Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity. No particular model is provided as a starting point for symbolic regression. Instead, initial expressions are formed by randomly combining mathematical building blocks such as mathematical operators, analytic functions, constants, and state variables. Usually, a subset of these primitives will be specified by the person operating it, but that's not a requirement of the technique. The symbolic regression problem for mathematical functions has been tackled with a variety of methods, including recombining equations most commonly using genetic programming, as well as more recent methods utilizing Bayesian methods and neural networks. Another non-classical alternative method to SR is called Universal Functions Originator (UFO), which has a different mechanism, search-space, and building strategy. Further methods such as Exact Learning attempt to transform the fitting problem into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function. By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge. It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures, thus ensuring that the resulting models reveal the data's underlying structure in a way that's understandable from a human perspective. This facilitates reasoning and favors the odds of getting insights about the data-generating system, as well as improving generalisability and extrapolation behaviour by preventing overfitting. Accuracy and simplicity may be left as two separate objectives of the regression—in which case the optimum solutions form a Pareto front—or they may be combined into a single objective by means of a model selection principle such as minimum description length. It has been proven that symbolic regression is an NP-hard problem. Nevertheless, if the sought-for equation is not too complex it is possible to solve the symbolic regression problem exactly by generating every possible function (built from some predefined set of operators) and evaluating them on the dataset in question. == Difference from classical regression == While conventional regression techniques seek to optimize the parameters for a pre-specified model structure, symbolic regression avoids imposing prior assumptions, and instead infers the model from the data. In other words, it attempts to discover both model structures and model parameters. This approach has the disadvantage of having a much larger space to search, because not only the search space in symbolic regression is infinite, but there are an infinite number of models which will perfectly fit a finite data set (provided that the model complexity isn't artificially limited). This means that it will possibly take a symbolic regression algorithm longer to find an appropriate model and parametrization, than traditional regression techniques. This can be attenuated by limiting the set of building blocks provided to the algorithm, based on existing knowledge of the system that produced the data; but in the end, using symbolic regression is a decision that has to be balanced with how much is known about the underlying system. Nevertheless, this characteristic of symbolic regression also has advantages: because the evolutionary algorithm requires diversity in order to effectively explore the search space, the result is likely to be a selection of high-scoring models (and their corresponding set of parameters). Examining this collection could provide better insight into the underlying process, and allows the user to identify an approximation that better fits their needs in terms of accuracy and simplicity. == Benchmarking == === SRBench === In 2021, SRBench was proposed as a large benchmark for symbolic regression. In its inception, SRBench featured 14 symbolic regression methods, 7 other ML methods, and 252 datasets from PMLB. The benchmark intends to be a living project: it encourages the submission of improvements, new datasets, and new methods, to keep track of the state of the art in SR. === SRBench Competition 2022 === In 2022, SRBench announced the competition Interpretable Symbolic Regression for Data Science, which was held at the GECCO conference in Boston, MA. The competition pitted nine leading symbolic regression algorithms against each other on a novel set of data problems and considered different evaluation criteria. The competition was organized in two tracks, a synthetic track and a real-world data track. ==== Synthetic Track ==== In the synthetic track, methods were compared according to five properties: re-discovery of exact expressions; feature selection; resistance to local optima; extrapolation; and sensitivity to noise. Rankings of the methods were: QLattice PySR (Python Symbolic Regression) uDSR (Deep Symbolic Optimization) ==== Real-world Track ==== In the real-world track, methods were trained to build interpretable predictive models for 14-day forecast counts of COVID-19 cases, hospitalizations, and deaths in New York State. These models were reviewed by a subject expert and assigned trust ratings and evaluated for accuracy and simplicity. The ranking of the methods was: uDSR (Deep Symbolic Optimization) QLattice geneticengine (Genetic Engine) == Non-standard methods == Most symbolic regression algorithms prevent combinatorial explosion by implementing evolutionary algorithms that iteratively improve the best-fit expression over many generations. Recently, researchers have proposed algorithms utilizing other tactics in AI. Silviu-Marian Udrescu and Max Tegmark developed the "AI Feynman" algorithm, which attempts symbolic regression by training a neural network to represent the mystery function, then runs tests against the neural network to attempt to break up the problem into smaller parts. For example, if f ( x 1 , . . . , x i , x i + 1 , . . . , x n ) = g ( x 1 , . . . , x i ) + h ( x i + 1 , . . . , x n ) {\displaystyle f(x_{1},...,x_{i},x_{i+1},...,x_{n})=g(x_{1},...,x_{i})+h(x_{i+1},...,x_{n})} , tests against the neural network can recognize the separation and proceed to solve for g {\displaystyle g} and h {\displaystyle h} separately and with different variables as inputs. This is an example of divide and conquer, which reduces the size of the problem to be more manageable. AI Feynman also transforms the inputs and outputs of the mystery function in order to produce a new function which can be solved with other techniques, and performs dimensional analysis to reduce the number of independent variables involved. The algorithm was able to "discover" 100 equations from The Feynman Lectures on Physics, while a leading software using evolutionary algorithms, Eureqa, solved only 71. AI Feynman, in contrast to classic symbolic regression methods, requires a very large dataset in order to first train the neural network and is naturally biased towards equations that are common in elementary physics.
JDoodle
JDoodle is a cloud-based online integrated development environment and compiler platform that supports execution of source code in 70+ programming languages including Java, Python, C/C++, PHP, Ruby, Perl, HTML, and more. It provides zero‑setup code for compilation, execution, and sharing via a web browser interface. == Features == Provides real‑time collaboration and code embedding via shareable URLs and APIs Offers an integrated terminal interface supporting database engines such as MySQL and MongoDB. JDroid — AI‑assistant to generate code snippets, optimize code, and assist debugging. == Languages and frameworks supported ==
Kindwise
FlowerChecker, also known as Kindwise, is a company that uses machine learning to identify natural objects from images. This includes plants and their diseases, but also insects and mushrooms. It is based in Brno, Czech Republic. It was founded in 2014 by Ondřej Veselý, Jiří Řihák, and Ondřej Vild, at the time Ph.D. students. == Features & Tools == FlowerChecker offers multiple products. Plant.id is a machine learning-based plant identification API launched in 2018, with the plant disease identification API, plant.health, released in April 2022. The plant.id API is suitable for integration into other software, such as mobile apps or urban trees from remote-sensing imagery. Other products include insect.id, mushroom.id and crop.health are machine learning-based identification APIs for the identification of insects, fungi and economically important plants, respectively, and include also online public demos. The FlowerChecker app was discontinued in October 2024 after 10 years of successful operation. == Recognition == In 2019, FlowerChecker won the Idea of the Year award in the AI Awards organized by the Confederation of Industry of the Czech Republic. In 2020, an academic study comparing ten free automated image recognition apps showed that plant.id's performance excelled in most of the parameters studied. In an independent study comparing different image-based species recognition models and their suitability for recognizing invasive alien species, the plant.id achieved the highest accuracy compared to other tools. In a subsequent study, plant.id was utilized to evaluate urban forest biodiversity using remote-sensing imagery, achieving the highest accuracy in tree species identification among compared methods. The technology has also been referenced as an example of practical integration of AI-based plant identification into cross-platform precision agriculture systems. == Research activities == Flowerchecker cooperates with the Nature Conservation Agency of the Czech Republic on a biodiversity mapping project. FlowerChecker plans to adapt its services to participate in the control of invasive species. In 2022, the company entered a consortium to develop a weeder capable of in-row weed detection and removal. In 2025, it received funding for the development of a technology for the removal of invasive species.
BRFplus
BRFplus (Business Rule Framework plus) is a business rule management system (BRMS) offered by SAP AG. BRFplus is part of the SAP NetWeaver ABAP stack. Therefore, all SAP applications that are based on SAP NetWeaver can access BRFplus within the boundaries of an SAP system. However, it is also possible to generate web services so that BRFplus rules can also be offered as a service in a SOA landscape, regardless of the software platform used by the service consumers. BRFplus development started as a supporting tool that was part of SAP Business ByDesign, an ERP solution targeted at small and medium size companies. By that time, the tool was called "Formula and Derivation Tool" (FDT). Later on, it was decided to maintain BRFplus on those codelines that serve as the basis for SAP Business Suite. With that, business rules that have been created for Business ByDesign can easily be taken over in a full-size SAP system where they are ready for use without any changes. == Overview == BRFplus offers a unified modeling and runtime environment for business rules that addresses both technical users (programmers, system administrators) as well as business users who take care of operational business processes (like procurement, bidding, tax form validation, etc.). The different requirements and usage scenarios of the different target groups can be covered with the help of the SAP authorization system and a user interface that can be individually customized. Being integrated into SAP NetWeaver, BRFplus-based applications can look at, and model, business rules from a strictly business-oriented perspective, rather than starting with the underlying technical artifacts. This is because the integration allows for direct access to the business objects available in the SAP dictionary (like customer, supplier, material, bill, etc.). In addition to the predefined expression types (decision table, decision tree, formula, database access, loops, etc.) and actions (sending e-mails, triggering a workflow, etc.), BRFplus can be extended by custom expression types. Also, direct calls of function modules as well as ABAP OO class methods are supported so that the entire range of the ABAP programming language is available for solving business tasks. BRFplus comes with an optional versioning mechanism. Versioning can be switched on and off for individual objects as well as for entire applications. Versioned business rules are needed in certain use cases for legal reasons, but they also allow for simulating the system behavior as it would have been at a particular point in time. Once the rule objects are in a consistent state and active, the system automatically generates ABAP OO classes that encapsulate the functional scope of the underlying rule object. This is done on an on-demand base and speeds up processing. The execution of functions as well as of single expressions can be simulated. The processing log of the simulation is useful for checking the implementation and for investigating problems. BRFplus applications can be exported and imported as an XML file. This is an easy way of creating a data backup. XML files can also be used for deploying rule applications throughout the company. == Main object types == === Application === The application object serves as a container for all the BRFplus objects that have been assembled to solve a particular business task. It is possible to define certain default settings on application level that are inherited by all objects that are created in the scope of that application. === Function === A function is used to connect a business application with the rule processing framework of BRFplus. The calling business application passes input values to the function which are then processed by the expressions and rulesets that are associated with the called function. The calculated result is then returned to the calling business application. === Expression types and action types === Boolean BRMS Connector Case Database Lookup Decision Table Decision Tree Formula Function Call Loop Procedure Call Random Number Search Tree Step Sequence Value Range1 XSL Transformation === Ruleset === A ruleset is a container for an arbitrary number of rule objects which in turn carry out the necessary calculations with the help of assigned expressions and actions. Instead of assigning an expression to a function, it is also possible to assign any number of rulesets to a function. When the function is called, all assigned rulesets are subsequently processed. === Data objects === BRFplus supports elementary data objects (text, number, boolean, time point, amount, quantity) as well as structures and tables. Structures can be nested. For all types of data objects it is possible to reference data objects that reside in the data dictionary of the backend system. With that, a BRFplus data object does not only inherit the type definition of the referenced object but can also access associated data like domain value lists or object documentation. === Other objects === With catalogs, it is possible to define business-specific subsets of the rule objects that reside in the system. This is helpful for hiding the complexity of a rule system, thus improving usability. Object filters are used by system administrators to ensure that for selected users, only a predefined subset of object types is visible. This is useful to enforce access rights as well as modeling policies. == Other BRM solutions offered by SAP == BRFplus is positioned as the successor product of an older business rule solution known as BRF (Business Rule Framework). For a longer transition phase, both solutions exist in parallel. However, an increasing number of SAP applications that used to be based on BRF are migrating to BRFplus. While BRFplus supports business rules for applications based on the SAP NetWeaver ABAP stack, SAP is offering another product named SAP NetWeaver Business Rules Management (BRM). BRM supports business rule modeling for the SAP NetWeaver Java stack. Both products do not compete. They are available in parallel and can be used in a collaborative approach to deal with use cases where both technology stacks are used in parallel. BRFplus comes with a special expression type that helps bridging the gap between the two different technologies. == Availability == BRFplus has been delivered to the public with SAP NetWeaver 7.0 Enhancement Package 1 for the first time. Being part of SAP NetWeaver, the usage of BRFplus is covered by the "SAP NetWeaver Foundation for Third Party Applications" license, with no additional costs. == Literature == Carsten Ziegler, Thomas Albrecht: BRFplus – Business Rule Management for ABAP Applications. Galileo Press 2011. ISBN 978-1-59229-293-6
Cerebellar model articulation controller
The cerebellar model arithmetic computer (CMAC) is a type of neural network based on a model of the mammalian cerebellum. It is also known as the cerebellar model articulation controller. It is a type of associative memory. The CMAC was first proposed as a function modeler for robotic controllers by James Albus in 1975 (hence the name), but has been extensively used in reinforcement learning and also as for automated classification in the machine learning community. The CMAC is an extension of the perceptron model. It computes a function for n {\displaystyle n} input dimensions. The input space is divided up into hyper-rectangles, each of which is associated with a memory cell. The contents of the memory cells are the weights, which are adjusted during training. Usually, more than one quantisation of input space is used, so that any point in input space is associated with a number of hyper-rectangles, and therefore with a number of memory cells. The output of a CMAC is the algebraic sum of the weights in all the memory cells activated by the input point. A change of value of the input point results in a change in the set of activated hyper-rectangles, and therefore a change in the set of memory cells participating in the CMAC output. The CMAC output is therefore stored in a distributed fashion, such that the output corresponding to any point in input space is derived from the value stored in a number of memory cells (hence the name associative memory). This provides generalisation. == Building blocks == In the adjacent image, there are two inputs to the CMAC, represented as a 2D space. Two quantising functions have been used to divide this space with two overlapping grids (one shown in heavier lines). A single input is shown near the middle, and this has activated two memory cells, corresponding to the shaded area. If another point occurs close to the one shown, it will share some of the same memory cells, providing generalisation. The CMAC is trained by presenting pairs of input points and output values, and adjusting the weights in the activated cells by a proportion of the error observed at the output. This simple training algorithm has a proof of convergence. It is normal to add a kernel function to the hyper-rectangle, so that points falling towards the edge of a hyper-rectangle have a smaller activation than those falling near the centre. One of the major problems cited in practical use of CMAC is the memory size required, which is directly related to the number of cells used. This is usually ameliorated by using a hash function, and only providing memory storage for the actual cells that are activated by inputs. == One-step convergent algorithm == Initially least mean square (LMS) method is employed to update the weights of CMAC. The convergence of using LMS for training CMAC is sensitive to the learning rate and could lead to divergence. In 2004, a recursive least squares (RLS) algorithm was introduced to train CMAC online. It does not need to tune a learning rate. Its convergence has been proved theoretically and can be guaranteed to converge in one step. The computational complexity of this RLS algorithm is O(N3). == Hardware implementation infrastructure == Based on QR decomposition, an algorithm (QRLS) has been further simplified to have an O(N) complexity. Consequently, this reduces memory usage and time cost significantly. A parallel pipeline array structure on implementing this algorithm has been introduced. Overall by utilizing QRLS algorithm, the CMAC neural network convergence can be guaranteed, and the weights of the nodes can be updated using one step of training. Its parallel pipeline array structure offers its great potential to be implemented in hardware for large-scale industry usage. == Continuous CMAC == Since the rectangular shape of CMAC receptive field functions produce discontinuous staircase function approximation, by integrating CMAC with B-splines functions, continuous CMAC offers the capability of obtaining any order of derivatives of the approximate functions. == Deep CMAC == In recent years, numerous studies have confirmed that by stacking several shallow structures into a single deep structure, the overall system could achieve better data representation, and, thus, more effectively deal with nonlinear and high complexity tasks. In 2018, a deep CMAC (DCMAC) framework was proposed and a backpropagation algorithm was derived to estimate the DCMAC parameters. Experimental results of an adaptive noise cancellation task showed that the proposed DCMAC can achieve better noise cancellation performance when compared with that from the conventional single-layer CMAC. == Summary ==
Machine translation software usability
The sections below give objective criteria for evaluating the usability of machine translation software output. == Stationarity or canonical form == Do repeated translations converge on a single expression in both languages? I.e. does the translation method show stationarity or produce a canonical form? Does the translation become stationary without losing the original meaning? This metric has been criticized as not being well correlated with BLEU (BiLingual Evaluation Understudy) scores. == Adaptive to colloquialism, argot or slang == Is the system adaptive to colloquialism, argot or slang? The French language has many rules for creating words in the speech and writing of popular culture. Two such rules are: (a) The reverse spelling of words such as femme to meuf. (This is called verlan.) (b) The attachment of the suffix -ard to a noun or verb to form a proper noun. For example, the noun faluche means "student hat". The word faluchard formed from faluche colloquially can mean, depending on context, "a group of students", "a gathering of students" and "behavior typical of a student". The Google translator as of 28 December 2006 doesn't derive the constructed words as for example from rule (b), as shown here: Il y a une chorale falucharde mercredi, venez nombreux, les faluchards chantent des paillardes! ==> There is a choral society falucharde Wednesday, come many, the faluchards sing loose-living women! French argot has three levels of usage: familier or friendly, acceptable among friends, family and peers but not at work grossier or swear words, acceptable among friends and peers but not at work or in family verlan or ghetto slang, acceptable among lower classes but not among middle or upper classes The United States National Institute of Standards and Technology conducts annual evaluations [1] Archived 2009-03-22 at the Wayback Machine of machine translation systems based on the BLEU-4 criterion [2]. A combined method called IQmt which incorporates BLEU and additional metrics NIST, GTM, ROUGE and METEOR has been implemented by Gimenez and Amigo [3]. == Well-formed output == Is the output grammatical or well-formed in the target language? Using an interlingua should be helpful in this regard, because with a fixed interlingua one should be able to write a grammatical mapping to the target language from the interlingua. Consider the following Arabic language input and English language translation result from the Google translator as of 27 December 2006 [4]. This Google translator output doesn't parse using a reasonable English grammar: وعن حوادث التدافع عند شعيرة رمي الجمرات -التي كثيرا ما يسقط فيها العديد من الضحايا- أشار الأمير نايف إلى إدخال "تحسينات كثيرة في جسر الجمرات ستمنع بإذن الله حدوث أي تزاحم". ==> And incidents at the push Carbuncles-throwing ritual, which often fall where many of the victims - Prince Nayef pointed to the introduction of "many improvements in bridge Carbuncles God would stop the occurrence of any competing." == Semantics preservation == Do repeated re-translations preserve the semantics of the original sentence? For example, consider the following English input passed multiple times into and out of French using the Google translator as of 27 December 2006: Better a day earlier than a day late. ==> Améliorer un jour plus tôt qu'un jour tard. ==> To improve one day earlier than a day late. ==> Pour améliorer un jour plus tôt qu'un jour tard. ==> To improve one day earlier than a day late. As noted above and in, this kind of round-trip translation is a very unreliable method of evaluation. == Trustworthiness and security == An interesting peculiarity of Google Translate as of 24 January 2008 (corrected as of 25 January 2008) is the following result when translating from English to Spanish, which shows an embedded joke in the English-Spanish dictionary which has some added poignancy given recent events: Heath Ledger is dead ==> Tom Cruise está muerto This raises the issue of trustworthiness when relying on a machine translation system embedded in a Life-critical system in which the translation system has input to a Safety Critical Decision Making process. Conjointly it raises the issue of whether in a given use the software of the machine translation system is safe from hackers. It is not known whether this feature of Google Translate was the result of a joke/hack or perhaps an unintended consequence of the use of a method such as statistical machine translation. Reporters from CNET Networks asked Google for an explanation on January 24, 2008; Google said only that it was an "internal issue with Google Translate". The mistranslation was the subject of much hilarity and speculation on the Internet. If it is an unintended consequence of the use of a method such as statistical machine translation, and not a joke/hack, then this event is a demonstration of a potential source of critical unreliability in the statistical machine translation method. In human translations, in particular on the part of interpreters, selectivity on the part of the translator in performing a translation is often commented on when one of the two parties being served by the interpreter knows both languages. This leads to the issue of whether a particular translation could be considered verifiable. In this case, a converging round-trip translation would be a kind of verification.
Leading the Future
Leading the Future is an American super PAC network focused on lobbying for policies friendly to the artificial intelligence industry. It was launched in 2025 with over $100 million from industry stakeholders including Andreessen Horowitz, OpenAI President Greg Brockman and Palantir co-founder Joe Lonsdale. The launch was preceded by talks between Collin McCune, head of government affairs at Andreessen Horowitz, and Chris Lehane, chief global affairs officer at OpenAI. Among the members of the network are the American Mission PAC, which supported Chris Gober, and the Think Big PAC, which targeted Alex Bores. Leading the Future is affiliated with the nonprofit Build American AI, which Axios describes as a dark money advocacy "offshoot" operating alongside the super PAC. NBC News states that the network’s efforts are modeled after the pro-cryptocurrency group Fairshake. Leading the Future is led by Zac Moffatt and Josh Vlasto, the latter of whom previously served as an advisor to Fairshake. In response to the creation of Leading the Future, former members of Congress Brad Carson and Chris Stewart co-founded the super PAC network Public First, aiming to counter the group’s influence. In April 2026, an investigation by Model Republic linked Leading the Future to The Wire By Acutus, an automated news website that allegedly used AI agents posing as human journalists to solicit interviews. The site's content was found to closely mirror the PAC's deregulatory policy goals while targeting researchers and advocates skeptical of rapid AI development. In May 2026, Wired revealed that Build American AI used a "dark money" campaign to pay TikTok and Instagram influencers $5,000 per video to promote scripted narratives framing Chinese AI as a "national security threat." According to internal documents and staff at the marketing agency managing the project, the campaign's explicit goal was to "subtly shift public debate" toward the deregulation of AI industries while intentionally avoiding technical discussions regarding AI quality or safety. During the 2026 primary season Leading the Future went on to endorse several candidates in both Democratic and Republican races with several of them going on to win.