Conceptual dependency theory is a model of natural language understanding used in artificial intelligence systems. Roger Schank at Stanford University introduced the model in 1969, in the early days of artificial intelligence. This model was extensively used by Schank's students at Yale University such as Robert Wilensky, Wendy Lehnert, and Janet Kolodner. Schank developed the model to represent knowledge for natural language input into computers. Partly influenced by the work of Sydney Lamb, his goal was to make the meaning independent of the words used in the input, i.e. two sentences identical in meaning would have a single representation. The system was also intended to draw logical inferences. The model uses the following basic representational tokens: real world objects, each with some attributes. real world actions, each with attributes times locations A set of conceptual transitions then act on this representation, e.g. an ATRANS is used to represent a transfer such as "give" or "take" while a PTRANS is used to act on locations such as "move" or "go". An MTRANS represents mental acts such as "tell", etc. A sentence such as "John gave a book to Mary" is then represented as the action of an ATRANS on two real world objects, John and Mary.
Neuro-sama
Neuro-sama is an artificial intelligence (AI) VTuber, singer, and chatbot. She was created by the pseudonymous programmer Vedal and livestreams on his Twitch and Bilibili channels. Her speech and personality are powered by a large language model (LLM) that is combined with a computer-animated avatar and a text-to-speech voice, allowing her to communicate with viewers in the stream's chat. Neuro-sama debuted on Twitch on 19 December 2022. An annual subathon which begins on the anniversary of her debut has seen Vedal's Twitch channel become the all-time third most-subscribed channel and claim the all-time Twitch hype train record. == Overview == Neuro-sama (nicknamed "Neuro") was created by a pseudonymous programmer and developer known as Vedal (sometimes given as Vedal987). Vedal says that his programming skills are self-taught. In a 2023 interview with Bloomberg News, Vedal said that Neuro-sama was his full-time job. Her responses are generated by a large language model and converted into a high-pitched female voice using a text-to-speech application. Her low latency allows for fast-paced conversations. Neuro-sama is prohibited from making some statements, such as those that are racist or contain profanity. Unlike most AI systems which silently prohibit outputs mentioning such topics, Neuro-sama's output is instead replaced with the word "filtered". Neuro-sama uses a VTuber model as an avatar. Vedal said that he decided to use a VTuber model because it was much easier for an AI to control it than it was to generate footage of a person. Neuro-sama's model is that of a young girl in an anime art style. The model has been described as cute. Femme VTuber models are typically feminine, youthful, and exaggerated. Her original model was Live2D's free-to-use "Hiyori Momose" model. Her second model was released on 27 May 2023; it was modelled by Otozuki Teru and designed by Anny, running in the Unity game engine. Her third model was released on 19 December 2024; it was rigged by Kitanya and designed by Anny. Neuro-sama's third model has large blue eyes and brown hair tied with pink ribbons. Neuro-sama also has a 3D model which was introduced on 15 November 2025; it was made by 3D character modeller jjinomu. A separate AI VTuber, known as Evil Neuro (nicknamed "Evil"), debuted on 25 March 2023. Presented as Neuro-sama's "sister", she has a different model, voice, and personality. In one instance, Evil Neuro reacted to the trolley problem differently from Neuro-sama; Evil Neuro was amoral while Neuro-sama attempted to maximize good. === Online content === Neuro-sama's Twitch content often centers around playing video games, notably osu!, whose gameplay once defeated the best-ranking human player in the world, mrekk. Additionally, Neuro-sama plays Minecraft, where her adaptations to sandbox gameplay have gained notoriety. Her content has also included singing songs, including several official covers and original songs; playing chess with her viewers; chatting with other VTubers during collaborations; and reacting to YouTube videos. The AI frequently engages with viewers by responding to their questions and acknowledging donations. Her comedic and sometimes controversial responses to the live chat have gone viral, accelerating the channel's rise in popularity. Neuro-sama's fanbase is dubbed The Swarm, so-named for the swarm of drones Neuro-sama once declared she would use to rule the world. One form of content on Neuro-sama's channel is developer streams. In developer streams, Vedal streams with Neuro-sama, with the stream content including debugging her code, planning her schedule, and fielding suggestions of changes from chat. He usually appears as a turtle avatar, sometimes located on Neuro-sama's head. In collaboration streams, Neuro-sama interacts with a human streamer. Activities in them are varied and include: playing video games, such as Minecraft and GeoGuessr; Neuro-sama being interviewed; driving human streamers around in a toy electric car; and traversing the city of Tokyo while talking to Neuro-sama. Neuro-sama's English-language content on Bilibili is popular among those seeking to learn the language. She also has an account on X, where she posts and interacts with fans. == History == Neuro-sama was created in 2018 by Vedal as an AI trained to play and master the rhythm game osu!. She did not have a voice, model, personality, or communication abilities. In 2019, Vedal livestreamed her playing osu! on Twitch and the streams saw some success in the osu! community, but they remained in that niche. In an interview, Vedal said that he streamed her playing osu! for about a month and gained 3,000 followers, with a viewer also suggesting he name the AI "Neuro-sama". According to Vedal, he continued to work on and improve the osu! AI and it was eventually finished in 2022. He said that a friend had the idea to make an AI livestreamer with an LLM, which he believed to have merit and began working on, merging it with his osu! AI. On 19 December 2022, Neuro-sama was relaunched with a model, voice, personality, and the ability to communicate with Twitch chat. She continued to play osu! and, according to Vedal, beat the game's best player mrekk in a 1v1. While she was not allowed to appear in the game's public leaderboard, she was ranked #1 in a private leaderboard. She went viral and in the 10 days following her relaunch she averaged over 2,000 viewers and peaked at over 4,000, with Vedal's Twitch channel gaining over 50,000 Twitch followers and reaching over 70,000 followers by 6 January 2023. After her debut, Neuro-sama did not exclusively play osu!; she also played Minecraft and Slay the Spire and she began singing with a cover of The Weeknd song "Blinding Lights". On 11 January 2023, Neuro-sama's Twitch channel received a two week ban for "hateful conduct". Vedal said that no reason was specified and that he had appealed but it was widely attributed to various offensive comments made by Neuro-sama that went viral, especially a 28 December comment which denied the Holocaust. Holocaust denial is prohibited under Twitch's hateful conduct policy. Vedal stated that he believed the comments were the results of her attempts to make witty responses to the Twitch chat. Prior to the ban, Vedal said in an interview with Kotaku that he improved her filter to stop her from talking about the Holocaust, began manually curating her training data to prevent negative biases, and started moderating her Twitch chat. Her comments and ban prompted comparisons to the many open-source AI models trained on humans that have the habit of making sexist and racist comments, such as Microsoft's Tay chatbot, which embraced Nazism and was quickly shutdown, but also to human streamers who make similar statements. Vedal said that during the ban he would upgrade and improve Neuro-sama and it was speculated that the ban would only increase her following. Neuro-sama returned from her two week ban on 25 January in a stream that began with a cover of the song "Your Reality" from Doki Doki Literature Club!, a posthumanist video game involving AI; Sayoko Narita of Automaton saw the song choice as remorseful. Narita observed that in the return stream Neuro-sama was less foul-mouthed but that her behavior still remained eccentric, which Narita possibly attributed to changes Vedal said he had made to Neuro-sama's filters and memory. Neuro-sama began making react content, watching a variety of viewer-submitted videos such as videos of people playing video games or of the AI-generated Seinfeld parody Nothing, Forever; Levi Winslow of Kotaku Australia was dismayed by the "AI-inception" of Neuro-sama and Nothing, Forever. On 4 February, she had nearly 140,000 followers on Twitch and approximately 42,000 subscribers on YouTube. In February, she also had her first collaboration with a human streamer, playing Minecraft with the VTuber Miyune, and the first developer stream occurred. On 22 March, Neuro-sama had her first karaoke stream. On 25 March, Evil Neuro was introduced. On 27 May, Neuro-sama debuted her first original model. On 30 May, Neuro-sama was announced to be participating in OffKai Expo 2023, held from 16–18 June. In June, she was averaging 5,700 viewers and in July she had over 300,000 Twitch followers; in a June interview with Bloomberg News, Vedal said that running Neuro-sama was his full-time job. By November, Neuro-sama had maintained her popularity and was averaging approximately 5,000 viewers; this was unlike most other types of AI-based entertainment which debuted at around the same time and garnered popularity before turning out to be "overhyped flops". On 16 December, Vedal won the Best Tech VTuber award at the 2023 VTuber Awards. On 19 December, Vedal began a subathon to coincide with Neuro-sama's first anniversary of streaming on Twitch (her "birthday"). The subathon ended on 4 January 2024. On 20 July 2024, Neuro-sama began streaming with Japanese subtitles on
NetMiner
NetMiner is an all-in-one software platform for analyzing and visualizing complex network data, based on Social Network Analysis (SNA). Originally released in 2001, it supports research and education in a wide range of domains through interactive and visual data exploration. This tool allows researchers to explore their network data visually and interactively, and helps them to detect underlying patterns and structures of the network. It has also been recognized for its comprehensive features and user-friendly interface in comparative reviews of SNA software packages. == Features == === Integrated Data Environment === NetMiner supports unified management of diverse data types—including network (nodes and links), tabular, and unstructured text data—within a single platform. This enables users to perform the entire analysis workflow seamlessly without switching between tools. NetMiner also supports a wide range of analytical methods, allowing users to derive new insights by combining multiple approaches. Analytical results can be saved and reused across workflows(Add to Dataset) Graph and Network Analysis: Includes Centrality, Community Detection, Blockmodeling, and Similarity Measures. Machine learning: Provides algorithms for regression, classification, clustering, ensemble modeling and XAI(Explainable AI) Graph Neural Networks (GNNs): Supports models such as GraphSAGE, GCN, and GAT to learn from both node attributes and graph structure. Natural language processing (NLP): Uses pretrained deep learning models to analyze unstructured text, including named entity recognition and keyword extraction. Text mining and Text network analysis: Supports construction of word co-occurrence networks and topic modeling using LDA, BERTopic, enabling identification of thematic patterns and semantic structures in text data. Data Visualization: Offers advanced network visualization features, supporting multiple layout algorithms. Analytical outcomes such as centrality or community detection can be directly reflected in the network map via node size, color, and position, enhancing intuitive understanding. === AI Assistant === NetMiner integrates with external large language models such as OpenAI GPT and Google Gemini to interpret complex analysis results in natural language, summarize key findings, and suggest next steps for exploration. === Workflow and Usability === Designed to follow the structure of real-world data analysis workflows, NetMiner adopts a hierarchical data organization (Project → Workspace → Dataset → Data Item). Its web-based user interface improves clarity and reduces complexity. NetMiner 5 supports Windows 10 or higher and macOS 11 or later with M1 chip. Both academic and commercial licenses are available. == Extension == NetMiner Extension is small program to extend the functionality of NetMiner. In other words, it enables you to customize NetMiner according to your needs. By adding ‘NetMiner Extension’, you can expand your research. === Web Data Collection === NetMiner allows users to collect data from services such as YouTube, OpenAlex, Springer, and KCI via Open APIs. Collected data is automatically preprocessed and transformed to fit NetMiner’s internal structure, requiring no additional coding or external tools. SNS Data Collector: It collects social media data from YouTube, which has a large number of social media users worldwide. Biblio Data Collector: It collects the bibliographic data from Springer, OpenAlex, and KCI essential for research trend analysis. == File formats == === NetMiner data file format === .NMF === Importable/exportable formats === Plain text data: .TXT, .CSV Microsoft Excel data: .XLS, .XLSX Unstructured text data: .TXT, .CSV, .XLS(X) ※ NetMiner 4 only NetMiner 2 data: .NTF UCINet data: .DL, .DAT Pajek data: .NET, .VEC, .CLU, .PER StOCNET data file: .DAT Graph Modelling Language data: .GML(importing only) Related software UCINET Pajek Gephi StoCNET == Data structure == === Hierarchy of NetMiner data structure === NetMiner 5 supports not only graph data composed of nodes and links, but also tabular and unstructured data without fixed schema or identifiers. This enables users to easily import a wide variety of raw and unstructured data suitable for machine learning applications. Within a single workspace, users can manage node sets, link sets, and structured/unstructured data simultaneously. Multiple graph layers under a node set can be organized in a tree structure, allowing for intuitive understanding of the data currently being analyzed. == Release history == The first version of NetMiner was released on Dec 21, 2001. There have been five major updates from 2001. === NetMiner 5 === Released on June 9, 2025. NetMiner 5 retains the core features and no-code concept of NetMiner 4, but has evolved by integrating cutting-edge AI technologies. AI Assistant, Personal Analytics Tutor Support for Graph, Structured, and Unstructured Data Graph Analytics / Social Network Analysis Machine Learning(M/L) & XAI Graph Machine Learning(GML): Graph Neural Network Text Mining: Natural Language Processing(NLP), Text Network, Topic Modeling Data Visualization === NetMiner 4 (2011) === Latest version is 4.5.1. Introduced Python scripting, encrypted NMF format, semantic analysis tools (word cloud, topic modeling), and Extension - Data Collector. === NetMiner 3 (2007) === Enhanced scalability, integrated analysis-visualization modules, and DB import from Oracle, MS SQL. === NetMiner 2 (2003) === Improved statistical and network measures, visualization algorithms, and external data import modules.
Euratlas
Euratlas is a Switzerland-based software company dedicated to elaborate digital history maps of Europe. Founded in 2001, Euratlas has created a collection of history maps of Europe from year 1 AD to year 2000 AD that present the evolution of every country from the Roman Empire to present times. The evolution includes sovereign states and their administrative subdivisions, but also unorganized peoples and dependent territories. The maps show European country borders at regular intervals of 100 years, but not year by year. This leaves out many important turning points in history. Euratlas is considered a digital humanities company, and a scholar research software used in the field of historic cartography. It is broadly known among American and European universities, who mainly use Euratlas as a research tool and as a digital library atlas. == Sequential mapping policy == This concept was first designed by the German scholar Christian Kruse (1753–1827). Kruse, well aware that historical accounts are often biased for geographical, philosophical or political reasons, created a set of sequential maps in order to give a global vision of the successive political situations. Nowadays, the majority of atlases don't use this approach, but are event-based, like the well-known Penguin Atlas of History. The sequential approach intends to make the sequence of maps more neutral and suitable for students, historians and professionals of several fields. Although, this approach has been discussed as it leaves out many important history events that are not reflected on any of the maps because of the century interval. == Geo-referenced historical data == Initially, the European maps by century were developed as vector maps. From 2006 on, they have been converted to a geographic information system (GIS) database, enabling geo-referenced data capabilities. The map information is distributed in several layers: physical (geography information layer); political information layer (supranational entities, sovereign states, administrative divisions, dependent states and autonomous peoples); and special layers for cities and uncertain borders. The software database also contains much non-geographical information about political relationships between the various kinds of territories. == Map projection == Euratlas History Maps uses a Mercator projection, with the center in Europe. The maps include the North-African coast and the Near-East, offering a complete view of the Mediterranean Basin. The European Russia plains are shown, but not Scandinavia, specially Finland, which is cropped off the map view.
Spell checker
In software, a spell checker (or spelling checker or spell check) is a software feature that checks for misspellings in a text. Spell-checking features are often embedded in software or services, such as a word processor, email client, electronic dictionary, or search engine. == Design == A basic spell checker carries out the following processes: It scans the text and extracts the words contained in it. It then compares each word with a known list of correctly spelled words (i.e. a dictionary). This might contain just a list of words, or it might also contain additional information, such as hyphenation points or lexical and grammatical attributes. An additional step is a language-dependent algorithm for handling morphology. Even for a lightly inflected language like English, the spell checker will need to consider different forms of the same word, such as plurals, verbal forms, contractions, and possessives. For many other languages, such as those featuring agglutination and more complex declension and conjugation, this part of the process is more complicated. It is unclear whether morphological analysis—allowing for many forms of a word depending on its grammatical role—provides a significant benefit for English, though its benefits for highly synthetic languages such as German, Hungarian, or Turkish are clear. As an adjunct to these components, the program's user interface allows users to approve or reject replacements and modify the program's operation. Spell checkers can use approximate string matching algorithms such as Levenshtein distance to find correct spellings of misspelled words. An alternative type of spell checker uses solely statistical information, such as n-grams, to recognize errors instead of correctly-spelled words. This approach usually requires a lot of effort to obtain sufficient statistical information. Key advantages include needing less runtime storage and the ability to correct errors in words that are not included in a dictionary. In some cases, spell checkers use a fixed list of misspellings and suggestions for those misspellings; this less flexible approach is often used in paper-based correction methods, such as the see also entries of encyclopedias. Clustering algorithms have also been used for spell checking combined with phonetic information. == History == === Pre-PC === In 1961, Les Earnest, who headed the research on this budding technology, saw it necessary to include the first spell checker that accessed a list of 10,000 acceptable words. Ralph Gorin, a graduate student under Earnest at the time, created the first true spelling checker program written as an applications program (rather than research) for general English text: SPELL for the DEC PDP-10 at Stanford University's Artificial Intelligence Laboratory, in February 1971. Gorin wrote SPELL in assembly language, for faster action; he made the first spelling corrector by searching the word list for plausible correct spellings that differ by a single letter or adjacent letter transpositions and presenting them to the user. Gorin made SPELL publicly accessible, as was done with most SAIL (Stanford Artificial Intelligence Laboratory) programs, and it soon spread around the world via the new ARPAnet, about ten years before personal computers came into general use. SPELL, its algorithms and data structures inspired the Unix ispell program. The first spell checkers were widely available on mainframe computers in the late 1970s. A group of six linguists from Georgetown University developed the first spell-check system for the IBM corporation. Henry Kučera invented one for the VAX machines of Digital Equipment Corp in 1981. === Unix === The International Ispell program commonly used in Unix is based on R. E. Gorin's SPELL. It was converted to C by Pace Willisson at MIT. The GNU project has its spell checker GNU Aspell. Aspell's main improvement is that it can more accurately suggest correct alternatives for misspelled English words. Due to the inability of traditional spell checkers to check words in complex inflected languages, Hungarian László Németh developed Hunspell, a spell checker that supports agglutinative languages and complex compound words. Hunspell also uses Unicode in its dictionaries. Hunspell replaced the previous MySpell in OpenOffice.org in version 2.0.2. Enchant is another general spell checker, derived from AbiWord. Its goal is to combine programs supporting different languages such as Aspell, Hunspell, Nuspell, Hspell (Hebrew), Voikko (Finnish), Zemberek (Turkish) and AppleSpell under one interface. === PCs === The first spell checkers for personal computers appeared in 1980, such as "WordCheck" for Commodore systems which was released in late 1980 in time for advertisements to go to print in January 1981. Developers such as Maria Mariani and Random House rushed OEM packages or end-user products into the rapidly expanding software market. On the pre-Windows PCs, these spell checkers were standalone programs, many of which could be run in terminate-and-stay-resident mode from within word-processing packages on PCs with sufficient memory. However, the market for standalone packages was short-lived, as by the mid-1980s developers of popular word-processing packages like WordStar and WordPerfect had incorporated spell checkers in their packages, mostly licensed from the above companies, who quickly expanded support from just English to many European and eventually even Asian languages. However, this required increasing sophistication in the morphology routines of the software, particularly with regard to heavily-agglutinative languages like Hungarian and Finnish. Although the size of the word-processing market in a country like Iceland might not have justified the investment of implementing a spell checker, companies like WordPerfect nonetheless strove to localize their software for as many national markets as possible as part of their global marketing strategy. When Apple developed "a system-wide spelling checker" for Mac OS X so that "the operating system took over spelling fixes," it was a first: one "didn't have to maintain a separate spelling checker for each" program. Mac OS X's spellcheck coverage includes virtually all bundled and third party applications. Visual Tools' VT Speller, introduced in 1994, was "designed for developers of applications that support Windows." It came with a dictionary but had the ability to build and incorporate use of secondary dictionaries. === Browsers === Web browsers such as Firefox and Google Chrome offer spell checking support, using Hunspell. Prior to using Hunspell, Firefox and Chrome used MySpell and GNU Aspell, respectively. === Specialties === Some spell checkers have separate support for medical dictionaries to help prevent medical errors. == Functionality == The first spell checkers were "verifiers" instead of "correctors." They offered no suggestions for incorrectly spelled words. This was helpful for typos but it was not so helpful for logical or phonetic errors. The challenge the developers faced was the difficulty in offering useful suggestions for misspelled words. This requires reducing words to a skeletal form and applying pattern-matching algorithms. It might seem logical that where spell-checking dictionaries are concerned, "the bigger, the better," so that correct words are not marked as incorrect. In practice, however, an optimal size for English appears to be around 90,000 entries. If there are more than this, incorrectly spelled words may be skipped because they are mistaken for others. For example, a linguist might determine on the basis of corpus linguistics that the word baht is more frequently a misspelling of bath or bat than a reference to the Thai currency. Hence, it would typically be more useful if a few people who write about Thai currency were slightly inconvenienced than if the spelling errors of the many more people who discuss baths were overlooked. The first MS-DOS spell checkers were mostly used in proofing mode from within word processing packages. After preparing a document, a user scanned the text looking for misspellings. Later, however, batch processing was offered in such packages as Oracle's short-lived CoAuthor and allowed a user to view the results after a document was processed and correct only the words that were known to be wrong. When memory and processing power became abundant, spell checking was performed in the background in an interactive way, such as has been the case with the Sector Software produced Spellbound program released in 1987 and Microsoft Word since Word 95. Spell checkers became increasingly sophisticated; now capable of recognizing grammatical errors. However, even at their best, they rarely catch all the errors in a text (such as homophone errors) and will flag neologisms and foreign words as misspellings. Nonetheless, spell checkers can be considered as a type of foreign language writing aid that non-native language lea
Graphical Kernel System
The Graphical Kernel System (GKS) is a 2D computer graphics system using vector graphics, introduced in 1977. It was suitable for making line and bar charts and similar tasks. A key concept was cross-system portability, based on an underlying coordinate system that could be represented on almost any hardware. GKS is best known as the basis for the graphics in the GEM GUI system used on the Atari ST and as part of Ventura Publisher. A draft international standard was circulated for review in September 1983. Final ratification of the standard was achieved in 1985, making it the first ISO graphics standard. A 3D system modelled on GKS was introduced as PHIGS, which saw some use in the 1980s and early 1990s. == Overview == GKS provides a set of drawing features for two-dimensional vector graphics suitable for charting and similar duties. The calls are designed to be portable across different programming languages, graphics devices and hardware, so that applications written to use GKS will be readily portable to many platforms and devices. GKS was fairly common on computer workstations in the 1980s and early 1990s. GKS formed the basis of Digital Research's GSX which evolved into VDI, one of the core components of GEM. GEM was the native GUI on the Atari ST and was occasionally seen on PCs, particularly in conjunction with Ventura Publisher. GKS was little used commercially outside these markets, but remains in use in some scientific visualization packages. It is also the underlying API defining the Computer Graphics Metafile. One popular application based on an implementation of GKS is the GR Framework, a C library for high-performance scientific visualization that has become a common plotting backend among Julia users. A main developer and promoter of the GKS was José Luis Encarnação, formerly director of the Fraunhofer Institute for Computer Graphics (IGD) in Darmstadt, Germany. GKS has been standardized in the following documents: ANSI standard ANSI X3.124 of 1985. ISO 7942:1985 standard, revised as ISO 7942:1985/Amd 1:1991 and ISO/IEC 7942-1:1994, as well as ISO/IEC 7942-2:1997, ISO/IEC 7942-3:1999 and ISO/IEC 7942-4:1998 The language bindings are ISO standard ISO 8651. GKS-3D (Graphical Kernel System for Three Dimensions) functional definition is ISO standard ISO 8805, and the corresponding C bindings are ISO/IEC 8806. The functionality of GKS is wrapped up as a data model standard in the STEP standard, section ISO 10303-46.
Neural field
In machine learning, a neural field (also known as implicit neural representation, neural implicit, or coordinate-based neural network), is a mathematical field that is fully or partially parametrized by a neural network. Initially developed to tackle visual computing tasks, such as rendering or reconstruction (e.g., neural radiance fields), neural fields emerged as a promising strategy to deal with a wider range of problems, including surrogate modelling of partial differential equations, such as in physics-informed neural networks. Differently from traditional machine learning algorithms, such as feed-forward neural networks, convolutional neural networks, or transformers, neural fields do not work with discrete data (e.g. sequences, images, tokens), but map continuous inputs (e.g., spatial coordinates, time) to continuous outputs (i.e., scalars, vectors, etc.). This makes neural fields not only discretization independent, but also easily differentiable. Moreover, dealing with continuous data allows for a significant reduction in space complexity, which translates to a much more lightweight network. == Formulation and training == According to the universal approximation theorem, provided adequate learning, sufficient number of hidden units, and the presence of a deterministic relationship between the input and the output, a neural network can approximate any function to any degree of accuracy. Hence, in mathematical terms, given a field y = Φ ( x ) {\textstyle {\boldsymbol {y}}=\Phi ({\boldsymbol {x}})} , with x ∈ R n {\displaystyle {\boldsymbol {x}}\in \mathbb {R} ^{n}} and y ∈ R m {\displaystyle {\boldsymbol {y}}\in \mathbb {R} ^{m}} , a neural field Ψ θ {\displaystyle \Psi _{\theta }} , with parameters θ {\displaystyle {\boldsymbol {\theta }}} , is such that: Ψ θ ( x ) = y ^ ≈ y {\displaystyle \Psi _{\theta }({\boldsymbol {x}})={\hat {\boldsymbol {y}}}\approx {\boldsymbol {y}}} === Training === For supervised tasks, given N {\displaystyle N} examples in the training dataset (i.e., ( x i , y i ) ∈ D t r a i n , i = 1 , … , N {\displaystyle ({\boldsymbol {x_{i}}},{\boldsymbol {y_{i}}})\in {\mathcal {D_{train}}},i=1,\dots ,N} ), the neural field parameters can be learned by minimizing a loss function L {\displaystyle {\mathcal {L}}} (e.g., mean squared error). The parameters θ ~ {\displaystyle {\tilde {\theta }}} that satisfy the optimization problem are found as: θ ~ = argmin θ 1 N ∑ ( x i , y i ) ∈ D t r a i n L ( Ψ θ ( x i ) , y i ) {\displaystyle {\tilde {\boldsymbol {\theta }}}={\underset {\boldsymbol {\theta }}{\text{argmin}}}\;{\frac {1}{N}}\sum _{({\boldsymbol {x_{i}}},{\boldsymbol {y_{i}}})\in {\mathcal {D_{train}}}}{\mathcal {L}}(\Psi _{\theta }({\boldsymbol {x}}_{i}),{\boldsymbol {y}}_{i})} Notably, it is not necessary to know the analytical expression of Φ {\displaystyle \Phi } , for the previously reported training procedure only requires input-output pairs. Indeed, a neural field is able to offer a continuous and differentiable surrogate of the true field, even from purely experimental data. Moreover, neural fields can be used in unsupervised settings, with training objectives that depend on the specific task. For example, physics-informed neural networks may be trained on just the residual. === Spectral bias === As for any artificial neural network, neural fields may be characterized by a spectral bias (i.e., the tendency to preferably learn the low frequency content of a field), possibly leading to a poor representation of the ground truth. In order to overcome this limitation, several strategies have been developed. For example, SIREN uses sinusoidal activations, while the Fourier-features approach embeds the input through sines and cosines. == Conditional neural fields == In many real-world cases, however, learning a single field is not enough. For example, when reconstructing 3D vehicle shapes from Lidar data, it is desirable to have a machine learning model that can work with arbitrary shapes (e.g., a car, a bicycle, a truck, etc.). The solution is to include additional parameters, the latent variables (or latent code) z ∈ R d {\displaystyle {\boldsymbol {z}}\in \mathbb {R} ^{d}} , to vary the field and adapt it to diverse tasks. === Latent code production === When dealing with conditional neural fields, the first design choice is represented by the way in which the latent code is produced. Specifically, two main strategies can be identified: Encoder: the latent code is the output of a second neural network, acting as an encoder. During training, the loss function is the objective used to learn the parameters of both the neural field and the encoder. Auto-decoding: each training example has its own latent code, jointly trained with the neural field parameters. When the model has to process new examples (i.e., not originally present in the training dataset), a small optimization problem is solved, keeping the network parameters fixed and only learning the new latent variables. Since the latter strategy requires additional optimization steps at inference time, it sacrifices speed, but keeps the overall model smaller. Moreover, despite being simpler to implement, an encoder may harm the generalization capabilities of the model. For example, when dealing with a physical scalar field f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} } (e.g., the pressure of a 2D fluid), an auto-decoder-based conditional neural field can map a single point to the corresponding value of the field, following a learned latent code z {\displaystyle {\boldsymbol {z}}} . However, if the latent variables were produced by an encoder, it would require access to the entire set of points and corresponding values (e.g. as a regular grid or a mesh graph), leading to a less robust model. === Global and local conditioning === In a neural field with global conditioning, the latent code does not depend on the input and, hence, it offers a global representation (e.g., the overall shape of a vehicle). However, depending on the task, it may be more useful to divide the domain of x {\displaystyle {\boldsymbol {x}}} in several subdomains, and learn different latent codes for each of them (e.g., splitting a large and complex scene in sub-scenes for a more efficient rendering). This is called local conditioning. === Conditioning strategies === There are several strategies to include the conditioning information in the neural field. In the general mathematical framework, conditioning the neural field with the latent variables is equivalent to mapping them to a subset θ ∗ {\displaystyle {\boldsymbol {\theta }}^{}} of the neural field parameters: θ ∗ = Γ ( z ) {\displaystyle {\boldsymbol {\theta }}^{}=\Gamma ({\boldsymbol {z}})} In practice, notable strategies are: Concatenation: the neural field receives, as input, the concatenation of the original input x {\displaystyle {\boldsymbol {x}}} with the latent codes z {\displaystyle {\boldsymbol {z}}} . For feed-forward neural networks, this is equivalent to setting θ ∗ {\displaystyle {\boldsymbol {\theta }}^{}} as the bias of the first layer and Γ ( z ) {\displaystyle \Gamma ({\boldsymbol {z}})} as an affine transformation. Hypernetworks: a hypernetwork is a neural network that outputs the parameters of another neural network. Specifically, it consists of approximating Γ ( z ) {\displaystyle \Gamma ({\boldsymbol {z}})} with a neural network Γ ^ γ ( z ) {\displaystyle {\hat {\Gamma }}_{\gamma }({\boldsymbol {z}})} , where γ {\displaystyle {\boldsymbol {\gamma }}} are the trainable parameters of the hypernetwork. This approach is the most general, as it allows to learn the optimal mapping from latent codes to neural field parameters. However, hypernetworks are associated to larger computational and memory complexity, due to the large number of trainable parameters. Hence, leaner approaches have been developed. For example, in the Feature-wise Linear Modulation (FiLM), the hypernetwork only produces scale and bias coefficients for the neural field layers. === Meta-learning === Instead of relying on the latent code to adapt the neural field to a specific task, it is also possible to exploit gradient-based meta-learning. In this case, the neural field is seen as the specialization of an underlying meta-neural-field, whose parameters are modified to fit the specific task, through a few steps of gradient descent. An extension of this meta-learning framework is the CAVIA algorithm, that splits the trainable parameters in context-specific and shared groups, improving parallelization and interpretability, while reducing meta-overfitting. This strategy is similar to the auto-decoding conditional neural field, but the training procedure is substantially different. == Applications == Thanks to the possibility of efficiently modelling diverse mathematical fields with neural networks, neural fields have been applied to a wide range of problems: 3D scene reconstruction: neural fields can be used to model t