Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.
","accessed":{"date-parts":[["2023",12,12]]},"author":[{"family":"Patterson","given":"Evan"},{"family":"Baas","given":"Andrew"},{"family":"Hosgood","given":"Timothy"},{"family":"Fairbanks","given":"James"}],"citation-key":"pattersonDiagrammaticViewDifferential2022","container-title":"Mathematics in Engineering","container-title-short":"MINE","DOI":"10.3934/mine.2023036","ISSN":"2640-3501","issue":"2","issued":{"date-parts":[["2022"]]},"page":"1-59","source":"DOI.org (Crossref)","title":"A diagrammatic view of differential equations in physics","type":"article-journal","URL":"http://www.aimspress.com/article/doi/10.3934/mine.2023036","volume":"5"},{"id":"baezOpenPetriNets2020","abstract":"The reachability semantics for Petri nets can be studied using open Petri nets. For us an \"open\" Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\\mathsf{Open}(\\mathsf{Petri})$, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\\mathbb{O}\\mathbf{pen}(\\mathsf{Petri})$. We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of $\\mathbb{O}\\mathbf{pen}(\\mathsf{Petri})$. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.","accessed":{"date-parts":[["2023",12,12]]},"author":[{"family":"Baez","given":"John C."},{"family":"Master","given":"Jade"}],"citation-key":"baezOpenPetriNets2020","container-title":"Mathematical Structures in Computer Science","container-title-short":"Math. Struct. Comp. 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More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams. We define the symmetric monoidal category W, from which we may construct an operad O(W), whose objects are black boxes with input and output ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic rules for interconnection. We then define two W-algebras, G and L, which associate semantic content to the structures in W. Respectively, they correspond to general and to linear systems of differential equations, in which an internal state is controlled by inputs and produces outputs. As an example, we use these algebras to formalize the classical problem of systems of tanks interconnected by pipes, and hence make explicit the algebraic relationships among systems at different levels of granularity.","accessed":{"date-parts":[["2023",12,8]]},"author":[{"family":"Vagner","given":"Dmitry"},{"family":"Spivak","given":"David I."},{"family":"Lerman","given":"Eugene"}],"citation-key":"vagnerAlgebrasOpenDynamical2015","DOI":"10.48550/arXiv.1408.1598","issued":{"date-parts":[["2015",10,2]]},"number":"arXiv:1408.1598","publisher":"arXiv","source":"arXiv.org","title":"Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams","type":"article","URL":"http://arxiv.org/abs/1408.1598"},{"id":"spivakCategoryTheorySciences2014","accessed":{"date-parts":[["2023",12,7]]},"author":[{"family":"Spivak","given":"David I."}],"citation-key":"spivakCategoryTheorySciences2014","event-place":"Cambridge, UNITED STATES","ISBN":"978-0-262-32052-8","issued":{"date-parts":[["2014"]]},"publisher":"MIT Press","publisher-place":"Cambridge, UNITED STATES","source":"ProQuest Ebook Central","title":"Category Theory for the Sciences","type":"book","URL":"http://ebookcentral.proquest.com/lib/pitt-ebooks/detail.action?docID=3339883"},{"id":"spivakOperadWiringDiagrams2013","abstract":"Wiring diagrams, as seen in digital circuits, can be nested hierarchically and thus have an aspect of self-similarity. We show that wiring diagrams form the morphisms of an operad $\\mcT$, capturing this self-similarity. We discuss the algebra $\\Rel$ of mathematical relations on $\\mcT$, and in so doing use wiring diagrams as a graphical language with which to structure queries on relational databases. We give the example of circuit diagrams as a special case. We move on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well. Throughout we include many examples and figures.","accessed":{"date-parts":[["2023",12,7]]},"author":[{"family":"Spivak","given":"David I."}],"citation-key":"spivakOperadWiringDiagrams2013","DOI":"10.48550/arXiv.1305.0297","issued":{"date-parts":[["2013",5,1]]},"number":"arXiv:1305.0297","publisher":"arXiv","source":"arXiv.org","title":"The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits","title-short":"The operad of wiring diagrams","type":"article","URL":"http://arxiv.org/abs/1305.0297"},{"id":"vagnerAlgebrasOpenDynamical2015a","abstract":"In this paper, we use the language of operads to study open dynamical systems. More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams. We define the symmetric monoidal category W, from which we may construct an operad O(W), whose objects are black boxes with input and output ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic rules for interconnection. We then define two W-algebras, G and L, which associate semantic content to the structures in W. Respectively, they correspond to general and to linear systems of differential equations, in which an internal state is controlled by inputs and produces outputs. As an example, we use these algebras to formalize the classical problem of systems of tanks interconnected by pipes, and hence make explicit the algebraic relationships among systems at different levels of granularity.","accessed":{"date-parts":[["2023",12,7]]},"author":[{"family":"Vagner","given":"Dmitry"},{"family":"Spivak","given":"David I."},{"family":"Lerman","given":"Eugene"}],"citation-key":"vagnerAlgebrasOpenDynamical2015a","DOI":"10.48550/arXiv.1408.1598","issued":{"date-parts":[["2015",10,2]]},"number":"arXiv:1408.1598","publisher":"arXiv","source":"arXiv.org","title":"Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams","type":"article","URL":"http://arxiv.org/abs/1408.1598"},{"id":"gouertArctyrEXAcceleratedEncrypted2023","abstract":"Fully Homomorphic Encryption (FHE) is a cryptographic method that guarantees the privacy and security of user data during computation. FHE algorithms can perform unlimited arithmetic computations directly on encrypted data without decrypting it. Thus, even when processed by untrusted systems, confidential data is never exposed. In this work, we develop new techniques for accelerated encrypted execution and demonstrate the significant performance advantages of our approach. Our current focus is the Fully Homomorphic Encryption over the Torus (CGGI) scheme, which is a current state-of-the-art method for evaluating arbitrary functions in the encrypted domain. CGGI represents a computation as a graph of homomorphic logic gates and each individual bit of the plaintext is transformed into a polynomial in the encrypted domain. Arithmetic on such data becomes very expensive: operations on bits become operations on entire polynomials. Therefore, evaluating even relatively simple nonlinear functions, such as a sigmoid, can take thousands of seconds on a single CPU thread. Using our novel framework for end-to-end accelerated encrypted execution called ArctyrEX, developers with no knowledge of complex FHE libraries can simply describe their computation as a C program that is evaluated over $40\\times$ faster on an NVIDIA DGX A100 and $6\\times$ faster with a single A100 relative to a 256-threaded CPU baseline.","accessed":{"date-parts":[["2023",12,6]]},"author":[{"family":"Gouert","given":"Charles"},{"family":"Joseph","given":"Vinu"},{"family":"Dalton","given":"Steven"},{"family":"Augonnet","given":"Cedric"},{"family":"Garland","given":"Michael"},{"family":"Tsoutsos","given":"Nektarios Georgios"}],"citation-key":"gouertArctyrEXAcceleratedEncrypted2023","DOI":"10.48550/arXiv.2306.11006","issued":{"date-parts":[["2023",6,19]]},"number":"arXiv:2306.11006","publisher":"arXiv","source":"arXiv.org","title":"ArctyrEX : Accelerated Encrypted Execution of General-Purpose Applications","title-short":"ArctyrEX","type":"article","URL":"http://arxiv.org/abs/2306.11006"},{"id":"jackyPyModelModelbasedTesting2011","abstract":"In unit testing, the programmer codes the test cases, and also codes assertions that check whether each test case passed. In model-based testing, the programmer codes a \"model\" that generates as many test cases as desired and also acts as the oracle that checks the cases. Model-based testing is recommended where so many test cases are needed that it is not feasible to code them all by hand. This need arises when testing behaviors that exhibit history-dependence and nondeterminism, so that many variations (data values, interleavings, etc.) should be tested for each scenario (or use case). Examples include communication protocols, web applications, control systems, and user interfaces. PyModel is a model-based testing framework in Python. PyModel supports on-the-fly testing, which can generate indefinitely long nonrepeating tests as the test run executes. PyModel can focus test cases on scenarios of interest by composition, a versatile technique that combines models by synchronizing shared actions and interleaving unshared actions. PyModel can guide test coverage according to programmable strategies coded by the programmer.","accessed":{"date-parts":[["2023",11,27]]},"author":[{"family":"Jacky","given":"Jonathan"}],"citation-key":"jackyPyModelModelbasedTesting2011","DOI":"10.25080/Majora-ebaa42b7-008","event-place":"Austin, Texas","event-title":"Python in Science Conference","issued":{"date-parts":[["2011"]]},"language":"en","page":"48-52","publisher-place":"Austin, Texas","source":"DOI.org (Crossref)","title":"PyModel: Model-based testing in Python","title-short":"PyModel","type":"paper-conference","URL":"https://conference.scipy.org/proceedings/scipy2011/jacky.html"},{"id":"sontagControlLyapunovFunctions1999","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Sontag","given":"Eduardo D."}],"citation-key":"sontagControlLyapunovFunctions1999","collection-editor":[{"family":"Dickinson","given":"B. W."},{"family":"Fettweis","given":"A."},{"family":"Massey","given":"J. L."},{"family":"Modestino","given":"J. W."},{"family":"Sontag","given":"E. D."},{"family":"Thoma","given":"M."}],"container-title":"Open Problems in Mathematical Systems and Control Theory","DOI":"10.1007/978-1-4471-0807-8_40","editor":[{"family":"Blondel","given":"Vincent"},{"family":"Sontag","given":"Eduardo D."},{"family":"Vidyasagar","given":"Mathukumalli"},{"family":"Willems","given":"Jan C."}],"event-place":"London","ISBN":"978-1-4471-1207-5 978-1-4471-0807-8","issued":{"date-parts":[["1999"]]},"page":"211-216","publisher":"Springer London","publisher-place":"London","source":"DOI.org (Crossref)","title":"Control-Lyapunov functions","type":"chapter","URL":"http://link.springer.com/10.1007/978-1-4471-0807-8_40"},{"id":"dicairanoStabilizingDynamicControllers2014","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Di Cairano","given":"Stefano"},{"family":"Heemels","given":"W. P. Maurice H."},{"family":"Lazar","given":"Mircea"},{"family":"Bemporad","given":"Alberto"}],"citation-key":"dicairanoStabilizingDynamicControllers2014","container-title":"IEEE Transactions on Automatic Control","container-title-short":"IEEE Trans. Automat. Contr.","DOI":"10.1109/TAC.2014.2324111","ISSN":"0018-9286, 1558-2523","issue":"10","issued":{"date-parts":[["2014",10]]},"page":"2629-2643","source":"DOI.org (Crossref)","title":"Stabilizing Dynamic Controllers for Hybrid Systems: A Hybrid Control Lyapunov Function Approach","title-short":"Stabilizing Dynamic Controllers for Hybrid Systems","type":"article-journal","URL":"http://ieeexplore.ieee.org/document/6816126/","volume":"59"},{"id":"asarinReachabilityAnalysisNonlinear2003","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Asarin","given":"Eugene"},{"family":"Dang","given":"Thao"},{"family":"Girard","given":"Antoine"}],"citation-key":"asarinReachabilityAnalysisNonlinear2003","collection-editor":[{"family":"Goos","given":"Gerhard"},{"family":"Hartmanis","given":"Juris"},{"family":"Van Leeuwen","given":"Jan"}],"container-title":"Hybrid Systems: Computation and Control","DOI":"10.1007/3-540-36580-X_5","editor":[{"family":"Maler","given":"Oded"},{"family":"Pnueli","given":"Amir"}],"event-place":"Berlin, Heidelberg","ISBN":"978-3-540-00913-9 978-3-540-36580-8","issued":{"date-parts":[["2003"]]},"page":"20-35","publisher":"Springer Berlin Heidelberg","publisher-place":"Berlin, Heidelberg","source":"DOI.org (Crossref)","title":"Reachability Analysis of Nonlinear Systems Using Conservative Approximation","type":"chapter","URL":"http://link.springer.com/10.1007/3-540-36580-X_5","volume":"2623"},{"id":"bartosiewiczLocalPositiveReachability2016","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Bartosiewicz","given":"Zbigniew"}],"citation-key":"bartosiewiczLocalPositiveReachability2016","container-title":"IEEE Transactions on Automatic Control","container-title-short":"IEEE Trans. Automat. Contr.","DOI":"10.1109/TAC.2015.2511921","ISSN":"0018-9286, 1558-2523","issue":"12","issued":{"date-parts":[["2016",12]]},"page":"4217-4221","source":"DOI.org (Crossref)","title":"Local Positive Reachability of Nonlinear Continuous-Time Systems","type":"article-journal","URL":"http://ieeexplore.ieee.org/document/7365428/","volume":"61"},{"id":"goubaultForwardInnerApproximatedReachability2017","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Goubault","given":"Eric"},{"family":"Putot","given":"Sylvie"}],"citation-key":"goubaultForwardInnerApproximatedReachability2017","container-title":"Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control","DOI":"10.1145/3049797.3049811","event-place":"Pittsburgh Pennsylvania USA","event-title":"HSCC '17: 20th International Conference on Hybrid Systems: Computation and Control","ISBN":"978-1-4503-4590-3","issued":{"date-parts":[["2017",4,13]]},"language":"en","page":"1-10","publisher":"ACM","publisher-place":"Pittsburgh Pennsylvania USA","source":"DOI.org (Crossref)","title":"Forward Inner-Approximated Reachability of Non-Linear Continuous Systems","type":"paper-conference","URL":"https://dl.acm.org/doi/10.1145/3049797.3049811"},{"id":"asarinRecentProgressContinuous2006","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Asarin","given":"Eugene"},{"family":"Dang","given":"Thao"},{"family":"Frehse","given":"Goran"},{"family":"Girard","given":"Antoine"},{"family":"Le Guernic","given":"Colas"},{"family":"Maler","given":"Oded"}],"citation-key":"asarinRecentProgressContinuous2006","container-title":"2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control","DOI":"10.1109/CACSD-CCA-ISIC.2006.4776877","event-place":"Munich, Germany","event-title":"2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control","issued":{"date-parts":[["2006",10]]},"page":"1582-1587","publisher":"IEEE","publisher-place":"Munich, Germany","source":"DOI.org (Crossref)","title":"Recent progress in continuous and hybrid reachability analysis","type":"paper-conference","URL":"http://ieeexplore.ieee.org/document/4776877/"},{"id":"michelStabilityDynamicalSystems2008","author":[{"family":"Michel","given":"Anthony N."},{"family":"Hou","given":"Ling"},{"family":"Liu","given":"Derong"}],"citation-key":"michelStabilityDynamicalSystems2008","collection-title":"Systems & control: foundations & applications","event-place":"Boston Basel Berlin","ISBN":"978-0-8176-4486-4","issued":{"date-parts":[["2008"]]},"language":"eng","number-of-pages":"501","publisher":"Birkhäuser","publisher-place":"Boston Basel Berlin","source":"K10plus ISBN","title":"Stability of dynamical systems: continuous, discontinuous, and discrete systems","title-short":"Stability of dynamical systems","type":"book"},{"id":"lewisOptimalControl2012","author":[{"family":"Lewis","given":"Frank L."},{"family":"Vrabie","given":"Draguna L."},{"family":"Syrmos","given":"Vassilis L."}],"call-number":"QA402.3 .L487 2012","citation-key":"lewisOptimalControl2012","edition":"3rd ed","event-place":"Hoboken","ISBN":"978-0-470-63349-6","issued":{"date-parts":[["2012"]]},"number-of-pages":"540","publisher":"Wiley","publisher-place":"Hoboken","source":"Library of Congress ISBN","title":"Optimal control","type":"book"},{"id":"lionsHamiltonJacobiBellmanEquations1983","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Lions","given":"P. L."}],"citation-key":"lionsHamiltonJacobiBellmanEquations1983","container-title":"Acta Applicandae Mathematicae","container-title-short":"Acta Appl Math","DOI":"10.1007/BF02433840","ISSN":"0167-8019, 1572-9036","issue":"1","issued":{"date-parts":[["1983",3]]},"language":"en","page":"17-41","source":"DOI.org (Crossref)","title":"On the Hamilton-Jacobi-Bellman equations","type":"article-journal","URL":"http://link.springer.com/10.1007/BF02433840","volume":"1"},{"id":"slothVerificationContinuousDynamical2011","accessed":{"date-parts":[["2023",11,22]]},"author":[{"family":"Sloth","given":"Christoffer"},{"family":"Wisniewski","given":"Rafael"}],"citation-key":"slothVerificationContinuousDynamical2011","container-title":"Formal Methods in System Design","container-title-short":"Form Methods Syst Des","DOI":"10.1007/s10703-011-0118-0","ISSN":"0925-9856, 1572-8102","issue":"1","issued":{"date-parts":[["2011",8]]},"language":"en","page":"47-82","source":"DOI.org (Crossref)","title":"Verification of continuous dynamical systems by timed automata","type":"article-journal","URL":"http://link.springer.com/10.1007/s10703-011-0118-0","volume":"39"},{"id":"marinoPROFIBUSFormalSpecification2001","abstract":"Formal description languages, like ESTELLE [7], Language Temporary Of Ordering Speci®cation (LOTOS) [8] or Speci®cation Description Language (SDL) [9], allow us to specify complex system requirements in an ambiguities free and complete way [26]. The choice of what formal language to use in a particular system speci®cation must be taken a priori by the system designer [5]. It would be useful to have comparative information about systems described with some of these languages, to know which one would ®t better to that system. The literature published on this topic [1,14,22] compares too simple systems. This paper deals with this aspect and compares the formal speci®cation of the PROcess FIeld BUS (PROFIBUS) communications protocol in both languages LOTOS and SDL. Ó 2001 Elsevier Science B.V. All rights reserved.","author":[{"family":"Marino","given":"P"},{"family":"Nogueira","given":"J"},{"family":"Sigu","given":"C"}],"citation-key":"marinoPROFIBUSFormalSpecification2001","container-title":"Computer Networks","issued":{"date-parts":[["2001"]]},"language":"en","source":"Zotero","title":"The PROFIBUS formal specification: a comparison between two FDTs","type":"article-journal"},{"id":"prestlBMWActiveCruise2000","abstract":"With series introduction of Adaptive Cruise Control (ACC) systems, automotive industry at present makes a step towards a new category of vehicle control systems. For the first time in automotive history these systems make use of information about the surrounding traffic situation. This information i","accessed":{"date-parts":[["2023",11,20]]},"author":[{"family":"Prestl","given":"Willibald"},{"family":"Sauer","given":"Thomas"},{"family":"Steinle","given":"Joachim"},{"family":"Tschernoster","given":"Oliver"}],"citation-key":"prestlBMWActiveCruise2000","DOI":"10.4271/2000-01-0344","event-place":"Warrendale, PA","genre":"SAE Technical Paper","ISSN":"0148-7191, 2688-3627","issued":{"date-parts":[["2000",3,6]]},"language":"English","number":"2000-01-0344","publisher":"SAE International","publisher-place":"Warrendale, PA","source":"www.sae.org","title":"The BMW Active Cruise Control ACC","type":"report","URL":"https://www.sae.org/publications/technical-papers/content/2000-01-0344/"},{"id":"saeedloeiLogicbasedModelingVerification2011","abstract":"Cyber-physical systems (CPS) consist of perpetually and concurrently executing physical and computational components. The presence of physical components require the computational components to deal with continuous quantities. A formalism that can model discrete and continuous quantities together with concurrent, perpetual execution is lacking. In this paper we report on the development of a formalism based on logic programming extended with co-induction, constraints over reals, and coroutining that allows CPS to be elegantly modeled. This logic programming realization can be used for verifying interesting properties as well as generating implementations of CPS. We illustrate this formalism by applying it to elegant modeling of the reactor temperature control system. Interesting properties of the system can be verified merely by posing appropriate queries to this model. Precise parametric analysis can also be performed.","accessed":{"date-parts":[["2023",10,10]]},"author":[{"family":"Saeedloei","given":"Neda"},{"family":"Gupta","given":"Gopal"}],"citation-key":"saeedloeiLogicbasedModelingVerification2011","container-title":"ACM SIGBED Review","container-title-short":"SIGBED Rev.","DOI":"10.1145/2000367.2000374","ISSN":"1551-3688","issue":"2","issued":{"date-parts":[["2011",6]]},"language":"en","page":"31-34","source":"DOI.org (Crossref)","title":"A logic-based modeling and verification of CPS","type":"article-journal","URL":"https://dl.acm.org/doi/10.1145/2000367.2000374","volume":"8"},{"id":"dangeloProbabilisticRobustnessAnalysis2019","abstract":"This work uses probabilistic robustness techniques to show how the stability margin of an uncertain controlled structure that operates in a harsh, potentially radioactive environment can be analyzed in order to find a less conservative destabilizing uncertainty perturbation. The uncertainty is quantified in terms of a measure on the size of the covariance matrix in a multivariate Gaussian distribution. This uncertainty is used to capture the aggregate effects on a structure’s dynamic behavior due to material changes resulting from radiation embrittlement and mechanical fatigue. A probabilistic-robust full-state feedback ℋ∞$${\\mathcal {H}_\\infty }$$controller is synthesized for a low-dimensional structural model using a technique known as scenario-based probabilistic-robust synthesis. A probabilistic-robust stability margin is defined and extracted from a stability degradation function, demonstrating that a fourfold increase in the amount of uncertainty in the model can be tolerated if the designer is willing to concede a small probability that the actively-controlled structure may be unstable for certain system configurations.","author":[{"family":"D’Angelo","given":"Christopher J."},{"family":"Cole","given":"Daniel G."},{"family":"Collinger","given":"John C."}],"citation-key":"dangeloProbabilisticRobustnessAnalysis2019","container-title":"Structural Health Monitoring, Photogrammetry & DIC, Volume 6","DOI":"10.1007/978-3-319-74476-6_17","editor":[{"family":"Niezrecki","given":"Christopher"},{"family":"Baqersad","given":"Javad"}],"event-place":"Cham","ISBN":"978-3-319-74476-6","issued":{"date-parts":[["2019"]]},"language":"en","page":"121-131","publisher":"Springer International Publishing","publisher-place":"Cham","source":"Springer Link","title":"Probabilistic Robustness Analysis of an Actively Controlled Structure that Operates in Harsh and Uncertain Environments","type":"paper-conference"},{"id":"nguyenFuzzyControlSystems2019","abstract":"More than 40 years after fuzzy logic control appeared as an effective tool to deal with complex processes, the research on fuzzy control systems has constantly evolved. Mamdani fuzzy control was originally introduced as a model-free control approach based on expert?s experience and knowledge. Due to the lack of a systematic framework to study Mamdani fuzzy systems, we have witnessed growing interest in fuzzy model-based approaches with Takagi-Sugeno fuzzy systems and singleton-type fuzzy systems (also called piecewise multiaffine systems) over the past decades. This paper reviews the key features of the three above types of fuzzy systems. Through these features, we point out the historical rationale for each type of fuzzy systems and its current research mainstreams. However, the focus is put on fuzzy model-based approaches developed via Lyapunov stability theorem and linear matrix inequality (LMI) formulations. Finally, our personal viewpoint on the perspectives and challenges of the future fuzzy control research is discussed.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Nguyen","given":"Anh-Tu"},{"family":"Taniguchi","given":"Tadanari"},{"family":"Eciolaza","given":"Luka"},{"family":"Campos","given":"Victor"},{"family":"Palhares","given":"Reinaldo"},{"family":"Sugeno","given":"Michio"}],"citation-key":"nguyenFuzzyControlSystems2019","container-title":"IEEE Computational Intelligence Magazine","DOI":"10.1109/MCI.2018.2881644","ISSN":"1556-6048","issue":"1","issued":{"date-parts":[["2019",2]]},"page":"56-68","source":"IEEE Xplore","title":"Fuzzy Control Systems: Past, Present and Future","title-short":"Fuzzy Control Systems","type":"article-journal","URL":"https://ieeexplore.ieee.org/abstract/document/8610273","volume":"14"},{"id":"huangFuzzyModelPredictive2000","abstract":"A fuzzy model predictive control (FMPC) approach is introduced to design a control system for a highly nonlinear process. In this approach, a process system is described by a fuzzy convolution model that consists of a number of quasi-linear fuzzy implications. In controller design, prediction errors and control energy are minimized through a two-layered iterative optimization process. At the lower layer, optimal local control policies are identified to minimize prediction errors in each subsystem. A near optimum is then identified through coordinating the subsystems to reach an overall minimum prediction error at the upper layer. The two-layered computing scheme avoids extensive online nonlinear optimization and permits the design of a controller based on linear control theory. The efficacy of the FMPC approach is demonstrated through three examples.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Huang","given":"Y.L."},{"family":"Lou","given":"H.H."},{"family":"Gong","given":"J.P."},{"family":"Edgar","given":"T.F."}],"citation-key":"huangFuzzyModelPredictive2000","container-title":"IEEE Transactions on Fuzzy Systems","DOI":"10.1109/91.890326","ISSN":"1941-0034","issue":"6","issued":{"date-parts":[["2000",12]]},"page":"665-678","source":"IEEE Xplore","title":"Fuzzy model predictive control","type":"article-journal","URL":"https://ieeexplore.ieee.org/abstract/document/890326","volume":"8"},{"id":"taoRobustFuzzyControl2005","abstract":"A robust complexity reduced proportional-integral-derivative (PID)-like fuzzy controllers is designed for a plant with fuzzy linear model. The plant model is described with the expert's linguistic information involved. The linguistic information for the plant model is represented as fuzzy sets. In order to design a robust fuzzy controller for a plant model with fuzzy sets, an approach is developed to implement the best crisp approximation of fuzzy sets into intervals. Then, Kharitonov's Theorem is applied to construct a robust fuzzy controller for the fuzzy uncertain plant with interval model. With the linear combination of input variables as a new input variable, the complexity of the fuzzy mechanism of PID-like fuzzy controller is significantly reduced. The parameters in the robust fuzzy controller are determined to satisfy the stability conditions. The robustness of the designed fuzzy controller is discussed. Also, with the provided definition of relative robustness, the robustness of the complexity reduced fuzzy controller is compared to the classical PID controller for a second-order plant with fuzzy linear model. The simulation results are included to show the effectiveness of the designed PID-like robust fuzzy controller with the complexity reduced fuzzy mechanism.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Tao","given":"C.W."},{"family":"Taur","given":"J.S."}],"citation-key":"taoRobustFuzzyControl2005","container-title":"IEEE Transactions on Fuzzy Systems","DOI":"10.1109/TFUZZ.2004.839653","ISSN":"1941-0034","issue":"1","issued":{"date-parts":[["2005",2]]},"page":"30-41","source":"IEEE Xplore","title":"Robust fuzzy control for a plant with fuzzy linear model","type":"article-journal","URL":"https://ieeexplore.ieee.org/abstract/document/1392998","volume":"13"},{"id":"kapinskiSimulationguidedLyapunovAnalysis2014","abstract":"Lyapunov functions are used to prove stability and to obtain performance bounds on system behaviors for nonlinear and hybrid dynamical systems, but discovering Lyapunov functions is a difficult task in general. We present a technique for discovering Lyapunov functions and barrier certificates for nonlinear and hybrid dynamical systems using a search-based approach. Our approach uses concrete executions, such as those obtained through simulation, to formulate a series of linear programming (LP) optimization problems; the solution to each LP creates a candidate Lyapunov function. Intermediate candidates are iteratively improved using a global optimizer guided by the Lie derivative of the candidate Lyapunov function. The analysis is refined using counterexamples from a Satisfiability Modulo Theories (SMT) solver. When no counterexamples are found, the soundness of the analysis is verified using an arithmetic solver. The technique can be applied to a broad class of nonlinear dynamical systems, including hybrid systems and systems with polynomial and even transcendental dynamics. We present several examples illustrating the efficacy of the technique, including two automotive powertrain control examples.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Kapinski","given":"James"},{"family":"Deshmukh","given":"Jyotirmoy V."},{"family":"Sankaranarayanan","given":"Sriram"},{"family":"Arechiga","given":"Nikos"}],"citation-key":"kapinskiSimulationguidedLyapunovAnalysis2014","collection-title":"HSCC '14","container-title":"Proceedings of the 17th international conference on Hybrid systems: computation and control","DOI":"10.1145/2562059.2562139","event-place":"New York, NY, USA","ISBN":"978-1-4503-2732-9","issued":{"date-parts":[["2014",4,15]]},"page":"133–142","publisher":"Association for Computing Machinery","publisher-place":"New York, NY, USA","source":"ACM Digital Library","title":"Simulation-guided lyapunov analysis for hybrid dynamical systems","type":"paper-conference","URL":"https://doi.org/10.1145/2562059.2562139"},{"id":"torbenAutomaticSimulationbasedTesting2023","abstract":"A methodology for automatic simulation-based testing of control systems for autonomous vessels is proposed. The work is motivated by the need for increased test coverage and formalism in the verification efforts. It aims to achieve this by formulating requirements in the formal logic Signal Temporal Logic (STL). This enables automatic evaluation of simulations against requirements using the STL robustness metric, resulting in a robustness score for requirements satisfaction. Furthermore, the proposed method uses a Gaussian Process (GP) model for estimating robustness scores including levels of uncertainty for untested cases. The GP model is updated by running simulations and observing the resulting robustness, and its estimates are used to automatically guide the test case selection toward cases with low robustness or high uncertainty. The main scientific contribution is the development of an automatic testing method which incrementally runs new simulations until the entire parameter space of the case is covered to the desired confidence level, or until a case which falsifies the requirement is identified. The methodology is demonstrated through a case study, where the test object is a Collision Avoidance (CA) system for a small high-speed vessel. STL requirements for safety distance, mission compliance, and COLREG compliance are developed. The proposed method shows promise, by both achieving verification in feasible time and identifying falsifying behaviors which would be difficult to detect manually or using brute-force methods. An additional contribution of this work is a formalization of COLREG using temporal logic, which appears to be an interesting direction for future work.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Torben","given":"Tobias Rye"},{"family":"Glomsrud","given":"Jon Arne"},{"family":"Pedersen","given":"Tom Arne"},{"family":"Utne","given":"Ingrid B"},{"family":"Sørensen","given":"Asgeir J"}],"citation-key":"torbenAutomaticSimulationbasedTesting2023","container-title":"Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability","container-title-short":"Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability","DOI":"10.1177/1748006X211069277","ISSN":"1748-006X","issue":"2","issued":{"date-parts":[["2023",4,1]]},"language":"en","page":"293-313","publisher":"SAGE Publications","source":"SAGE Journals","title":"Automatic simulation-based testing of autonomous ships using Gaussian processes and temporal logic","type":"article-journal","URL":"https://doi.org/10.1177/1748006X211069277","volume":"237"},{"id":"kapinskiSimulationguidedApproachesVerification2015","abstract":"Automotive embedded control systems are a vital aspect of modern automotive development, but the considerable complexity of these systems has made quality checking a challenging endeavor. Simulation-based checking approaches are attractive, as they often scale well with the complexity of the system design. This paper presents an overview of simulation-guided techniques that can be used to increase the confidence in the quality of an automotive powertrain control system design. We discuss the relationship between simulation-based approaches and the broader areas of verification and powertrain control design. Also, we discuss new software tools that use simulation-guided approaches to address various aspects of automotive powertrain control design verification. We conclude by considering ongoing challenges in developing new simulation-guided tools and applying them in a powertrain control development context.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Kapinski","given":"James"},{"family":"Deshmukh","given":"Jyotirmoy"},{"family":"Jin","given":"Xiaoqing"},{"family":"Ito","given":"Hisahiro"},{"family":"Butts","given":"Ken"}],"citation-key":"kapinskiSimulationguidedApproachesVerification2015","container-title":"2015 American Control Conference (ACC)","DOI":"10.1109/ACC.2015.7171968","event-title":"2015 American Control Conference (ACC)","ISSN":"2378-5861","issued":{"date-parts":[["2015",7]]},"page":"4086-4095","source":"IEEE Xplore","title":"Simulation-guided approaches for verification of automotive powertrain control systems","type":"paper-conference","URL":"https://ieeexplore.ieee.org/abstract/document/7171968"},{"id":"crespoComputationalFrameworkControl2010","abstract":"This paper presents a methodology for evaluating the robustness of a controller based on its ability to satisfy the design requirements. The framework proposed is generic since it allows for high-fidelity models, arbitrary control structures and arbitrary functional dependencies between the requirements and the uncertain parameters. The cornerstone of this contribution is the ability to bound the region of the uncertain parameter space where the degradation in closed-loop performance remains acceptable. The size of this bounding set, whose geometry can be prescribed according to deterministic or probabilistic uncertainty models, is a measure of robustness. The robustness metrics proposed herein are the parametric safety margin, the reliability index, the failure probability and upper bounds to this probability. The performance observed at the control verification setting, where the assumptions and approximations used for control design may no longer hold, will fully determine the proposed control assessment.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Crespo","given":"Luis G."},{"family":"Kenny","given":"Sean P."},{"family":"Giesy","given":"Daniel P."}],"citation-key":"crespoComputationalFrameworkControl2010","issued":{"date-parts":[["2010",1,1]]},"note":"NTRS Author Affiliations: National Inst. of Aerospace, NASA Langley Research Center\nNTRS Report/Patent Number: L-19786\nNTRS Document ID: 20100006918\nNTRS Research Center: Langley Research Center (LaRC)","source":"NASA NTRS","title":"A Computational Framework to Control Verification and Robustness Analysis","type":"paper-conference","URL":"https://ntrs.nasa.gov/citations/20100006918"},{"id":"aminiLearningRobustControl2020","abstract":"In this work, we present a data-driven simulation and training engine capable of learning end-to-end autonomous vehicle control policies using only sparse rewards. By leveraging real, human-collected trajectories through an environment, we render novel training data that allows virtual agents to drive along a continuum of new local trajectories consistent with the road appearance and semantics, each with a different view of the scene. We demonstrate the ability of policies learned within our simulator to generalize to and navigate in previously unseen real-world roads, without access to any human control labels during training. Our results validate the learned policy onboard a full-scale autonomous vehicle, including in previously un-encountered scenarios, such as new roads and novel, complex, near-crash situations. Our methods are scalable, leverage reinforcement learning, and apply broadly to situations requiring effective perception and robust operation in the physical world.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Amini","given":"Alexander"},{"family":"Gilitschenski","given":"Igor"},{"family":"Phillips","given":"Jacob"},{"family":"Moseyko","given":"Julia"},{"family":"Banerjee","given":"Rohan"},{"family":"Karaman","given":"Sertac"},{"family":"Rus","given":"Daniela"}],"citation-key":"aminiLearningRobustControl2020","container-title":"IEEE Robotics and Automation Letters","DOI":"10.1109/LRA.2020.2966414","ISSN":"2377-3766","issue":"2","issued":{"date-parts":[["2020",4]]},"page":"1143-1150","source":"IEEE Xplore","title":"Learning Robust Control Policies for End-to-End Autonomous Driving From Data-Driven Simulation","type":"article-journal","URL":"https://ieeexplore.ieee.org/abstract/document/8957584","volume":"5"},{"id":"blanchiniModelFreePlantTuning2017","abstract":"Given a static plant described by a differentiable input-output function, which is completely unknown, but whose Jacobian takes values in a known polytope in the matrix space, this paper considers the problem of tuning (i.e., driving to a desired value) the output, by suitably choosing the input. It is shown that, if the polytope is robustly nonsingular (or has full rank, in the nonsquare case), then a suitable tuning scheme drives the output to the desired point. The proof exploits a Lyapunov-like function and applies a well-known game-theoretic result, concerning the existence of a saddle point for a min-max zero-sum game. When the plant output is represented in an implicit form, it is shown that the same result can be obtained, resorting to a different Lyapunov-like function. The case in which proper input or output constraints must be enforced during the transient is considered as well. Some application examples are proposed to show the effectiveness of the approach.","accessed":{"date-parts":[["2024",7,10]]},"author":[{"family":"Blanchini","given":"Franco"},{"family":"Fenu","given":"Gianfranco"},{"family":"Giordano","given":"Giulia"},{"family":"Pellegrino","given":"Felice Andrea"}],"citation-key":"blanchiniModelFreePlantTuning2017","container-title":"IEEE Transactions on Automatic Control","DOI":"10.1109/TAC.2016.2616025","ISSN":"1558-2523","issue":"6","issued":{"date-parts":[["2017",6]]},"page":"2623-2634","source":"IEEE Xplore","title":"Model-Free Plant Tuning","type":"article-journal","URL":"https://ieeexplore.ieee.org/abstract/document/7586127","volume":"62"},{"id":"FormalMethodsSafetyCritical2023","accessed":{"date-parts":[["2024",7,9]]},"citation-key":"FormalMethodsSafetyCritical2023","container-title":"MIT LIDS","issued":{"date-parts":[["2023",10,26]]},"language":"en","title":"Formal Methods for Safety-Critical Control","type":"webpage","URL":"https://lids.mit.edu/news-and-events/events/formal-methods-safety-critical-control"},{"id":"hincheyIntroductionFormalMethods2006","accessed":{"date-parts":[["2024",7,9]]},"author":[{"family":"Hinchey","given":"Michael"},{"family":"Bowen","given":"Jonathan P."},{"family":"Rouff","given":"Christopher A."}],"citation-key":"hincheyIntroductionFormalMethods2006","container-title":"Agent Technology from a Formal Perspective","DOI":"10.1007/1-84628-271-3_2","editor":[{"family":"Rouff","given":"Christopher A."},{"family":"Hinchey","given":"Michael"},{"family":"Rash","given":"James"},{"family":"Truszkowski","given":"Walter"},{"family":"Gordon-Spears","given":"Diana"}],"event-place":"London","ISBN":"978-1-85233-947-0","issued":{"date-parts":[["2006"]]},"language":"en","page":"25-64","publisher":"Springer-Verlag","publisher-place":"London","source":"DOI.org (Crossref)","title":"Introduction to Formal Methods","type":"chapter","URL":"http://link.springer.com/10.1007/1-84628-271-3_2"},{"id":"vorosIntroductionFormalMethods2004","abstract":"This chapter begins with an introduction to the main concepts of formal methods. Languages and tools for developing formal System modeis are also described, while the use of semi formal notations and their integration with formal methods is covered as well. At the end of the chapter, an overview of the current Status of formal methods in embedded System design is presented.","accessed":{"date-parts":[["2024",7,9]]},"author":[{"family":"Voros","given":"Nikolaos S."},{"family":"Mueller","given":"Wolfgang"},{"family":"Snook","given":"Colin"}],"citation-key":"vorosIntroductionFormalMethods2004","container-title":"UML-B Specification for Proven Embedded Systems Design","DOI":"10.1007/978-1-4020-2867-0_1","editor":[{"family":"Bernin","given":"Fredrik"},{"family":"Butler","given":"Michael"},{"family":"Cansell","given":"Dominique"},{"family":"Hallerstede","given":"Stefan"},{"family":"Kronlöf","given":"Klaus"},{"family":"Krupp","given":"Alexander"},{"family":"Lecomte","given":"Thierry"},{"family":"Lundell","given":"Michael"},{"family":"Lundkvist","given":"Ola"},{"family":"Marchetti","given":"Michele"},{"family":"Mueller","given":"Wolfgang"},{"family":"Oliver","given":"Ian"},{"family":"Sabatier","given":"Denis"},{"family":"Schattkowsky","given":"Tim"},{"family":"Snook","given":"Colin"},{"family":"Voros","given":"Nikolaos S."},{"family":"Zimmermann","given":"Yann"},{"family":"Mermet","given":"Jean"}],"event-place":"Boston, MA","ISBN":"978-1-4020-2867-0","issued":{"date-parts":[["2004"]]},"language":"en","page":"1-20","publisher":"Springer US","publisher-place":"Boston, MA","source":"Springer Link","title":"An Introduction to Formal Methods","type":"chapter","URL":"https://doi.org/10.1007/978-1-4020-2867-0_1"},{"id":"wooldridgeLECTUREINTRODUCTIONFORMAL","author":[{"family":"Wooldridge","given":"Mike"}],"citation-key":"wooldridgeLECTUREINTRODUCTIONFORMAL","container-title":"Software Engineering","language":"en","source":"Zotero","title":"LECTURE 6: INTRODUCTION TO FORMAL METHODS","type":"article-journal"},{"id":"FormalMethodsa","accessed":{"date-parts":[["2024",7,9]]},"citation-key":"FormalMethodsa","title":"Formal Methods","type":"webpage","URL":"https://users.ece.cmu.edu/~koopman/des_s99/formal_methods/"},{"id":"weizenbaumELIZAComputerProgram1966","accessed":{"date-parts":[["2024",7,2]]},"author":[{"family":"Weizenbaum","given":"Joseph"}],"citation-key":"weizenbaumELIZAComputerProgram1966","container-title":"Commun. ACM","DOI":"10.1145/365153.365168","ISSN":"0001-0782","issue":"1","issued":{"date-parts":[["1966",1,1]]},"page":"36–45","source":"ACM Digital Library","title":"ELIZA—a computer program for the study of natural language communication between man and machine","type":"article-journal","URL":"https://doi.org/10.1145/365153.365168","volume":"9"},{"id":"ZoteroConnectors","accessed":{"date-parts":[["2024",7,2]]},"citation-key":"ZoteroConnectors","title":"Zotero | Connectors","type":"webpage","URL":"https://www.zotero.org/download/connectors"},{"id":"ExplainableVerificationSurvey2024","abstract":"This report focuses on potential changes in software development practice and research that would help tools used for formal methods explain their output, making software practitioners more likely to trust them.","accessed":{"date-parts":[["2024",6,26]]},"citation-key":"ExplainableVerificationSurvey2024","issued":{"date-parts":[["2024",4,16]]},"language":"en","title":"Explainable Verification: Survey, Situations, and New Ideas","title-short":"Explainable Verification","type":"webpage","URL":"https://insights.sei.cmu.edu/library/explainable-verification-survey-situations-and-new-ideas/"},{"id":"berztissFormalVerificationPrograms","author":[{"family":"Berztiss","given":"Alfs T"},{"family":"Ardis","given":"Mark A"}],"citation-key":"berztissFormalVerificationPrograms","language":"en","source":"Zotero","title":"Formal Verification of Programs","type":"article-journal"},{"id":"zhangUnderstandingUncertaintyCyberPhysical2016","abstract":"Uncertainty is intrinsic in most technical systems, including Cyber-Physical Systems (CPS). Therefore, handling uncertainty in a graceful manner during the real operation of CPS is critical. Since designing, developing, and testing modern and highly sophisticated CPS is an expanding field, a step towards dealing with uncertainty is to identify, define, and classify uncertainties at various levels of CPS. This will help develop a systematic and comprehensive understanding of uncertainty. To that end, we propose a conceptual model for uncertainty specifically designed for CPS. Since the study of uncertainty in CPS development and testing is still irrelatively unexplored, this conceptual model was derived in a large part by reviewing existing work on uncertainty in other fields, including philosophy, physics, statistics, and healthcare. The conceptual model is mapped to the three logical levels of CPS: Application, Infrastructure, and Integration. It is captured using UML class diagrams, including relevant OCL constraints. To validate the conceptual model, we identified, classified, and specified uncertainties in two distinct industrial case studies.","author":[{"family":"Zhang","given":"Man"},{"family":"Selic","given":"Bran"},{"family":"Ali","given":"Shaukat"},{"family":"Yue","given":"Tao"},{"family":"Okariz","given":"Oscar"},{"family":"Norgren","given":"Roland"}],"citation-key":"zhangUnderstandingUncertaintyCyberPhysical2016","container-title":"Modelling Foundations and Applications","DOI":"10.1007/978-3-319-42061-5_16","editor":[{"family":"Wąsowski","given":"Andrzej"},{"family":"Lönn","given":"Henrik"}],"event-place":"Cham","ISBN":"978-3-319-42061-5","issued":{"date-parts":[["2016"]]},"language":"en","page":"247-264","publisher":"Springer International Publishing","publisher-place":"Cham","source":"Springer Link","title":"Understanding Uncertainty in Cyber-Physical Systems: A Conceptual Model","title-short":"Understanding Uncertainty in Cyber-Physical Systems","type":"paper-conference"},{"id":"oquendoDealingUncertaintySoftware2019","abstract":"When architecting Software-intensive Systems-of-Systems (SoS) on the Internet-of-Things (IoT), architects face two sorts of uncertainties. First, they have only limited knowledge about the operational environment where the SoS will actually be deployed. Second, the constituent systems which will compose the SoS might not be known a priori (at design-time) or their availability (at runtime) is affected by dynamic factors, due to the openness of the IoT. The consequent research question is thereby how to deal with uncertainty in the design of an SoS architecture on the IoT. To tackle this challenging issue, this paper addresses the notion of uncertainty due to partial information in SoS and proposes an enhanced SoS Architecture Description language (SosADL) for expressing SoS architectures on the IoT under uncertainty. The core SosADL is extended with concurrent constraints and the concept of digital twins coupling the physical and virtual worlds. This novel approach is supported by an integrated toolset, the SosADL Studio. Validation results demonstrate its effectiveness in an SoS architecture for platooning of self-driving vehicles.","accessed":{"date-parts":[["2022",3,2]]},"author":[{"family":"Oquendo","given":"Flavio"}],"citation-key":"oquendoDealingUncertaintySoftware2019","container-title":"Computational Science and Its Applications – ICCSA 2019","DOI":"10.1007/978-3-030-24289-3_57","editor":[{"family":"Misra","given":"Sanjay"},{"family":"Gervasi","given":"Osvaldo"},{"family":"Murgante","given":"Beniamino"},{"family":"Stankova","given":"Elena"},{"family":"Korkhov","given":"Vladimir"},{"family":"Torre","given":"Carmelo"},{"family":"Rocha","given":"Ana Maria A.C."},{"family":"Taniar","given":"David"},{"family":"Apduhan","given":"Bernady O."},{"family":"Tarantino","given":"Eufemia"}],"event-place":"Cham","ISBN":"978-3-030-24288-6 978-3-030-24289-3","issued":{"date-parts":[["2019"]]},"language":"en","page":"770-786","publisher":"Springer International Publishing","publisher-place":"Cham","source":"DOI.org (Crossref)","title":"Dealing with Uncertainty in Software Architecture on the Internet-of-Things with Digital Twins","type":"chapter","URL":"http://link.springer.com/10.1007/978-3-030-24289-3_57","volume":"11619"},{"id":"kochunasDigitalTwinConcepts2021","abstract":"Digital Twins (DTs) are receiving considerable attention from multiple disciplines. Much of the literature at this time is dedicated to the conceptualization of digital twins, and associated enabling technologies and challenges. In this paper, we consider these propositions for the specific application of nuclear power. Our review finds that the current DT concepts are amenable to nuclear power systems, but benefit from some modifications and enhancements. Further, some areas of the existing modeling and simulation infrastructure around nuclear power systems are adaptable to DT development, while more recent efforts in advanced modeling and simulation are less suitable at this time. For nuclear power applications, DT development should rely first on mechanistic model-based methods to leverage the extensive experience and understanding of these systems. Model-free techniques can then be adopted to selectively, and correctively, augment limitations in the model-based approaches. Challenges to the realization of a DT are also discussed, with some being unique to nuclear engineering, however most are broader. A challenging aspect we discuss in detail for DTs is the incorporation of uncertainty quantification (UQ). Forward UQ enables the propagation of uncertainty from the digital representations to predict behavior of the physical asset. Similarly, inverse UQ allows for the incorporation of data from new measurements obtained from the physical asset back into the DT. Optimization under uncertainty facilitates decision support through the formal methods of optimal experimental design and design optimization that maximize information gain, or performance, of the physical asset in an uncertain environment.","accessed":{"date-parts":[["2022",3,2]]},"author":[{"family":"Kochunas","given":"Brendan"},{"family":"Huan","given":"Xun"}],"citation-key":"kochunasDigitalTwinConcepts2021","container-title":"Energies","container-title-short":"Energies","DOI":"10.3390/en14144235","ISSN":"1996-1073","issue":"14","issued":{"date-parts":[["2021",7,14]]},"language":"en","page":"4235","source":"DOI.org (Crossref)","title":"Digital Twin Concepts with Uncertainty for Nuclear Power Applications","type":"article-journal","URL":"https://www.mdpi.com/1996-1073/14/14/4235","volume":"14"},{"id":"RevModPhys","accessed":{"date-parts":[["2024",5,21]]},"citation-key":"RevModPhys","title":"Rev. Mod. Phys. 83, 943 (2011) - Bayesian inference in physics","type":"webpage","URL":"https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.83.943"},{"id":"kennedyBayesianCalibrationComputer2001","abstract":"We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models. Such models, implemented as computer codes, are often generic in the sense that by a suitable choice of some of the model's input parameters the code can be used to predict the behaviour of the system in a variety of specific applications. However, in any specific application the values of necessary parameters may be unknown. In this case, physical observations of the system in the specific context are used to learn about the unknown parameters. The process of fitting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc fitting, and after calibration the model is used, with the fitted input values, to predict the future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for all sources of uncertainty, including the remaining uncertainty over the fitted parameters. Second, they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best-fitting parameter values. The method is illustrated by using data from a nuclear radiation release at Tomsk, and from a more complex simulated nuclear accident exercise.","accessed":{"date-parts":[["2024",5,21]]},"author":[{"family":"Kennedy","given":"Marc C."},{"family":"O'Hagan","given":"Anthony"}],"citation-key":"kennedyBayesianCalibrationComputer2001","container-title":"Journal of the Royal Statistical Society Series B: Statistical Methodology","container-title-short":"Journal of the Royal Statistical Society Series B: Statistical Methodology","DOI":"10.1111/1467-9868.00294","ISSN":"1369-7412","issue":"3","issued":{"date-parts":[["2001",9,1]]},"page":"425-464","source":"Silverchair","title":"Bayesian Calibration of Computer Models","type":"article-journal","URL":"https://doi.org/10.1111/1467-9868.00294","volume":"63"},{"id":"oberkampfVerificationValidationPredictive2004","abstract":"Developers of computer codes, analysts who use the codes, and decision makers who rely on the results of the analyses face a critical question: How should confidence in modeling and simulation be critically assessed? Verification and validation (V&V) of computational simulations are the primary methods for building and quantifying this confidence. Briefly, verification is the assessment of the accuracy of the solution to a computational model. Validation is the assessment of the accuracy of a computational simulation by comparison with experimental data. In verification, the relationship of the simulation to the real world is not an issue. In validation, the relationship between computation and the real world, ie, experimental data, is the issue. This paper presents our viewpoint of the state of the art in V&V in computational physics. (In this paper we refer to all fields of computational engineering and physics, eg, computational fluid dynamics, computational solid mechanics, structural dynamics, shock wave physics, computational chemistry, etc, as computational physics.) We describe our view of the framework in which predictive capability relies on V&V, as well as other factors that affect predictive capability. Our opinions about the research needs and management issues in V&V are very practical: What methods and techniques need to be developed and what changes in the views of management need to occur to increase the usefulness, reliability, and impact of computational physics for decision making about engineering systems? We review the state of the art in V&V over a wide range of topics, for example, prioritization of V&V activities using the Phenomena Identification and Ranking Table (PIRT), code verification, software quality assurance (SQA), numerical error estimation, hierarchical experiments for validation, characteristics of validation experiments, the need to perform nondeterministic computational simulations in comparisons with experimental data, and validation metrics. We then provide an extensive discussion of V&V research and implementation issues that we believe must be addressed for V&V to be more effective in improving confidence in computational predictive capability. Some of the research topics addressed are development of improved procedures for the use of the PIRT for prioritizing V&V activities, the method of manufactured solutions for code verification, development and use of hierarchical validation diagrams, and the construction and use of validation metrics incorporating statistical measures. Some of the implementation topics addressed are the needed management initiatives to better align and team computationalists and experimentalists in conducting validation activities, the perspective of commercial software companies, the key role of analysts and decision makers as code customers, obstacles to the improved effectiveness of V&V, effects of cost and schedule constraints on practical applications in industrial settings, and the role of engineering standards committees in documenting best practices for V&V. There are 207 references cited in this review article.","accessed":{"date-parts":[["2024",5,21]]},"author":[{"family":"Oberkampf","given":"William L"},{"family":"Trucano","given":"Timothy G"},{"family":"Hirsch","given":"Charles"}],"citation-key":"oberkampfVerificationValidationPredictive2004","container-title":"Applied Mechanics Reviews","container-title-short":"Applied Mechanics Reviews","DOI":"10.1115/1.1767847","ISSN":"0003-6900","issue":"5","issued":{"date-parts":[["2004",12,21]]},"page":"345-384","source":"Silverchair","title":"Verification, validation, and predictive capability in computational engineering and physics","type":"article-journal","URL":"https://doi.org/10.1115/1.1767847","volume":"57"},{"id":"DigitalTwins","abstract":"Digital Twins","accessed":{"date-parts":[["2024",5,21]]},"citation-key":"DigitalTwins","container-title":"NRC Web","language":"en-US","title":"Digital Twins","type":"webpage","URL":"https://www.nrc.gov/reactors/power/digital-twins.html"},{"id":"hadiControlCOVID19System2021","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Hadi","given":"Musadaq A."},{"family":"Ali","given":"Hazem I."}],"citation-key":"hadiControlCOVID19System2021","container-title":"Biomedical Signal Processing and Control","container-title-short":"Biomedical Signal Processing and Control","DOI":"10.1016/j.bspc.2020.102317","ISSN":"17468094","issued":{"date-parts":[["2021",2]]},"language":"en","license":"https://www.elsevier.com/tdm/userlicense/1.0/","page":"102317","source":"DOI.org (Crossref)","title":"Control of COVID-19 system using a novel nonlinear robust control algorithm","type":"article-journal","URL":"https://linkinghub.elsevier.com/retrieve/pii/S1746809420304341","volume":"64"},{"id":"wangAdaptiveCriticNonlinear2017","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Wang","given":"Ding"},{"family":"He","given":"Haibo"},{"family":"Liu","given":"Derong"}],"citation-key":"wangAdaptiveCriticNonlinear2017","container-title":"IEEE Transactions on Cybernetics","container-title-short":"IEEE Trans. Cybern.","DOI":"10.1109/TCYB.2017.2712188","ISSN":"2168-2267, 2168-2275","issue":"10","issued":{"date-parts":[["2017",10]]},"license":"https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html","page":"3429-3451","source":"DOI.org (Crossref)","title":"Adaptive Critic Nonlinear Robust Control: A Survey","title-short":"Adaptive Critic Nonlinear Robust Control","type":"article-journal","URL":"http://ieeexplore.ieee.org/document/7967695/","volume":"47"},{"id":"durethConditionalDiffusionbasedMicrostructure2023","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Düreth","given":"Christian"},{"family":"Seibert","given":"Paul"},{"family":"Rücker","given":"Dennis"},{"family":"Handford","given":"Stephanie"},{"family":"Kästner","given":"Markus"},{"family":"Gude","given":"Maik"}],"citation-key":"durethConditionalDiffusionbasedMicrostructure2023","container-title":"Materials Today Communications","container-title-short":"Materials Today Communications","DOI":"10.1016/j.mtcomm.2023.105608","ISSN":"23524928","issued":{"date-parts":[["2023",6]]},"language":"en","page":"105608","source":"DOI.org (Crossref)","title":"Conditional diffusion-based microstructure reconstruction","type":"article-journal","URL":"https://linkinghub.elsevier.com/retrieve/pii/S2352492823002982","volume":"35"},{"id":"kongDiffWaveVersatileDiffusion2020","abstract":"In this work, we propose DiffWave, a versatile diffusion probabilistic model for conditional and unconditional waveform generation. The model is non-autoregressive, and converts the white noise signal into structured waveform through a Markov chain with a constant number of steps at synthesis. It is efficiently trained by optimizing a variant of variational bound on the data likelihood. DiffWave produces high-fidelity audios in different waveform generation tasks, including neural vocoding conditioned on mel spectrogram, class-conditional generation, and unconditional generation. We demonstrate that DiffWave matches a strong WaveNet vocoder in terms of speech quality (MOS: 4.44 versus 4.43), while synthesizing orders of magnitude faster. In particular, it significantly outperforms autoregressive and GAN-based waveform models in the challenging unconditional generation task in terms of audio quality and sample diversity from various automatic and human evaluations.","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Kong","given":"Zhifeng"},{"family":"Ping","given":"Wei"},{"family":"Huang","given":"Jiaji"},{"family":"Zhao","given":"Kexin"},{"family":"Catanzaro","given":"Bryan"}],"citation-key":"kongDiffWaveVersatileDiffusion2020","DOI":"10.48550/ARXIV.2009.09761","issued":{"date-parts":[["2020"]]},"license":"arXiv.org perpetual, non-exclusive license","publisher":"arXiv","source":"DOI.org (Datacite)","title":"DiffWave: A Versatile Diffusion Model for Audio Synthesis","title-short":"DiffWave","type":"article","URL":"https://arxiv.org/abs/2009.09761","version":"3"},{"id":"wangDiffuseBotBreedingSoft2023","author":[{"family":"Wang","given":"Tsun-Hsuan Johnson"},{"family":"Zheng","given":"Juntian"},{"family":"Ma","given":"Pingchuan"},{"family":"Du","given":"Yilun"},{"family":"Kim","given":"Byungchul"},{"family":"Spielberg","given":"Andrew"},{"family":"Tenenbaum","given":"Josh"},{"family":"Gan","given":"Chuang"},{"family":"Rus","given":"Daniela"}],"citation-key":"wangDiffuseBotBreedingSoft2023","container-title":"Advances in Neural Information Processing Systems","editor":[{"family":"Oh","given":"A."},{"family":"Naumann","given":"T."},{"family":"Globerson","given":"A."},{"family":"Saenko","given":"K."},{"family":"Hardt","given":"M."},{"family":"Levine","given":"S."}],"issued":{"date-parts":[["2023"]]},"page":"44398–44423","publisher":"Curran Associates, Inc.","title":"DiffuseBot: Breeding Soft Robots With Physics-Augmented Generative Diffusion Models","type":"paper-conference","URL":"https://proceedings.neurips.cc/paper_files/paper/2023/file/8b1008098947ad59144c18a78337f937-Paper-Conference.pdf","volume":"36"},{"id":"liSyntheticLagrangianTurbulence2024","abstract":"Abstract\n Lagrangian turbulence lies at the core of numerous applied and fundamental problems related to the physics of dispersion and mixing in engineering, biofluids, the atmosphere, oceans and astrophysics. Despite exceptional theoretical, numerical and experimental efforts conducted over the past 30 years, no existing models are capable of faithfully reproducing statistical and topological properties exhibited by particle trajectories in turbulence. We propose a machine learning approach, based on a state-of-the-art diffusion model, to generate single-particle trajectories in three-dimensional turbulence at high Reynolds numbers, thereby bypassing the need for direct numerical simulations or experiments to obtain reliable Lagrangian data. Our model demonstrates the ability to reproduce most statistical benchmarks across time scales, including the fat-tail distribution for velocity increments, the anomalous power law and the increased intermittency around the dissipative scale. Slight deviations are observed below the dissipative scale, particularly in the acceleration and flatness statistics. Surprisingly, the model exhibits strong generalizability for extreme events, producing events of higher intensity and rarity that still match the realistic statistics. This paves the way for producing synthetic high-quality datasets for pretraining various downstream applications of Lagrangian turbulence.","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Li","given":"T."},{"family":"Biferale","given":"L."},{"family":"Bonaccorso","given":"F."},{"family":"Scarpolini","given":"M. A."},{"family":"Buzzicotti","given":"M."}],"citation-key":"liSyntheticLagrangianTurbulence2024","container-title":"Nature Machine Intelligence","container-title-short":"Nat Mach Intell","DOI":"10.1038/s42256-024-00810-0","ISSN":"2522-5839","issue":"4","issued":{"date-parts":[["2024",4,17]]},"language":"en","page":"393-403","source":"DOI.org (Crossref)","title":"Synthetic Lagrangian turbulence by generative diffusion models","type":"article-journal","URL":"https://www.nature.com/articles/s42256-024-00810-0","volume":"6"},{"id":"esmaeiliEnhancingDigitalRock2024","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Esmaeili","given":"Mohammad"}],"citation-key":"esmaeiliEnhancingDigitalRock2024","container-title":"Neurocomputing","container-title-short":"Neurocomputing","DOI":"10.1016/j.neucom.2024.127676","ISSN":"09252312","issued":{"date-parts":[["2024",6]]},"language":"en","page":"127676","source":"DOI.org (Crossref)","title":"Enhancing digital rock analysis through generative artificial intelligence: Diffusion models","title-short":"Enhancing digital rock analysis through generative artificial intelligence","type":"article-journal","URL":"https://linkinghub.elsevier.com/retrieve/pii/S0925231224004478","volume":"587"},{"id":"yangDiffusionModelsComprehensive2024","abstract":"Diffusion models have emerged as a powerful new family of deep generative models with record-breaking performance in many applications, including image synthesis, video generation, and molecule design. In this survey, we provide an overview of the rapidly expanding body of work on diffusion models, categorizing the research into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. We also discuss the potential for combining diffusion models with other generative models for enhanced results. We further review the wide-ranging applications of diffusion models in fields spanning from computer vision, natural language processing, temporal data modeling, to interdisciplinary applications in other scientific disciplines. This survey aims to provide a contextualized, in-depth look at the state of diffusion models, identifying the key areas of focus and pointing to potential areas for further exploration. Github:\n https://github.com/YangLing0818/Diffusion-Models-Papers-Survey-Taxonomy","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Yang","given":"Ling"},{"family":"Zhang","given":"Zhilong"},{"family":"Song","given":"Yang"},{"family":"Hong","given":"Shenda"},{"family":"Xu","given":"Runsheng"},{"family":"Zhao","given":"Yue"},{"family":"Zhang","given":"Wentao"},{"family":"Cui","given":"Bin"},{"family":"Yang","given":"Ming-Hsuan"}],"citation-key":"yangDiffusionModelsComprehensive2024","container-title":"ACM Computing Surveys","container-title-short":"ACM Comput. Surv.","DOI":"10.1145/3626235","ISSN":"0360-0300, 1557-7341","issue":"4","issued":{"date-parts":[["2024",4,30]]},"language":"en","page":"1-39","source":"DOI.org (Crossref)","title":"Diffusion Models: A Comprehensive Survey of Methods and Applications","title-short":"Diffusion Models","type":"article-journal","URL":"https://dl.acm.org/doi/10.1145/3626235","volume":"56"},{"id":"jagvaralUnifiedFrameworkDiffusion2024","abstract":"Diffusion-based generative models represent the current state-of-the-art for image generation. However, standard diffusion models are based on Euclidean geometry and do not translate directly to manifold-valued data. In this work, we develop extensions of both score-based generative models (SGMs) and Denoising Diffusion Probabilistic Models (DDPMs) to the Lie group of 3D rotations, SO(3). SO(3) is of particular interest in many disciplines such as robotics, biochemistry and astronomy/cosmology science. Contrary to more general Riemannian manifolds, SO(3) admits a tractable solution to heat diffusion, and allows us to implement efficient training of diffusion models. We apply both SO(3) DDPMs and SGMs to synthetic densities on SO(3) and demonstrate state-of-the-art results. Additionally, we demonstrate the practicality of our model on pose estimation tasks and in predicting correlated galaxy orientations for astrophysics/cosmology.","accessed":{"date-parts":[["2024",5,20]]},"author":[{"family":"Jagvaral","given":"Yesukhei"},{"family":"Lanusse","given":"Francois"},{"family":"Mandelbaum","given":"Rachel"}],"citation-key":"jagvaralUnifiedFrameworkDiffusion2024","container-title":"Proceedings of the AAAI Conference on Artificial Intelligence","container-title-short":"AAAI","DOI":"10.1609/aaai.v38i11.29171","ISSN":"2374-3468, 2159-5399","issue":"11","issued":{"date-parts":[["2024",3,24]]},"page":"12754-12762","source":"DOI.org (Crossref)","title":"Unified Framework for Diffusion Generative Models in SO(3): Applications in Computer Vision and Astrophysics","title-short":"Unified Framework for Diffusion Generative Models in SO(3)","type":"article-journal","URL":"https://ojs.aaai.org/index.php/AAAI/article/view/29171","volume":"38"},{"id":"avigadFORMALSYSTEMEUCLID2009","abstract":"We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.","accessed":{"date-parts":[["2024",5,16]]},"author":[{"family":"Avigad","given":"Jeremy"},{"family":"Dean","given":"Edward"},{"family":"Mumma","given":"John"}],"citation-key":"avigadFORMALSYSTEMEUCLID2009","container-title":"The Review of Symbolic Logic","container-title-short":"The Review of Symbolic Logic","DOI":"10.1017/S1755020309990098","ISSN":"1755-0203, 1755-0211","issue":"4","issued":{"date-parts":[["2009",12]]},"language":"en","license":"https://www.cambridge.org/core/terms","page":"700-768","source":"DOI.org (Crossref)","title":"A FORMAL SYSTEM FOR EUCLID’S The core of the demonstration of this paper is to interpret the forward propagation process of machine learning as a parameter estimation problem of nonlinear dynamical systems. This process is to establish a connection between the Recurrent Neural Network and the discrete differential equation, so as to construct a new network structure: ODE-RU. At the same time, under the inspiration of the theory of ordinary differential equations, we propose a new forward propagation mode. In a large number of simulations and experiments, the forward propagation not only shows the trainability of the new architecture, but also achieves a low training error on the basis of main-taining the stability of the network. For the problem requiring long-term memory, we specifically study the obstacle shape reconstruction problem using the backscattering far-field features data set, and demonstrate the effectiveness of the proposed architecture using the data set. The results show that the network can effectively reduce the sensitivity to small changes in the input feature. And the error generated by the ordinary differential equation cyclic unit network in inverting the shape and position of obstacles is less than $ 10^{-2} $.
","accessed":{"date-parts":[["2024",1,30]]},"author":[{"family":"Meng","given":"Pinchao"},{"family":"Wang","given":"Xinyu"},{"family":"Yin","given":"Weishi"}],"citation-key":"mengODERUDynamicalSystem2022","container-title":"Electronic Research Archive","container-title-short":"era","DOI":"10.3934/era.2022014","ISSN":"2688-1594","issue":"1","issued":{"date-parts":[["2022"]]},"page":"257-271","source":"DOI.org (Crossref)","title":"ODE-RU: a dynamical system view on recurrent neural networks","title-short":"ODE-RU","type":"article-journal","URL":"http://www.aimspress.com/article/doi/10.3934/era.2022014","volume":"30"},{"id":"stiasnyPhysicsInformedNeuralNetworks2023","abstract":"The simulation of power system dynamics poses a computationally expensive task. Considering the growing uncertainty of generation and demand patterns, thousands of scenarios need to be continuously assessed to ensure the safety of power systems. Physics-Informed Neural Networks (PINNs) have recently emerged as a promising solution for drastically accelerating computations of non-linear dynamical systems. This work investigates the applicability of these methods for power system dynamics, focusing on the dynamic response to load disturbances. Comparing the prediction of PINNs to the solution of conventional solvers, we find that PINNs can be 10 to 1000 times faster than conventional solvers. At the same time, we find them to be sufficiently accurate and numerically stable even for large time steps. To facilitate a deeper understanding, this paper also present a new regularisation of Neural Network (NN) training by introducing a gradient-based term in the loss function. The resulting NNs, which we call dtNNs, help us deliver a comprehensive analysis about the strengths and weaknesses of the NN based approaches, how incorporating knowledge of the underlying physics affects NN performance, and how this compares with conventional solvers for power system dynamics.","accessed":{"date-parts":[["2024",1,30]]},"author":[{"family":"Stiasny","given":"Jochen"},{"family":"Chatzivasileiadis","given":"Spyros"}],"citation-key":"stiasnyPhysicsInformedNeuralNetworks2023","container-title":"Electric Power Systems Research","container-title-short":"Electric Power Systems Research","DOI":"10.1016/j.epsr.2023.109748","ISSN":"03787796","issued":{"date-parts":[["2023",11]]},"page":"109748","source":"arXiv.org","title":"Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibility","title-short":"Physics-Informed Neural Networks for Time-Domain Simulations","type":"article-journal","URL":"http://arxiv.org/abs/2303.08994","volume":"224"},{"id":"cuomoScientificMachineLearning2022","abstract":"Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.","accessed":{"date-parts":[["2024",1,30]]},"author":[{"family":"Cuomo","given":"Salvatore"},{"family":"Cola","given":"Vincenzo Schiano","non-dropping-particle":"di"},{"family":"Giampaolo","given":"Fabio"},{"family":"Rozza","given":"Gianluigi"},{"family":"Raissi","given":"Maziar"},{"family":"Piccialli","given":"Francesco"}],"citation-key":"cuomoScientificMachineLearning2022","issued":{"date-parts":[["2022",6,7]]},"number":"arXiv:2201.05624","publisher":"arXiv","source":"arXiv.org","title":"Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next","title-short":"Scientific Machine Learning through Physics-Informed Neural Networks","type":"article","URL":"http://arxiv.org/abs/2201.05624"},{"id":"wangPINNsBasedUncertaintyQuantification2023","abstract":"This paper addresses the challenge of transient stability in power systems with missing parameters and uncertainty propagation in swing equations. We introduce a novel application of Physics-Informed Neural Networks (PINNs), specifically an Ensemble of PINNs (E-PINNs), to estimate critical parameters like rotor angle and inertia coefficient with enhanced accuracy and reduced computational load. E-PINNs capitalize on the underlying physical principles of swing equations to provide a robust solution. Our approach not only facilitates efficient parameter estimation but also quantifies uncertainties, delivering probabilistic insights into the system behavior. The efficacy of E-PINNs is demonstrated through the analysis of $1$-bus and $2$-bus systems, highlighting the model's ability to handle parameter variability and data scarcity. The study advances the application of machine learning in power system stability, paving the way for reliable and computationally efficient transient stability analysis.","accessed":{"date-parts":[["2024",1,30]]},"author":[{"family":"Wang","given":"Ren"},{"family":"Zhong","given":"Ming"},{"family":"Xu","given":"Kaidi"},{"family":"Sánchez-Cortés","given":"Lola Giráldez"},{"family":"Guerra","given":"Ignacio de Cominges"}],"citation-key":"wangPINNsBasedUncertaintyQuantification2023","issued":{"date-parts":[["2023",11,21]]},"number":"arXiv:2311.12947","publisher":"arXiv","source":"arXiv.org","title":"PINNs-Based Uncertainty Quantification for Transient Stability Analysis","type":"article","URL":"http://arxiv.org/abs/2311.12947"},{"id":"stiasnyTransientStabilityAnalysis2023","abstract":"We explore the possibility to use physics-informed neural networks to drastically accelerate the solution of ordinary differential-algebraic equations that govern the power system dynamics. When it comes to transient stability assessment, the traditionally applied methods either carry a significant computational burden, require model simplifications, or use overly conservative surrogate models. Conventional neural networks can circumvent these limitations but are faced with high demand of high-quality training datasets, while they ignore the underlying governing equations. Physics-informed neural networks are different: they incorporate the power system differential algebraic equations directly into the neural network training and drastically reduce the need for training data. This paper takes a deep dive into the performance of physics-informed neural networks for power system transient stability assessment. Introducing a new neural network training procedure to facilitate a thorough comparison, we explore how physics-informed neural networks compare with conventional differential-algebraic solvers and classical neural networks in terms of computation time, requirements in data, and prediction accuracy. We illustrate the findings on the Kundur two-area system, and assess the opportunities and challenges of physics-informed neural networks to serve as a transient stability analysis tool, highlighting possible pathways to further develop this method.","accessed":{"date-parts":[["2024",1,30]]},"author":[{"family":"Stiasny","given":"Jochen"},{"family":"Misyris","given":"Georgios S."},{"family":"Chatzivasileiadis","given":"Spyros"}],"citation-key":"stiasnyTransientStabilityAnalysis2023","issued":{"date-parts":[["2023",3,15]]},"number":"arXiv:2106.13638","publisher":"arXiv","source":"arXiv.org","title":"Transient Stability Analysis with Physics-Informed Neural Networks","type":"article","URL":"http://arxiv.org/abs/2106.13638"},{"id":"alwanTheoryHybridSystems2018","abstract":"This book is the first to present the application of the hybrid system theory to systems with EPCA (equations with piecewise continuous arguments). The hybrid system paradigm is a valuable modeling tool for describing a wide range of real-world applications. Moreover, although new technology has produced, and continues to produce highly hierarchical sophisticated machinery that cannot be analyzed as a whole system, hybrid system representation can be used to reduce the structural complexity of these systems. That is to say, hybrid systems have become a modeling priority, which in turn has led to the creation of a promising research field with several application areas. As such, the book explores recent developments in the area of deterministic and stochastic hybrid systems using the Lyapunov and Razumikhin-Lyapunov methods to investigate the systems' properties. It also describes properties such as stability, stabilization, reliable control, H-infinity optimal control, input-to-state stability (ISS)/stabilization, state estimation, and large-scale singularly perturbed systems","author":[{"family":"Alwan","given":"Mohamad S."},{"family":"Liu","given":"Xinzhi"}],"call-number":"629.8","citation-key":"alwanTheoryHybridSystems2018","collection-title":"Nonlinear Physical Science","DOI":"10.1007/978-981-10-8046-3","edition":"1st ed. 2018","event-place":"Singapore","ISBN":"978-981-10-8046-3","issued":{"date-parts":[["2018"]]},"number-of-pages":"1","publisher":"Springer Singapore : Imprint: Springer","publisher-place":"Singapore","source":"Library of Congress ISBN","title":"Theory of Hybrid Systems: Deterministic and Stochastic","title-short":"Theory of Hybrid Systems","type":"book"},{"id":"communityLeanMathematicalLibrary2020","abstract":"This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large- and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led us here.","accessed":{"date-parts":[["2024",1,30]]},"author":[{"family":"Community","given":"The","dropping-particle":"mathlib"}],"citation-key":"communityLeanMathematicalLibrary2020","container-title":"Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs","DOI":"10.1145/3372885.3373824","issued":{"date-parts":[["2020",1,20]]},"page":"367-381","source":"arXiv.org","title":"The Lean mathematical library","type":"paper-conference","URL":"http://arxiv.org/abs/1910.09336"},{"id":"voevodskyOriginsMotivationsUnivalent","author":[{"family":"Voevodsky","given":"Vladimir"}],"citation-key":"voevodskyOriginsMotivationsUnivalent","language":"en","source":"Zotero","title":"The Origins and Motivations of Univalent Foundations","type":"article-journal"},{"id":"sorensenLecturesCurryHowardIsomorphism","author":[{"family":"Sorensen","given":"Morten Heine B."},{"family":"Urzyczyn","given":"Pawel"}],"citation-key":"sorensenLecturesCurryHowardIsomorphism","title":"Lectures on the Curry-Howard Isomorphism","type":"document"},{"id":"rojasTutorialIntroductionLambda","abstract":"This paper is a short and painless introduction to the λ calculus. Originally developed in order to study some mathematical properties of effectively computable functions, this formalism has provided a strong theoretical foundation for the family of functional programming languages. We show how to perform some arithmetical computations using the λ calculus and how to define recursive functions, even though functions in λ calculus are not given names and thus cannot refer explicitly to themselves.","author":[{"family":"Rojas","given":"Raul"}],"citation-key":"rojasTutorialIntroductionLambda","language":"en","source":"Zotero","title":"A Tutorial Introduction to the Lambda Calculus","type":"article-journal"},{"id":"viteriExplosiveProofsMathematical2022","abstract":"Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because their probability of error grows exponentially as the argument expands. Here we show that under a cognitively-plausible belief formation mechanism that combines deductive and abductive reasoning, mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with four hand-constructed cases from Euclid, Apollonius, Spinoza, and Andrew Wiles. Our results bear both on recent work in the history and philosophy of mathematics, and on a question, basic to cognitive science, of how we form beliefs, and justify them to others.","accessed":{"date-parts":[["2024",1,29]]},"author":[{"family":"Viteri","given":"Scott"},{"family":"DeDeo","given":"Simon"}],"citation-key":"viteriExplosiveProofsMathematical2022","container-title":"Cognition","container-title-short":"Cognition","DOI":"10.1016/j.cognition.2022.105120","ISSN":"00100277","issued":{"date-parts":[["2022",8]]},"language":"en","page":"105120","source":"arXiv.org","title":"Explosive Proofs of Mathematical Truths","type":"article-journal","URL":"http://arxiv.org/abs/2004.00055","volume":"225"},{"id":"hanDeepLinkEquating2023","abstract":"Mathematical logic and the code of computer programs are, in an exact way, mirror images of each other.","accessed":{"date-parts":[["2024",1,28]]},"author":[{"family":"Han","given":"Sheon"}],"citation-key":"hanDeepLinkEquating2023","issued":{"date-parts":[["2023",10,11]]},"title":"The Deep Link Equating Math Proofs and Computer Programs","type":"webpage","URL":"https://www.quantamagazine.org/the-deep-link-equating-math-proofs-and-computer-programs-20231011/"},{"id":"TLA2023","abstract":"TLA+ is a formal specification language developed by Leslie Lamport. It is used for designing, modelling, documentation, and verification of programs, especially concurrent systems and distributed systems. TLA+ is considered to be exhaustively-testable pseudocode, and its use likened to drawing blueprints for software systems; TLA is an acronym for Temporal Logic of Actions.\nFor design and documentation, TLA+ fulfills the same purpose as informal technical specifications. However, TLA+ specifications are written in a formal language of logic and mathematics, and the precision of specifications written in this language is intended to uncover design flaws before system implementation is underway.Since TLA+ specifications are written in a formal language, they are amenable to finite model checking. The model checker finds all possible system behaviours up to some number of execution steps, and examines them for violations of desired invariance properties such as safety and liveness. TLA+ specifications use basic set theory to define safety (bad things won't happen) and temporal logic to define liveness (good things eventually happen).\nTLA+ is also used to write machine-checked proofs of correctness both for algorithms and mathematical theorems. The proofs are written in a declarative, hierarchical style independent of any single theorem prover backend. Both formal and informal structured mathematical proofs can be written in TLA+; the language is similar to LaTeX, and tools exist to translate TLA+ specifications to LaTeX documents.TLA+ was introduced in 1999, following several decades of research into a verification method for concurrent systems. Ever since, a toolchain has been developed, including an IDE and a distributed model checker. The pseudocode-like language PlusCal was created in 2009; it transpiles to TLA+ and is useful for specifying sequential algorithms. TLA+2 was announced in 2014, expanding language support for proof constructs. The current TLA+ reference is The TLA+ Hyperbook by Leslie Lamport.","accessed":{"date-parts":[["2024",1,28]]},"citation-key":"TLA2023","container-title":"Wikipedia","issued":{"date-parts":[["2023",12,24]]},"license":"Creative Commons Attribution-ShareAlike License","note":"Page Version ID: 1191589904","source":"Wikipedia","title":"TLA