the film thickness is only several hundred nanometers, which would be analogous to supporting a large boulder on a thin sheet of paper. == Digital volume correlation == Digital Volume Correlation (DVC, and sometimes called Volumetric-DIC) extends the 2D-DIC algorithms into three dimensions to calculate the full-field 3D deformation from a pair of 3D images. This technique is distinct from 3D-DIC, which only calculates the 3D deformation of an exterior surface using conventional optical images. The DVC algorithm is able to track full-field displacement information in the form of voxels instead of pixels. The theory is similar to above except that another dimension is added: the z-dimension. The displacement is calculated from the correlation of 3D subsets of the reference and deformed volumetric images, which is analogous to the correlation of 2D subsets described above. DVC can be performed using volumetric image datasets. These images can be obtained using confocal microscopy, X-ray computed tomography, Magnetic Resonance Imaging or other techniques. Similar to the other DIC techniques, the images must exhibit a distinct, high-contrast 3D "speckle pattern" to ensure accurate displacement measurement. DVC was first developed in 1999 to study the deformation of trabecular bone using X-ray computed tomography images. Since then, applications of DVC have grown to include granular materials, metals, foams, composites and biological materials. To date it has been used with images acquired by MRI imaging, Computer Tomography (CT), micro-CT, confocal microscopy, and lightsheet microscopy. DVC is currently considered to be ideal in the research world for 3D quantification of local displacements, strains, and stress in biological specimens. It is preferred because of the non-invasiveness of the method over traditional experimental methods. Two of the key challenges are improving the speed and reliability of the DVC measurement. The 3D imaging techniques produce noisier images than conventional 2D optical images, which reduces the quality of the displacement measurement. Computational speed is restricted by the file sizes of 3D images, which are significantly larger than 2D images. For example, an

Neural operators

Neural operators are a class of deep learning architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial neural networks, marking a departure from the typical focus on learning mappings between finite-dimensional Euclidean spaces or finite sets. Neural operators directly learn operators between function spaces; they can receive input functions, and the output function can be evaluated at any discretization. The primary application of neural operators is in learning surrogate maps for the solution operators of partial differential equations (PDEs), which are critical tools in modeling the natural environment. Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs compared to existing machine learning methodologies while being significantly faster than numerical solvers. Neural operators have also been applied to various scientific and engineering disciplines such as turbulent flow modeling, computational mechanics, graph-structured data, and the geosciences. In particular, they have been applied to learning stress-strain fields in materials, classifying complex data like spatial transcriptomics, predicting multiphase flow in porous media, and carbon dioxide migration simulations. Finally, the operator learning paradigm allows learning maps between function spaces, and is different from parallel ideas of learning maps from finite-dimensional spaces to function spaces, and subsumes these settings as special cases when limited to a fixed input resolution. == Operator learning == Understanding and mapping relationships between function spaces has many applications in engineering and the sciences. In particular, one can cast the problem of solving partial differential equations as identifying a map between function spaces, such as from an initial condition to a time-evolved state. In other PDEs this map takes an input coefficient function and outputs a solution function. Operator learning is a machine learning paradigm to learn solution operators mapping the input function to the output function . Using traditional machine learning methods, addressing this problem would involve discretizing the infinite-dimensional input and output function spaces into finite-dimensional grids and applying standard learning models, such as neural networks. This approach reduces the operator learning to finite-dimensional function learning and has some limitations, such as generalizing to discretizations beyond the grid used in training. The primary properties of neural operators that differentiate them from traditional neural networks is discretization invariance and discretization convergence. Unlike conventional neural networks, which are fixed on the discretization of training data, neural operators can adapt to various discretizations without re-training. This property improves the robustness and applicability of neural operators in different scenarios, providing consistent performance across different resolutions and grids. == Definition and formulation == Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are composed of alternating linear maps and non-linearities. Since neural operators act on and output functions, neural operators have been instead formulated as a sequence of alternating linear integral operators on function spaces and point-wise non-linearities. Using an analogous architecture to finite-dimensional neural networks, similar universal approximation theorems have been proven for neural operators. In particular, it has been shown that neural operators can approximate any continuous operator on a compact set. Neural operators seek to approximate some operator G : A → U {\displaystyle {\mathcal {G}}:{\mathcal {A}}\to {\mathcal {U}}} between function spaces A {\displaystyle {\mathcal {A}}} and U {\displaystyle {\mathcal {U}}} by building a parametric map G ϕ : A → U {\displaystyle {\mathcal {G}}_{\phi }:{\mathcal {A}}\to {\mathcal {U}}} . Such parametric maps G ϕ {\displaystyle {\mathcal {G}}_{\phi }} can generally be defined in the form G ϕ := Q ∘ σ ( W T + K T + b T ) ∘ ⋯ ∘ σ ( W 1 + K 1 + b 1 ) ∘ P , {\displaystyle {\mathcal {G}}_{\phi }:={\mathcal {Q}}\circ \sigma (W_{T}+{\mathcal {K}}_{T}+b_{T})\circ \cdots \circ \sigma (W_{1}+{\mathcal {K}}_{1}+b_{1})\circ {\mathcal {P}},} where P , Q {\displaystyle {\mathcal {P}},{\mathcal {Q}}} are the lifting (lifting the codomain of the input function to a higher dimensional space) and projection (projecting the codomain of the intermediate function to the output dimension) operators, respectively. These operators act pointwise on functions and are typically parametrized as multilayer perceptrons. σ {\displaystyle \sigma } is a pointwise nonlinearity, such as a rectified linear unit (ReLU), or a Gaussian error linear unit (GeLU). Each layer t = 1 , … , T {\displaystyle t=1,\dots ,T} has a respective local operator W t {\displaystyle W_{t}} (usually parameterized by a pointwise neural network), a kernel integral operator K t {\displaystyle {\mathcal {K}}_{t}} , and a bias function b t {\displaystyle b_{t}} . Given some intermediate functional representation v t {\displaystyle v_{t}} with domain D {\displaystyle D} in the t {\displaystyle t} -th hidden layer, a kernel integral operator K ϕ {\displaystyle {\mathcal {K}}_{\phi }} is defined as ( K ϕ v t ) ( x ) := ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x):=\int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy,} where the kernel κ ϕ {\displaystyle \kappa _{\phi }} is a learnable implicit neural network, parametrized by ϕ {\displaystyle \phi } . In practice, one is often given the input function to the neural operator at a specific resolution. For instance, consider the setting where one is given the evaluation of v t {\displaystyle v_{t}} at n {\displaystyle n} points { y j } j n {\displaystyle \{y_{j}\}_{j}^{n}} . Borrowing from Nyström integral approximation methods such as Riemann sum integration and Gaussian quadrature, the above integral operation can be computed as follows: ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y ≈ ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j , {\displaystyle \int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy\approx \sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}},} where Δ y j {\displaystyle \Delta _{y_{j}}} is the sub-area volume or quadrature weight associated to the point y j {\displaystyle y_{j}} . Thus, a simplified layer can be computed as v t + 1 ( x ) ≈ σ ( ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j + W t ( v t ( y j ) ) + b t ( x ) ) . {\displaystyle v_{t+1}(x)\approx \sigma \left(\sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}}+W_{t}(v_{t}(y_{j}))+b_{t}(x)\right).} The above approximation, along with parametrizing κ ϕ {\displaystyle \kappa _{\phi }} as an implicit neural network, results in the graph neural operator (GNO). There have been various parameterizations of neural operators for different applications. These typically differ in their parameterization of κ {\displaystyle \kappa } . The most popular instantiation is the Fourier neural operator (FNO). FNO takes κ ϕ ( x , y , v t ( x ) , v t ( y ) ) := κ ϕ ( x − y ) {\displaystyle \kappa _{\phi }(x,y,v_{t}(x),v_{t}(y)):=\kappa _{\phi }(x-y)} and by applying the convolution theorem, arrives at the following parameterization of the kernel integral operator: ( K ϕ v t ) ( x ) = F − 1 ( R ϕ ⋅ ( F v t ) ) ( x ) , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x)={\mathcal {F}}^{-1}(R_{\phi }\cdot ({\mathcal {F}}v_{t}))(x),} where F {\displaystyle {\mathcal {F}}} represents the Fourier transform and R ϕ {\displaystyle R_{\phi }} represents the Fourier transform of some periodic function κ ϕ {\displaystyle \kappa _{\phi }} . That is, FNO parameterizes the kernel integration directly in Fourier space, using a prescribed number of Fourier modes. When the grid at which the input function is presented is uniform, the Fourier transform can be approximated using the discrete Fourier transform (DFT) with frequencies below some specified threshold. The discrete Fourier transform can be computed using a fast Fourier transform (FFT) implementation. == Training == Training neural operators is similar to the training process for a traditional neural network. Neural operators are typically trained in some Lp norm or Sobolev norm. In particular, for a dataset { ( a i , u i ) } i = 1 N {\displaystyle \{(a_{i},u_{i})\}_{i=1}^{N}} of size N {\displaystyle N} , neural operators minimize (a discretization of) L U ( { ( a i , u i ) } i = 1 N ) := ∑ i = 1 N ‖ u i − G θ ( a i ) ‖ U 2 {\displaystyle {\mathcal {L}}_{\mathca