In machine learning, physics-informed neural networks (PINNs), also referred to as theory-trained neural networks (TTNs), are a type of universal function approximator that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples. Because they process continuous spatial and time coordinates and output continuous PDE solutions, they can be categorized as neural fields. == Function approximation == Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization. Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. To remedy this, Physics-Informed Neural Networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training datasets that do not necessarily need to be large or complete. An accurate solution of partial differential equations can potentially be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINNs may be used for finding an optimal solution with high fidelity. PINNs can be applied to a wide range of problems in computational science, and are a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a mesh-free alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, a trained PINN network can be used to predict values on simulation grids of different resolutions without needing to be retrained. Additionally, the derivatives used in the partial differential equations can be computed using automatic differentiation (AD), which is assessed to be superior to numerical or symbolic differentiation. == Modeling and computation == A general nonlinear partial differential equation can be written as: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} where u ( t , x ) {\displaystyle u(t,x)} denotes the solution, N [ ⋅ ; λ ] {\displaystyle {\mathcal {N}}[\cdot ;\lambda ]} is a nonlinear operator parameterized by λ {\displaystyle \lambda } , and Ω {\displaystyle \Omega } is a subset of R D {\displaystyle \mathbb {R} ^{D}} . This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems: data-driven solutions of partial differential equations data-driven discovery of partial differential equations === Data-driven solution of partial differential equations === The data-driven solution of PDE computes the hidden state u ( t , x ) {\displaystyle u(t,x)} of the system given boundary data and/or measurements z {\displaystyle z} , and fixed model parameters λ {\displaystyle \lambda } . We solve: u t + N [ u ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u]=0,\quad x\in \Omega ,\quad t\in [0,T]} . by defining the residual f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ] {\displaystyle f:=u_{t}+{\mathcal {N}}[u]} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} is the error between the PINN u ( t , x ) {\displaystyle u(t,x)} and the set of boundary conditions and measured data on the set of points Γ {\displaystyle \Gamma } where the boundary conditions and data are defined. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the PDE during the training process. This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE. === Data-driven discovery of partial differential equations === Given noisy and incomplete measurements z {\displaystyle z} of the state of the system, the data-driven discovery of PDEs results in computing the unknown state u ( t , x ) {\displaystyle u(t,x)} and learning model parameters λ {\displaystyle \lambda } that best describe the observed data: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} By defining f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ; λ ] = 0 {\displaystyle f:=u_{t}+{\mathcal {N}}[u;\lambda ]=0} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network, f ( t , x ) {\displaystyle f(t,x)} results in a PINN. This network can be derived using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} , together with the parameter λ {\displaystyle \lambda } of the differential operator can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} , with u {\displaystyle u} and z {\displaystyle z} state solutions and measurements at sparse location Γ {\displaystyle \Gamma } , respectively. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation. == Extensions and applications == === For piece-wise function approximation === PINNs are unable to approximate PDEs that have strong non-linearity or sharp gradients (such as those that commonly occur in practical fluid flow problems). Piecewise approximation has been an old practic
Transparency in the software supply chain
Transparency in the software supply chain is a condition in which participants involved in the development, procurement, operation, auditing, or regulation of software can determine which components, dependencies, build stages, identifiers, and relationships within the supply chain make up the delivered product. The disclosure of information about software components, their interrelationships, origins, and development methods—for the purposes of risk management, vulnerability detection, and compliance—takes place throughout the software lifecycle. Transparency is one of the key security attributes of the software supply chain, as a deeper understanding of the chain enables participants to identify vulnerabilities and mitigate threats. Problems in the software supply chain can cause billions in losses and create operational challenges for government and commercial entities, as demonstrated by incidents involving SolarWinds, Bybit, 3CX, Jaguar Land Rover, GitHub, and NotPetya. Modern software is often assembled from third-party libraries and open-source components. According to research by the Linux Foundation and Synopsys, 96% of the commercial codebases analyzed contained open-source software, and 70–90% of a typical codebase may consist of open-source components. Without transparency, any software component can become a threat. As a result, companies may spend billions of dollars building robust external defenses, but this will not protect against vulnerabilities in legitimate software inside the perimeter. At the same time, supply chain attacks also erode trust between customers and their IT providers, as malicious code is often embedded in official updates with certificates and digital signatures. One of the primary ways to ensure transparency is through a software bill of materials, which documents the components used to create the software and the relationships within the supply chain. == Concept == The software supply chain is the collection of systems, devices, people, artifacts, and processes involved in the creation of the final software product. Attacks on the software supply chain differ from conventional attacks in that they follow a four-stage pattern: compromise, modification, distribution, and subsequent exploitation of the compromised or modified component. A defining feature of a supply chain attack is the introduction or manipulation of a change at an upstream stage, which is subsequently exploited at a downstream stage. Transparency refers to the availability of knowledge about the chain, while validity concerns the integrity of operations and artifacts and the authentication of participants, and separation involves reducing unnecessary trust relationships and the radius of impact through compartmentalization. In this framework, transparency primarily helps during the pre-compromise and detection phases, as a clearer understanding of participants, operations, and artifacts makes it easier to identify weak links before attackers exploit them. Current major attack vectors include dependencies and containers, build infrastructure, and human participants, such as maintainers or developers. == History == Software supply-chain transparency developed from earlier efforts to document software components, long before the term came into widespread use in the cybersecurity field. Early component-documentation formats included SPDX, first published in 2011, and CycloneDX, first published in 2017. Initially, these formats were created to support license compliance, package identification, and tool compatibility. Their development helped shape a broader concept of software supply chain transparency, encompassing component documentation, disclosure practices, risk management, security analysis, and regulatory compliance. In 2018, the U.S. National Telecommunications and Information Administration launched a multistakeholder process on promoting software component transparency. This process helped move work on SBOMs from a specialized technical practice into the realm of policy and procurement to identify components used in software products. The 2020 compromise of the SolarWinds Orion platform made software supply chain security a central issue in government cybersecurity policy. An analysis of the “Sunburst” campaign prepared by the Atlantic Council noted that the vulnerability of the software supply chain had become a realized risk for national-security agencies. In May 2021, U.S. President Joe Biden issued Executive Order 14028, which directed federal agencies to improve cybersecurity and increase transparency in the software supply chain, including requirements related to SBOMs. Reuters reported that the executive order required software developers selling their products to the federal government to provide greater visibility into their software and make security data available. In July 2021, the NTIA published the document “The Minimum Elements for a Software Bill of Materials (SBOM)”, defining the basic data fields and practices for creating SBOMs. Between 2021 and 2025, the U.S. Cybersecurity and Infrastructure Security Agency updated its guidance on “Framing Software Component Transparency”, expanding the set of SBOM attributes, metadata requirements, and operational recommendations for the creation, exchange, and use of SBOMs. Major incidents that occurred following the SolarWinds attack have underscored the importance of transparency in vulnerability management and supply chain security. The Log4Shell vulnerability in the Log4j library, disclosed in December 2021, demonstrated how difficult it can be for organizations to identify a vulnerable component deeply embedded within applications and services. In 2024, an attempt to plant a backdoor in XZ Utils showed how attackers could exploit trust in open-source maintenance processes to introduce malicious code into widely used infrastructure software. By the mid-2020s, software supply chain transparency had become part of international cybersecurity coordination and regulation. On September 3, 2025, Japan's Ministry of Economy, Trade and Industry and the National Cybersecurity Office, in collaboration with cybersecurity agencies from 15 countries, released the document “A Shared Vision of Software Bill of Materials (SBOM) for Cybersecurity.” In the European Union, the Cyber Resilience Act required manufacturers of products with digital elements to create, maintain, and retain SBOMs as part of the technical documentation for software placed on the EU market. == Transparency mechanisms == The primary mechanism for ensuring transparency is the software bill of materials (SBOM). An SBOM is a structured list of components, libraries, and tools used to build and distribute a software product, and it records dependencies in a way that helps organizations understand and assess their software supply chains. It can also be described as a formal record of components and their interdependencies, which gives users insight into their actual exposure to risks and threats. Five key areas of SBOM application in software supply chain security have been identified: vulnerability management, ensuring transparency, component evaluation, risk assessment, and ensuring supply chain integrity. In software supply chains, an SBOM documents all components, both open-source and proprietary. Under Executive Order 14028, U.S. federal agencies require software suppliers to provide SBOMs for government-procured software. The list of minimum required SBOM elements defined by NTIA includes three main categories: required data fields for describing each component (name, version, identifiers), automation support (machine-readable format, generation tools), and recommendations for creating SBOMs during development and purchasing. The post-2021 push for SBOMs was intended to provide visibility into the components used within software and to expose parts of an application that would otherwise remain hidden. This information can be used to prioritize patches, manage vulnerabilities, and support compliance work. Transparency also supports software traceability, which is becoming a standard feature of developer platforms. Traceability has become important because organizations are increasingly required to demonstrate how software was created, rather than simply listing its components. Higher levels of assurance require signed, tamper-proof traceability and more isolated, verifiable build environments. A related mechanism is build reproducibility. Reproducible builds are defined as build processes that make the compilation process deterministic, ensuring that the same source code always produces the same binary file. These builds are considered a foundational element for distributed verification, transparency-log maintenance, supply-chain workflow integration, and the creation of keyless signatures based on verifiable logs. Although reproducibility does not replace inventory or attestation, it gives external par
Knowledge processing for robots
KnowRob (Knowledge processing for robots) is a system which combines knowledge representation and reasoning methods to acquire and ground knowledge. This system is the backbone of openEASE. both under developing at the Institute for Artificial Intelligence at the University of Bremen, Germany. == The framework == KnowRob can serve as a common sense framework for the integration of knowledge. This knowledge can be static encyclopedic knowledge, common sense knowledge, task descriptions, environment models, object information, observed actions, etc., which can come from different sources, like manually axiomatized, derived from observations, or imported from the web. KnowRob has been used by different research groups, as the Rice University using the ontological knowledge base in a robotic platform. As well by the Eindhoven University of Technology research group competing in the RoboCup league, in the "at Home" category, with the RoboEarth project. As well, KnowRob is mentioned in the work of some research groups from the Lucian Blaga University of Sibiu, Middle East Technical University in their combination of different knowledge bases, Keio University as related work because of the ontology service, University of Texas at Austin as related work as well because of the relation with the work presented, Hanyang University as related work as an OWL based knowledge processing framework. == Representations == To represent the knowledge, KnowRob uses the OWL ontology language and an extended first-order logic knowledge representation with computable predicates. To give the order of subactions, KnowRob includes a pair-wise ordering constrain, which gives a partial ordering. KnowRob adopts the closed-world assumption Prolog, and an open-world assumption by the use of computables. To include reasoning rules into Prolog, KnowRob uses an inference procedure beyond the capabilities of OWL to extract information about tasks executions. In its second version, KnowRob provides a logic interface to the hybrid reasoning kernel as a logic based language. This language presents the hybrid reasoning kernel as if everything were entities retrievable by providing partial descriptions for them. This entities descriptions include objects, their parts, and articulation models, environments composed of objects, software components, actions, and events. === Episodic memories === Episodic memory is related to the experience information, which is organized temporally and spatially, alongside combined with context information. In KnowRob, an episodic memory is understood as a recording that the agent makes of the ongoing activity, which includes very detailed information about the actions, motions, their purposes, effects and the behavior they generate, it also includes the images captured during execution, etc. == Usage == The knowledge is computed by external methods using Prolog queries. In the second version of the KnowRob system, is included a better structure of the packages and documentations. Which includes some extensions from the previous version, as well as a logic based language. For example, a cup description from perception can be represented in this language as: entity(Cup,[an, object, [type, cup], [shape, cylinder], [color, orange]]) As well, a controller could represent the same object as: entity(Cup, [an, object, [type, cup], [proper_physical_parts, [an, object, [type, handle], [grasp−pose, G−pose]]]]) The interface language is comparable to other query languages for symbolic knowledge bases. KnowRob's query language integrates reasoning methods, such as the simulation-based reasoning. == Goals == The goal of the KnowRob framework is to make semantic knowledge available for service robots. It is able to answer queries about missing information in vague instructions for tasks. This is possible with the actions hierarchical representation and information about objects which can be included in certain action.
Sarah Guo
Sarah Guo is an American tech investor. She is the founder of the venture capital firm Conviction and formerly a general partner at Greylock Partners. == Early life and education == Guo grew up in Wisconsin. Her parents worked for Bell Labs. After attending Phillips Academy, she graduated from the University of Pennsylvania and its Wharton School. She received a Bachelor of Arts, a Bachelor of Science, a Master of Business Administration (M.B.A.), and a Master of Arts from the University of Pennsylvania. == Career == As a teenager, Guo worked at Casa Systems, a cloud networking company founded by her parents that launched in 2003 and went public in 2017. She then worked at Goldman Sachs. In 2013, Guo joined Greylock Partners. While still in her twenties, she became the firm's youngest General Partner. Guo left Greylock in July 2022, and in October of that year, launched a new early-stage venture capital firm focused on AI with $101 million. In 2025, Conviction raised a second fund in late 2024 with Mike Vernal. Conviction's investments include early investments in Baseten, Cognition AI, OpenEvidence, Harvey, HeyGen, Mistral AI, Sierra Platform, Sunday Robotics, and Thinking Machines Lab. Guo appears in media outlets, as an expert in AI, infrastructure, business software, cybersecurity, technology policy and software engineering. Guo is on the Midas List and the Midas Seed List of top investors. She co-hosts the podcast No Priors with tech founder and super angel Elad Gil. == Personal life == Guo is married to Pat Grady of Sequoia Capital.
Ilya Sutskever
Ilya Sutskever (Hebrew: איליה סוצקבר; born 1986) is a computer scientist who specializes in machine learning. He has made several major contributions to the field of deep learning, including sequence-to-sequence learning, reasoning models, GPT models, and contributions to CLIP, DALL-E, and AlphaGo. With Alex Krizhevsky and Geoffrey Hinton, he co-created AlexNet, a convolutional neural network. One of the most highly cited computer scientists in history, he has won the NeurIPS Test of Time Award for his lasting impact on AI research three times in a row (2022–2024) and received the National Academy of Sciences Award for the Industrial Application of Science in 2026. Sutskever co-founded and was chief scientist at OpenAI, where he oversaw the research breakthroughs that led to large language models and to the launch of ChatGPT. He also led the research that led to reasoning models such as o1. In 2023, he was one of the members of OpenAI's board that ousted Sam Altman as its CEO; Altman was reinstated a week later, and Sutskever stepped down from the board. In June 2024, Sutskever co-founded the company Safe Superintelligence Inc., alongside Daniel Gross and Daniel Levy. Within a year, the company was valued at more than $30 billion. == Early life and education == Sutskever was born in 1986 into a Jewish family in Nizhny Novgorod, Russia (then Gorky, Russian SFSR, Soviet Union). At the age of 5, he immigrated to Israel with his family and grew up in Jerusalem. Sutskever proved to be a good student in school, and in eighth grade started taking classes at the Open University of Israel. At 16, he moved with his family to Canada, where he attended high school for a month before being admitted to the University of Toronto in Ontario as a third-year undergraduate student. At the University of Toronto, Sutskever received a bachelor's degree in mathematics in 2005, a master's degree in computer science in 2007, and a PhD in computer science in 2013. His doctoral advisor was Geoffrey Hinton. In 2012, Sutskever built AlexNet in collaboration with Geoffrey Hinton and Alex Krizhevsky. == Career and research == In 2012, Sutskever spent about two months as a postdoc with Andrew Ng at Stanford University. He then returned to the University of Toronto and joined Hinton's new research company DNNResearch, a spinoff of Hinton's research group. In 2013, Google acquired DNNResearch and hired Sutskever as a research scientist at Google Brain. At Google Brain, Sutskever worked with Oriol Vinyals and Quoc Viet Le to create the sequence-to-sequence learning algorithm, and worked on TensorFlow. He is also one of the AlphaGo paper's many co-authors. At the end of 2015, Sutskever left Google to become cofounder and chief scientist of the newly founded organization OpenAI. In 2022, Sutskever tweeted, "it may be that today's large neural networks are slightly conscious", which triggered debates about AI consciousness. He is considered to have played a key role in the development of ChatGPT, and later in leading the research that led to reasoning models. He is credited with establishing OpenAI’s scaling ethos. In 2023, he announced that he would co-lead OpenAI's new "Superalignment" project, which was trying to solve the alignment of superintelligences within four years. He wrote that even if superintelligence seems far off, it could happen this decade. Sutskever was formerly one of the six board members of the nonprofit entity that controlled OpenAI. In November 2023, the board fired Sam Altman, saying that "he was not consistently candid in his communications with the board". He authored a 52-page memo that relied heavily on information from Mira Murati, accusing Altman of lying, manipulating executives, and fostering internal division. Sutskever submitted the memo to the board after months of tension and dissatisfaction with Altman's leadership style, and ultimately joined the board in voting for Altman's termination. In an all-hands company meeting shortly after the board meeting, Sutskever said that firing Altman was "the board doing its duty", but the next week, he expressed regret at having participated in Altman's ouster. Altman's firing and OpenAI's co-founder Greg Brockman's resignation led three senior researchers to resign from OpenAI. After that, Sutskever stepped down from the OpenAI board and was absent from OpenAI's office. Some sources suggested he was leading the team remotely, while others said he no longer had access to the team's work. In May 2024, Sutskever announced his departure from OpenAI to focus on a new project that was "very personally meaningful" to him. His decision followed a turbulent period at OpenAI marked by leadership crises and internal debates about the direction of AI development and alignment protocols. Jan Leike, the other leader of the superalignment project, announced his departure hours later, citing an erosion of safety and trust in OpenAI's leadership. In June 2024, Sutskever announced Safe Superintelligence Inc., a new company he founded with Daniel Gross and Daniel Levy with offices in Palo Alto and Tel Aviv. In contrast to OpenAI, which releases revenue-generating products, Sutskever said the new company's "first product will be the safe superintelligence, and it will not do anything else up until then". In September 2024, the company announced that it had raised $1 billion from venture capital firms including Andreessen Horowitz, Sequoia Capital, DST Global, and SV Angel. In March 2025, Safe Superintelligence Inc. raised $2 billion more and reportedly reached a $32 billion valuation, notably due to Sutskever's reputation. In June 2025, SSI rejected an offer from Meta Platforms to buy the company. Sutskever became CEO of SSI shortly thereafter, after co-founder and CEO Gross left for Meta. In an October 2024 interview after winning the Nobel Prize in Physics, Geoffrey Hinton expressed support for Sutskever's decision to fire Altman, emphasizing concerns about AI safety. During the Musk v. Altman trial in 2026, Sutskever confirmed he had a $7 billion stake in OpenAI. === Awards and honors === In 2015, Sutskever was named in MIT Technology Review's 35 Innovators Under 35. In 2018, he was the keynote speaker at Nvidia Ntech 2018 and AI Frontiers Conference 2018. In 2022, he was elected a Fellow of the Royal Society (FRS). In 2023 and 2024, included in Time's list of the 100 most influential people in AI In 2022, 2023, and 2024, he won Neural Information Processing Systems’ Test of Time award, which recognizes papers that significantly shaped the AI field over at least ten years. In 2025, he received an honorary doctorate from his alma mater, the University of Toronto In 2026, he received the National Academy of Sciences Award for the Industrial Application of Science, presented for the first time in artificial intelligence.
Level-set method
The Level-set method (LSM) is a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. LSM makes it easier to perform computations on shapes with sharp corners and shapes that change topology (such as by splitting in two or developing holes). These characteristics make LSM effective for modeling objects that vary in time, such as an airbag inflating or a drop of oil floating in water. == Overview == The figure on the right illustrates several ideas about LSM. In the upper left corner is a bounded region with a well-behaved boundary. Below it, the red surface is the graph of a level set function φ {\displaystyle \varphi } determining this shape, and the flat blue region represents the X-Y plane. The boundary of the shape is then the zero-level set of φ {\displaystyle \varphi } , while the shape itself is the set of points in the plane for which φ {\displaystyle \varphi } is positive (interior of the shape) or zero (at the boundary). In the top row, the shape's topology changes as it is split in two. It is challenging to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. An algorithm can be used to detect the moment the shape splits in two and then construct parameterizations for the two newly obtained curves. On the bottom row, however, the plane at which the level set function is sampled is translated upwards, on which the shape's change in topology is described. It is less challenging to work with a shape through its level-set function rather than with itself directly, in which a method would need to consider all the possible deformations the shape might undergo. Thus, in two dimensions, the level-set method amounts to representing a closed curve Γ {\displaystyle \Gamma } (such as the shape boundary in our example) using an auxiliary function φ {\displaystyle \varphi } , called the level-set function. The curve Γ {\displaystyle \Gamma } is represented as the zero-level set of φ {\displaystyle \varphi } by Γ = { ( x , y ) ∣ φ ( x , y ) = 0 } , {\displaystyle \Gamma =\{(x,y)\mid \varphi (x,y)=0\},} and the level-set method manipulates Γ {\displaystyle \Gamma } implicitly through the function φ {\displaystyle \varphi } . This function φ {\displaystyle \varphi } is assumed to take positive values inside the region delimited by the curve Γ {\displaystyle \Gamma } and negative values outside. == The level-set equation == If the curve Γ {\displaystyle \Gamma } moves in the normal direction with a speed v {\displaystyle v} , then by chain rule and implicit differentiation, it can be determined that the level-set function φ {\displaystyle \varphi } satisfies the level-set equation ∂ φ ∂ t = v | ∇ φ | . {\displaystyle {\frac {\partial \varphi }{\partial t}}=v|\nabla \varphi |.} Here, | ⋅ | {\displaystyle |\cdot |} is the Euclidean norm (denoted customarily by single bars in partial differential equations), and t {\displaystyle t} is time. This is a partial differential equation, in particular a Hamilton–Jacobi equation, and can be solved numerically, for example, by using finite differences on a Cartesian grid. However, the numerical solution of the level set equation may require advanced techniques. Simple finite difference methods fail quickly. Upwinding methods such as the Godunov method are considered better; however, the level set method does not guarantee preservation of the volume and shape of the set level in an advection field that maintains shape and size, for example, a uniform or rotational velocity field. Instead, the shape of the level set may become distorted, and the level set may disappear over a few time steps. Therefore, high-order finite difference schemes, such as high-order essentially non-oscillatory (ENO) schemes, are often required, and even then, the feasibility of long-term simulations is questionable. More advanced methods have been developed to overcome this; for example, combinations of the leveling method with tracking marker particles suggested by the velocity field. == Example == Consider a unit circle in R 2 {\textstyle \mathbb {R} ^{2}} , shrinking in on itself at a constant rate, i.e. each point on the boundary of the circle moves along its inwards pointing normally at some fixed speed. The circle will shrink and eventually collapse down to a point. If an initial distance field is constructed (i.e. a function whose value is the signed Euclidean distance to the boundary, positive interior, negative exterior) on the initial circle, the normalized gradient of this field will be the circle normal. If the field has a constant value subtracted from it in time, the zero level (which was the initial boundary) of the new fields will also be circular and will similarly collapse to a point. This is due to this being effectively the temporal integration of the Eikonal equation with a fixed front velocity. == Applications == In mathematical modeling of combustion, LSM is used to describe the instantaneous flame surface, known as the G equation. Level-set data structures have been developed to facilitate the use of the level-set method in computer applications. Computational fluid dynamics Trajectory planning Optimization Image processing Computational biophysics Discrete complex dynamics (visualization of the parameter plane and the dynamic plane) == History == The level-set method was developed in 1979 by Alain Dervieux, and subsequently popularized by Stanley Osher and James Sethian. It has since become popular in many disciplines, such as image processing, computer graphics, computational geometry, optimization, computational fluid dynamics, and computational biology.
OpenAI Five
OpenAI Five is a computer program by OpenAI that plays the five-on-five video game Dota 2. Its first public appearance occurred in 2017, where it was demonstrated in a live one-on-one game against the professional player Dendi, who lost to it. The following year, the system had advanced to the point of performing as a full team of five, and began playing against and showing the capability to defeat professional teams. By choosing a game as complex as Dota 2 to study machine learning, OpenAI thought they could more accurately capture the unpredictability and continuity seen in the real world, thus constructing more general problem-solving systems. The algorithms and code used by OpenAI Five were eventually borrowed by another neural network in development by the company, one which controlled a physical robotic hand. OpenAI Five has been compared to other similar cases of artificial intelligence (AI) playing against and defeating humans, such as AlphaStar in the video game StarCraft II, AlphaGo in the board game Go, Deep Blue in chess, and Watson on the television game show Jeopardy!. == History == Development on the algorithms used for the bots began in November 2016. OpenAI decided to use Dota 2, a competitive five-on-five video game, as a base due to it being popular on the live streaming platform Twitch, having native support for Linux, and had an application programming interface (API) available. Before becoming a team of five, the first public demonstration occurred at The International 2017 in August, the annual premiere championship tournament for the game, where Dendi, a Ukrainian professional player, lost against an OpenAI bot in a live one-on-one matchup. After the match, CTO Greg Brockman explained that the bot had learned by playing against itself for two weeks of real time, and that the learning software was a step in the direction of creating software that can handle complex tasks "like being a surgeon". OpenAI used a methodology called reinforcement learning, as the bots learn over time by playing against itself hundreds of times a day for months, in which they are rewarded for actions such as killing an enemy and destroying towers. By June 2018, the ability of the bots expanded to play together as a full team of five and were able to defeat teams of amateur and semi-professional players. At The International 2018, OpenAI Five played in two games against professional teams, one against the Brazilian-based paiN Gaming and the other against an all-star team of former Chinese players. Although the bots lost both matches, OpenAI still considered it a successful venture, stating that playing against some of the best players in Dota 2 allowed them to analyze and adjust their algorithms for future games. The bots' final public demonstration occurred in April 2019, where they won a best-of-three series against The International 2018 champions OG at a live event in San Francisco. A four-day online event to play against the bots, open to the public, occurred the same month. There, the bots played in 42,729 public games, winning 99.4% of those games. == Architecture == Each OpenAI Five bot is a neural network containing a single layer with a 4096-unit LSTM that observes the current game state extracted from the Dota developer's API. The neural network conducts actions via numerous possible action heads (no human data involved), and every head has meaning. For instance, the number of ticks to delay an action, what action to select – the X or Y coordinate of this action in a grid around the unit. In addition, action heads are computed independently. The AI system observes the world as a list of 20,000 numbers and takes an action by conducting a list of eight enumeration values. Also, it selects different actions and targets to understand how to encode every action and observe the world. OpenAI Five has been developed as a general-purpose reinforcement learning training system on the "Rapid" infrastructure. Rapid consists of two layers: it spins up thousands of machines and helps them 'talk' to each other and a second layer runs software. By 2018, OpenAI Five had played around 180 years worth of games in reinforcement learning running on 256 GPUs and 128,000 CPU cores, using Proximal Policy Optimization, a policy gradient method. == Comparisons with other game AI systems == Prior to OpenAI Five, other AI versus human experiments and systems have been successfully used before, such as Jeopardy! with Watson, chess with Deep Blue, and Go with AlphaGo. In comparison with other games that have used AI systems to play against human players, Dota 2 differs as explained below: Long run view: The bots run at 30 frames per second for an average match time of 45 minutes, which results in 80,000 ticks per game. OpenAI Five observes every fourth frame, generating 20,000 moves. By comparison, chess usually ends before 40 moves, while Go ends before 150 moves. Partially observed state of the game: Players and their allies can only see the map directly around them. The rest of it is covered in a fog of war which hides enemies units and their movements. Thus, playing Dota 2 requires making inferences based on this incomplete data, as well as predicting what their opponent could be doing at the same time. By comparison, Chess and Go are "full-information games", as they do not hide elements from the opposing player. Continuous action space: Each playable character in a Dota 2 game, known as a hero, can take dozens of actions that target either another unit or a position. The OpenAI Five developers allow the space into 170,000 possible actions per hero. Without counting the perpetual aspects of the game, there are an average of ~1,000 valid actions each tick. By comparison, the average number of actions in chess is 35 and 250 in Go. Continuous observation space: Dota 2 is played on a large map with ten heroes, five on each team, along with dozens of buildings and non-player character (NPC) units. The OpenAI system observes the state of a game through developers' bot API, as 20,000 numbers that constitute all information a human is allowed to get access to. A chess board is represented as about 70 lists, whereas a Go board has about 400 enumerations. == Reception == OpenAI Five have received acknowledgement from the AI, tech, and video game community at large. Microsoft founder Bill Gates called it a "big deal", as their victories "required teamwork and collaboration". Chess champion Garry Kasparov, who lost against the Deep Blue AI in 1997, stated that despite their losing performance at The International 2018, the bots would eventually "get there, and sooner than expected". In a conversation with MIT Technology Review, AI experts also considered OpenAI Five system as a significant achievement, as they noted that Dota 2 was an "extremely complicated game", so even beating non-professional players was impressive. PC Gamer wrote that their wins against professional players was a significant event in machine learning. In contrast, Motherboard wrote that the victory was "basically cheating" due to the simplified hero pools on both sides, as well as the fact that bots were given direct access to the API, as opposed to using computer vision to interpret pixels on the screen. The Verge wrote that the bots were evidence that the company's approach to reinforcement learning and its general philosophy about AI was "yielding milestones". In 2019, DeepMind unveiled a similar bot for StarCraft II, AlphaStar. Like OpenAI Five, AlphaStar used reinforcement learning and self-play. The Verge reported that "the goal with this type of AI research is not just to crush humans in various games just to prove it can be done. Instead, it's to prove that — with enough time, effort, and resources — sophisticated AI software can best humans at virtually any competitive cognitive challenge, be it a board game or a modern video game." They added that the DeepMind and OpenAI victories were also a testament to the power of certain uses of reinforcement learning. It was OpenAI's hope that the technology could have applications outside of the digital realm. In 2018, they were able to reuse the same reinforcement learning algorithms and training code from OpenAI Five for Dactyl, a human-like robot hand with a neural network built to manipulate physical objects. In 2019, Dactyl solved the Rubik's Cube.