In artificial intelligence research, the situated approach builds agents that are designed to behave effectively successfully in their environment. This requires designing AI "from the bottom-up" by focussing on the basic perceptual and motor skills required to survive. The situated approach gives a much lower priority to abstract reasoning or problem-solving skills. The approach was originally proposed as an alternative to traditional approaches (that is, approaches popular before 1985 or so). After several decades, classical AI technologies started to face intractable issues (e.g. combinatorial explosion) when confronted with real-world modeling problems. All approaches to address these issues focus on modeling intelligences situated in an environment. They have become known as the situated approach to AI. == Emergence of a concept == === From traditional AI to Nouvelle AI === During the late 1980s, the approach now known as Nouvelle AI (Nouvelle means new in French) was pioneered at the MIT Artificial Intelligence Laboratory by Rodney Brooks. As opposed to classical or traditional artificial intelligence, Nouvelle AI purposely avoided the traditional goal of modeling human-level performance, but rather tries to create systems with intelligence at the level of insects, closer to real-world robots. But eventually, at least at MIT new AI did lead to an attempt for humanoid AI in the Cog Project. === From Nouvelle AI to behavior-based and situated AI === The conceptual shift introduced by nouvelle AI flourished in the robotics area, given way to behavior-based robotics (BBR), a methodology for developing AI based on a modular decomposition of intelligence. It was made famous by Rodney Brooks: his subsumption architecture was one of the earliest attempts to describe a mechanism for developing BBAI. It is extremely popular in robotics and to a lesser extent to implement intelligent virtual agents because it allows the successful creation of real-time dynamic systems that can run in complex environments. For example, it underlies the intelligence of the Sony Aibo and many RoboCup robot teams. Realizing that in fact all these approaches were aiming at building not an abstract intelligence, but rather an intelligence situated in a given environment, they have come to be known as the situated approach. In fact, this approach stems out from early insights of Alan Turing, describing the need to build machines equipped with sense organs to learn directly from the real-world instead of focusing on abstract activities, such as playing chess. == Definitions == Classically, a software entity is defined as a simulated element, able to act on itself and on its environment, and which has an internal representation of itself and of the outside world. An entity can communicate with other entities, and its behavior is the consequence of its perceptions, its representations, and its interactions with the other entities. === AI loop === Simulating entities in a virtual environment requires simulating the entire process that goes from a perception of the environment, or more generally from a stimulus, to an action on the environment. This process is called the AI loop and technology used to simulate it can be subdivided in two categories. Sensorimotor or low-level AI deals with either the perception problem (what is perceived?) or the animation problem (how are actions executed?). Decisional or high-level AI deals with the action selection problem (what is the most appropriate action in response to a given perception, i.e. what is the most appropriate behavior?). === Traditional or symbolic AI === There are two main approaches in decisional AI. The vast majority of the technologies available on the market, such as planning algorithms, finite-state machines (FSA), or expert systems, are based on the traditional or symbolic AI approach. Its main characteristics are: It is top-down: it subdivides, in a recursive manner, a given problem into a series of sub-problems that are supposedly easier to solve. It is knowledge-based: it relies on a symbolic description of the world, such as a set of rules. However, the limits of traditional AI, which goal is to build systems that mimic human intelligence, are well-known: inevitably, a combinatorial explosion of the number of rules occurs due to the complexity of the environment. In fact, it is impossible to predict all the situations that will be encountered by an autonomous entity. === Situated or behavioral AI === In order to address these issues, another approach to decisional AI, also known as situated or behavioral AI, has been proposed. It does not attempt to model systems that produce deductive reasoning processes, but rather systems that behave realistically in their environment. The main characteristics of this approach are the following: It is bottom-up: it relies on elementary behaviors, which can be combined to implement more complex behaviors. It is behavior-based: it does not rely on a symbolic description of the environment, but rather on a model of the interactions of the entities with their environment. The goal of situated AI is to model entities that are autonomous in their environment. This is achieved thanks to both the intrinsic robustness of the control architecture, and its adaptation capabilities to unforeseen situations. === Situated agents === In artificial intelligence and cognitive science, the term situated refers to an agent which is embedded in an environment. The term situated is commonly used to refer to robots, but some researchers argue that software agents can also be situated if: they exist in a dynamic (rapidly changing) environment, which they can manipulate or change through their actions, and which they can sense or perceive. Examples might include web-based agents, which can alter data or trigger processes (such as purchases) over the Internet, or virtual-reality bots which inhabit and change virtual worlds, such as Second Life. Being situated is generally considered to be part of being embodied, but it is useful to consider each perspective individually. The situated perspective emphasizes that intelligent behavior derives from the environment and the agent's interactions with it. The nature of these interactions are defined by an agent's embodiment. == Implementation principles == === Modular decomposition === The most important attribute of a system driven by situated AI is that the intelligence is controlled by a set of independent semi-autonomous modules. In the original systems, each module was actually a separate device or was at least conceived of as running on its own processing thread. Generally, though, the modules are just abstractions. In this respect, situated AI may be seen as a software engineering approach to AI, perhaps akin to object oriented design. Situated AI is often associated with reactive planning, but the two are not synonymous. Brooks advocated an extreme version of cognitive minimalism which required initially that the behavior modules were finite-state machines and thus contained no conventional memory or learning. This is associated with reactive AI because reactive AI requires reacting to the current state of the world, not to an agent's memory or preconception of that world. However, learning is obviously key to realistic strong AI, so this constraint has been relaxed, though not entirely abandoned. === Action selection mechanism === The situated AI community has presented several solutions to modeling decision-making processes, also known as action selection mechanisms. The first attempt to solve this problem goes back to subsumption architectures, which were in fact more an implementation technique than an algorithm. However, this attempt paved the way to several others, in particular the free-flow hierarchies and activation networks. A comparison of the structure and performances of these two mechanisms demonstrated the advantage of using free-flow hierarchies in solving the action selection problem. However, motor schemas and process description languages are two other approaches that have been used with success for autonomous robots. == Notes and references == Arsenio, Artur M. (2004) Towards an embodied and situated AI, In: Proceedings of the International FLAIRS conference, 2004. (online) The Artificial Life Route To Artificial Intelligence: Building Embodied, Situated Agents, Luc Steels and Rodney Brooks Eds., Lawrence Erlbaum Publishing, 1995. (ISBN 978-0805815184) Rodney A. Brooks Cambrian Intelligence (MIT Press, 1999) ISBN 0-262-52263-2; collection of early papers including "Intelligence without representation" and "Intelligence without reason", from 1986 & 1991 respectively. Ronald C. Arkin Behavior-Based Robotics (MIT Press, 1998) ISBN 0-262-01165-4 Hendriks-Jansen, Horst (1996) Catching Ourselves in the Act: Situated Activity, Interactive Emergence, Evolution, and Human Thought. Cambridge, Mass.: MIT Press.
Automatic acquisition of sense-tagged corpora
The knowledge acquisition bottleneck is perhaps the major impediment to solving the word-sense disambiguation (WSD) problem. Unsupervised learning methods rely on knowledge about word senses, which is barely formulated in dictionaries and lexical databases. Supervised learning methods depend heavily on the existence of manually annotated examples for every word sense, a requisite that can so far be met only for a handful of words for testing purposes, as it is done in the Senseval exercises. == Existing methods == Therefore, one of the most promising trends in WSD research is using the largest corpus ever accessible, the World Wide Web, to acquire lexical information automatically. WSD has been traditionally understood as an intermediate language engineering technology which could improve applications such as information retrieval (IR). In this case, however, the reverse is also true: Web search engines implement simple and robust IR techniques that can be successfully used when mining the Web for information to be employed in WSD. The most direct way of using the Web (and other corpora) to enhance WSD performance is the automatic acquisition of sense-tagged corpora, the fundamental resource to feed supervised WSD algorithms. Although this is far from being commonplace in the WSD literature, a number of different and effective strategies to achieve this goal have already been proposed. Some of these strategies are: acquisition by direct Web searching (searches for monosemous synonyms, hypernyms, hyponyms, parsed gloss' words, etc.), Yarowsky algorithm (bootstrapping), acquisition via Web directories, and acquisition via cross-language meaning evidences. == Summary == === Optimistic results === The automatic extraction of examples to train supervised learning algorithms reviewed has been, by far, the best explored approach to mine the web for word-sense disambiguation. Some results are certainly encouraging: In some experiments, the quality of the Web data for WSD equals that of human-tagged examples. This is the case of the monosemous relatives plus bootstrapping with Semcor seeds technique and the examples taken from the ODP Web directories. In the first case, however, Semcor-size example seeds are necessary (and only available for English), and it has only been tested with a very limited set of nouns; in the second case, the coverage is quite limited, and it is not yet clear whether it can be grown without compromising the quality of the examples retrieved. It has been shown that a mainstream supervised learning technique trained exclusively with web data can obtain better results than all unsupervised WSD systems which participated at Senseval-2. Web examples made a significant contribution to the best Senseval-2 English all-words system. === Difficulties === There are, however, several open research issues related to the use of Web examples in WSD: High precision in the retrieved examples (i.e., correct sense assignments for the examples) does not necessarily lead to good supervised WSD results (i.e., the examples are possibly not useful for training). The most complete evaluation of Web examples for supervised WSD indicates that learning with Web data improves over unsupervised techniques, but the results are nevertheless far from those obtained with hand-tagged data, and do not even beat the most-frequent-sense baseline. Results are not always reproducible; the same or similar techniques may lead to different results in different experiments. Compare, for instance, Mihalcea (2002) with Agirre and Martínez (2004), or Agirre and Martínez (2000) with Mihalcea and Moldovan (1999). Results with Web data seem to be very sensitive to small differences in the learning algorithm, to when the corpus was extracted (search engines change continuously), and on small heuristic issues (e.g., differences in filters to discard part of the retrieved examples). Results are strongly dependent on bias (i.e., on the relative frequencies of examples per word sense). It is unclear whether this is simply a problem of Web data, or an intrinsic problem of supervised learning techniques, or just a problem of how WSD systems are evaluated (indeed, testing with rather small Senseval data may overemphasize sense distributions compared to sense distributions obtained from the full Web as corpus). In any case, Web data has an intrinsic bias, because queries to search engines directly constrain the context of the examples retrieved. There are approaches that alleviate this problem, such as using several different seeds/queries per sense or assigning senses to Web directories and then scanning directories for examples; but this problem is nevertheless far from being solved. Once a Web corpus of examples is built, it is not entirely clear whether its distribution is safe from a legal perspective. === Future === Besides automatic acquisition of examples from the Web, there are some other WSD experiments that have profited from the Web: The Web as a social network has been successfully used for cooperative annotation of a corpus (OMWE, Open Mind Word Expert project), which has already been used in three Senseval-3 tasks (English, Romanian and Multilingual). The Web has been used to enrich WordNet senses with domain information: topic signatures and Web directories, which have in turn been successfully used for WSD. Also, some research benefited from the semantic information that the Wikipedia maintains on its disambiguation pages. It is clear, however, that most research opportunities remain largely unexplored. For instance, little is known about how to use lexical information extracted from the Web in knowledge-based WSD systems; and it is also hard to find systems that use Web-mined parallel corpora for WSD, even though there are already efficient algorithms that use parallel corpora in WSD.
Albert One
Albert One is an artificial intelligence chatbot created by Robby Garner and designed to mimic the way humans make conversations using a multi-faceted approach in natural language programming. == History == In both 1998 and 1999, Albert One won the Loebner Prize Contest, a competition between chatterbots. Some parts of Albert were deployed on the internet beginning in 1995, to gather information about what kinds of things people would say to a chatterbot. Another element of Albert One involved the building of a large database of human statements, and associated replies. This portion of the project was tested at the 1994-1997 Loebner Prize contests. Albert was the first of Robby Garner's multifaceted bots. The Albert One system was composed of several subsystems. Among those were a version of Eliza, the therapist, Elivs, another Eliza-like bot, and several other helper applications working together in a hierarchical arrangement. As a continuation of the stimulus-response library, various other database queries and assertions were tested to arrive at each of Albert's responses. Robby went on to develop networked examples of this kind of hierarchical "glue" at The Turing Hub.
Chinchilla (language model)
Chinchilla is a family of large language models (LLMs) developed by the research team at Google DeepMind, presented in March 2022. == Models == It is named "chinchilla" because it is a further development over a previous model family named Gopher. Both model families were trained in order to investigate the scaling laws of large language models. It claimed to outperform GPT-3. It considerably simplifies downstream utilization because it requires much less computer power for inference and fine-tuning. Based on the training of previously employed language models, it has been determined that if one doubles the model size, one must also have twice the number of training tokens. This hypothesis has been used to train Chinchilla by DeepMind. Similar to Gopher in terms of cost, Chinchilla has 70B parameters and four times as much data. Chinchilla has an average accuracy of 67.5% on the Measuring Massive Multitask Language Understanding (MMLU) benchmark, which is 7% higher than Gopher's performance. Chinchilla was still in the testing phase as of January 12, 2023. Chinchilla contributes to developing an effective training paradigm for large autoregressive language models with limited compute resources. The Chinchilla team recommends that the number of training tokens is twice for every model size doubling, meaning that using larger, higher-quality training datasets can lead to better results on downstream tasks. It has been used for the Flamingo vision-language model. == Architecture == Both the Gopher family and Chinchilla family are families of transformer models. In particular, they are essentially the same as GPT-2, with different sizes and minor modifications. Gopher family uses RMSNorm instead of LayerNorm; relative positional encoding rather than absolute positional encoding. The Chinchilla family is the same as the Gopher family, but trained with AdamW instead of Adam optimizer. The Gopher family contains six models of increasing size, from 44 million parameters to 280 billion parameters. They refer to the largest one as "Gopher" by default. Similar naming conventions apply for the Chinchilla family. Table 1 of shows the entire Gopher family: Table 4 of compares the 70-billion-parameter Chinchilla with Gopher 280B.
Normal distributions transform
The normal distributions transform (NDT) is a point cloud registration algorithm introduced by Peter Biber and Wolfgang Straßer in 2003, while working at University of Tübingen. The algorithm registers two point clouds by first associating a piecewise normal distribution to the first point cloud, that gives the probability of sampling a point belonging to the cloud at a given spatial coordinate, and then finding a transform that maps the second point cloud to the first by maximising the likelihood of the second point cloud on such distribution as a function of the transform parameters. Originally introduced for 2D point cloud map matching in simultaneous localization and mapping (SLAM) and relative position tracking, the algorithm was extended to 3D point clouds and has wide applications in computer vision and robotics. NDT is very fast and accurate, making it suitable for application to large scale data, but it is also sensitive to initialisation, requiring a sufficiently accurate initial guess, and for this reason it is typically used in a coarse-to-fine alignment strategy. == Formulation == The NDT function associated to a point cloud is constructed by partitioning the space in regular cells. For each cell, it is possible to define the mean q = 1 n ∑ i x i {\displaystyle \textstyle \mathbf {q} ={\frac {1}{n}}\sum _{i}\mathbf {x_{i}} } and covariance S = 1 n ∑ i ( x i − q ) ( x i − q ) ⊤ {\displaystyle \textstyle \mathbf {S} ={\frac {1}{n}}\sum _{i}\left(\mathbf {x} _{i}-\mathbf {q} \right)\left(\mathbf {x} _{i}-\mathbf {q} \right)^{\top }} of the n {\displaystyle n} points of the cloud x 1 , … , x n {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{n}} that fall within the cell. The probability density of sampling a point at a given spatial location x {\displaystyle \mathbf {x} } within the cell is then given by the normal distribution e − 1 2 ( x − q ) ⊤ S − 1 ( x − q ) {\displaystyle e^{-{\frac {1}{2}}\left(\mathbf {x} -\mathbf {q} \right)^{\top }\mathbf {S} ^{-1}\left(\mathbf {x} -\mathbf {q} \right)}} . Two point clouds can be mapped by a Euclidean transformation f {\displaystyle f} with rotation matrix R {\displaystyle \mathbf {R} } and translation vector t {\displaystyle \mathbf {t} } f R , t ( x ) = R x + t {\displaystyle f_{\mathbf {R} ,\mathbf {t} }(\mathbf {x} )=\mathbf {R} \mathbf {x} +\mathbf {t} } that maps from the second cloud to the first, parametrised by the rotation angles and translation components. The algorithm registers the two point clouds by optimising the parameters of the transformation that maps the second cloud to the first, with respect to a loss function based on the NDT of the first point cloud, solving the following problem arg min R , t { − ∑ i NDT ( f R , t ( x i ) ) } {\displaystyle \arg \min _{\mathbf {R} ,\mathbf {t} }\left\{-\sum _{i}\operatorname {NDT} \left(f_{\mathbf {R} ,\mathbf {t} }\left(\mathbf {x_{i}} \right)\right)\right\}} where the loss function represents the negated likelihood, obtained by applying the transformation to all points in the second cloud and summing the value of the NDT at each transformed point f R , t ( x ) {\displaystyle f_{\mathbf {R} ,\mathbf {t} }(\mathbf {x} )} . The loss is piecewise continuous and differentiable, and can be optimised with gradient-based methods (in the original formulation, the authors use Newton's method). In order to reduce the effect of cell discretisation, a technique consists of partitioning the space into multiple overlapping grids, shifted by half cell size along the spatial directions, and computing the likelihood at a given location as the sum of the NDTs induced by each grid.
Distributed file system for cloud
A distributed file system for cloud is a file system that allows many clients to have access to data and supports operations (create, delete, modify, read, write) on that data. Each data file may be partitioned into several parts called chunks. Each chunk may be stored on different remote machines, facilitating the parallel execution of applications. Typically, data is stored in files in a hierarchical tree, where the nodes represent directories. There are several ways to share files in a distributed architecture: each solution must be suitable for a certain type of application, depending on how complex the application is. Meanwhile, the security of the system must be ensured. Confidentiality, availability and integrity are the main keys for a secure system. Users can share computing resources through the Internet thanks to cloud computing which is typically characterized by scalable and elastic resources – such as physical servers, applications and any services that are virtualized and allocated dynamically. Synchronization is required to make sure that all devices are up-to-date. Distributed file systems enable many big, medium, and small enterprises to store and access their remote data as they do local data, facilitating the use of variable resources. == Overview == === History === Today, there are many implementations of distributed file systems. The first file servers were developed by researchers in the 1970s. Sun Microsystem's Network File System became available in the 1980s. Before that, people who wanted to share files used the sneakernet method, physically transporting files on storage media from place to place. Once computer networks started to proliferate, it became obvious that the existing file systems had many limitations and were unsuitable for multi-user environments. Users initially used FTP to share files. FTP first ran on the PDP-10 at the end of 1973. Even with FTP, files needed to be copied from the source computer onto a server and then from the server onto the destination computer. Users were required to know the physical addresses of all computers involved with the file sharing. === Supporting techniques === Modern data centers must support large, heterogenous environments, consisting of large numbers of computers of varying capacities. Cloud computing coordinates the operation of all such systems, with techniques such as data center networking (DCN), the MapReduce framework, which supports data-intensive computing applications in parallel and distributed systems, and virtualization techniques that provide dynamic resource allocation, allowing multiple operating systems to coexist on the same physical server. === Applications === Cloud computing provides large-scale computing thanks to its ability to provide the needed CPU and storage resources to the user with complete transparency. This makes cloud computing particularly suited to support different types of applications that require large-scale distributed processing. This data-intensive computing needs a high performance file system that can share data between virtual machines (VM). Cloud computing dynamically allocates the needed resources, releasing them once a task is finished, requiring users to pay only for needed services, often via a service-level agreement. Cloud computing and cluster computing paradigms are becoming increasingly important to industrial data processing and scientific applications such as astronomy and physics, which frequently require the availability of large numbers of computers to carry out experiments. == Architectures == Most distributed file systems are built on the client-server architecture, but other, decentralized, solutions exist as well. === Client-server architecture === Network File System (NFS) uses a client-server architecture, which allows sharing of files between a number of machines on a network as if they were located locally, providing a standardized view. The NFS protocol allows heterogeneous clients' processes, probably running on different machines and under different operating systems, to access files on a distant server, ignoring the actual location of files. Relying on a single server results in the NFS protocol suffering from potentially low availability and poor scalability. Using multiple servers does not solve the availability problem since each server is working independently. The model of NFS is a remote file service. This model is also called the remote access model, which is in contrast with the upload/download model: Remote access model: Provides transparency, the client has access to a file. He sends requests to the remote file (while the file remains on the server). Upload/download model: The client can access the file only locally. It means that the client has to download the file, make modifications, and upload it again, to be used by others' clients. The file system used by NFS is almost the same as the one used by Unix systems. Files are hierarchically organized into a naming graph in which directories and files are represented by nodes. === Cluster-based architectures === A cluster-based architecture ameliorates some of the issues in client-server architectures, improving the execution of applications in parallel. The technique used here is file-striping: a file is split into multiple chunks, which are "striped" across several storage servers. The goal is to allow access to different parts of a file in parallel. If the application does not benefit from this technique, then it would be more convenient to store different files on different servers. However, when it comes to organizing a distributed file system for large data centers, such as Amazon and Google, that offer services to web clients allowing multiple operations (reading, updating, deleting,...) to a large number of files distributed among a large number of computers, then cluster-based solutions become more beneficial. Note that having a large number of computers may mean more hardware failures. Two of the most widely used distributed file systems (DFS) of this type are the Google File System (GFS) and the Hadoop Distributed File System (HDFS). The file systems of both are implemented by user level processes running on top of a standard operating system (Linux in the case of GFS). ==== Design principles ==== ===== Goals ===== Google File System (GFS) and Hadoop Distributed File System (HDFS) are specifically built for handling batch processing on very large data sets. For that, the following hypotheses must be taken into account: High availability: the cluster can contain thousands of file servers and some of them can be down at any time A server belongs to a rack, a room, a data center, a country, and a continent, in order to precisely identify its geographical location The size of a file can vary from many gigabytes to many terabytes. The file system should be able to support a massive number of files The need to support append operations and allow file contents to be visible even while a file is being written Communication is reliable among working machines: TCP/IP is used with a remote procedure call RPC communication abstraction. TCP allows the client to know almost immediately when there is a problem and a need to make a new connection. ===== Load balancing ===== Load balancing is essential for efficient operation in distributed environments. It means distributing work among different servers, fairly, in order to get more work done in the same amount of time and to serve clients faster. In a system containing N chunkservers in a cloud (N being 1000, 10000, or more), where a certain number of files are stored, each file is split into several parts or chunks of fixed size (for example, 64 megabytes), the load of each chunkserver being proportional to the number of chunks hosted by the server. In a load-balanced cloud, resources can be efficiently used while maximizing the performance of MapReduce-based applications. ===== Load rebalancing ===== In a cloud computing environment, failure is the norm, and chunkservers may be upgraded, replaced, and added to the system. Files can also be dynamically created, deleted, and appended. That leads to load imbalance in a distributed file system, meaning that the file chunks are not distributed equitably between the servers. Distributed file systems in clouds such as GFS and HDFS rely on central or master servers or nodes (Master for GFS and NameNode for HDFS) to manage the metadata and the load balancing. The master rebalances replicas periodically: data must be moved from one DataNode/chunkserver to another if free space on the first server falls below a certain threshold. However, this centralized approach can become a bottleneck for those master servers, if they become unable to manage a large number of file accesses, as it increases their already heavy loads. The load rebalance problem is NP-hard. In order to get a large number of chunkservers to work in collaboration, and to
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