In this work we present a novel, accurate, and robust physics-informed method for solving problems involving parametric differential equations (DEs) called the Extreme Theory of Functional Connections, or X-TFC. The proposed method is a synergy of two recently developed frameworks for solving problems involving parametric DEs, 1) the Theory of Functional Connections, TFC, and the Physics-Informed Neural Networks, PINN. Although this paper focuses on the solution of exact problems involving parametric DEs (i.e. problems where the modeling error is negligible) with known parameters, X-TFC can also be used for data-driven solutions and data-driven discovery of parametric DEs. In the proposed method, the latent solution of the parametric DEs is approximated by a TFC constrained expression that uses a Neural Network (NN) as the free-function. This approximate solution form always analytically satisfies the constraints of the DE, while maintaining a NN with unconstrained parameters, like the Deep-TFC method. X-TFC differs from PINN and Deep-TFC; whereas PINN and Deep-TFC use a deep-NN, X-TFC uses a single-layer NN, or more precisely, an Extreme Learning Machine, ELM. This choice is based on the properties of the ELM algorithm. In order to numerically validate the method, it was tested over a range of problems including the approximation of solutions to linear and non-linear ordinary DEs (ODEs), systems of ODEs (SODEs), and partial DEs (PDEs). Furthermore, a few of these problems are of interest in physics and engineering such as the Classic Emden-Fowler equation, the Radiative Transfer (RT) equation, and the Heat-Transfer (HT) equation. The results show that X-TFC achieves high accuracy with low computational time and thus it is comparable with the other state-of-the-art methods.
Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution
Tongji University, College of Civil Engineering（同济大学，土木工程学院）；Tongji University, State Key Laboratory of Disaster Reduction in Civil Engineering（同济大学，灾害防治工程重点实验室）
On the training of physics-informed neural operators for solving parametric partial differential equations
College of Civil Engineering, Tongji University（同济大学土木工程学院）；Department of Engineering Science, University of Oxford（牛津大学工程科学系）；Institute for Foundations of Data Science, Yale University（耶鲁大学数据科学基础研究所）
Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning
Oden Institute for Computational Sciences and Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算科学与工程研究所）；College of Computing, Georgia Institute of Technology（佐治亚理工学院计算学院）；Walker Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校沃克机械工程系）
LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEs
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China（上海交通大学航空宇航科学与技术学院）；School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China（浙江大学航空宇航科学与技术学院）；Key Laboratory for Satellite Digitalization Technology, Innovation Academy for Microsatellites of CAS, Shanghai, China（中国科学院卫星数字化技术重点实验室）
Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks
Institute of Applied Mechanics（应用力学研究所）；Technische Universität Braunschweig（Braunschweig技术大学）；Clausthal University of Technology（克莱斯特哈大学）；Institute for Acoustics and Dynamics（声学与动力学研究所）；Eidgenössische Technische Hochschule Zürich（苏黎世联邦理工学院）
A Rapid Physics-Informed Machine Learning Framework Based on Extreme Learning Machine for Inverse Stefan Problems
School of Qilu Transportation, Shandong University（山东大学齐鲁交通学院）；School of Mechanical Engineering, Qilu University of Technology (Shandong Academy of Sciences)（齐鲁工业大学（山东省科学院）机械工程学院）
Interpretable Physics-Informed Load Forecasting for U.S. Grid Resilience: SHAP-Guided Ensemble Validation in Hybrid Deep Learning Under Extreme Weather
1Dept. of Computer Science, Midwestern State University, Dallas, TX, USA 2O\&M Analytics, AES Clean Energy, The AES Corporation, Louisville, CO, USA 3Dept. of Information；Communication Engineering, Pabna University of Science
Physics-informed extreme learning machine for Terzaghi consolidation problems and interpretation of coefficient of consolidation based on CPTu data
Shandong University（山东大学）；China University of Mining and Technology（中国矿业大学）；Qilu University of Technology (Shandong Academy of Sciences)（齐鲁工业大学（山东省科学院））；University of Warwick（华威大学）；University of Leeds（利兹大学）
Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs
Oden Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；McKelvey School of Engineering, Washington University in St. Louis（圣路易斯华盛顿大学麦凯维工程学院）；Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校机械工程系）
Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems
Ansys Inc.（安西斯公司）；North Carolina State University（北卡罗来纳州立大学）；Sandia National Laboratories（桑迪亚国家实验室）；Arizona State University（亚利桑那州立大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries
Graduate School of Science and Technology, Keio University（Keio大学科学技术研究生院）；Department of Mechanical Engineering, Keio University（Keio大学机械工程系）；Access e.V.
Physics-Informed Extreme Learning Machine (PIELM) for Tunnelling-Induced Soil-Pile Interactions
School of Qilu Transportation, Shandong University（山东大学齐鲁交通学院）；School of Mechanical Engineering, Shandong Key Laboratory of CNC Machine Tool Functional Components, Qilu University of Technology（齐鲁工业大学机械工程学院，山东省数控机床功能部件重点实验室）
Physics-Informed Extreme Learning Machine (PIELM): Opportunities and Challenges
Shandong University（山东大学）；Qilu University of Technology (Shandong Academy of Sciences)（齐鲁工业大学（山东省科学院））；University of Naples Federico II（那不勒斯费德里科二世大学）；University of Leeds（利兹大学）；Shandong Institute of Mechanical Design and Research（山东省机械设计研究院）
General Fourier Feature Physics-Informed Extreme Learning Machine (GFF-PIELM) for High-Frequency PDEs
Qilu University of Technology (Shandong Academy of Sciences)（齐鲁工业大学（山东省科学院））；Shandong Institute of Mechanical Design and Research（山东省机械设计研究院）；Yale University（耶鲁大学）；Shandong University（山东大学）；University of Leeds（利兹大学）
Physics-informed reservoir characterization from bulk and extreme pressure events with a differentiable simulator
Earth and Environmental Sciences Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室地球与环境科学部）；Courant Institute School of Mathematics, Computing and Data Science, New York University（纽约大学柯朗研究所数学、计算与数据科学学院）
Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；College of Science, Shijiazhuang University（石家庄学院理学院）；Department of Engineering Mechanics, Northwestern Polytechnical University（西北工业大学工程力学系）；Department of Aeronautical Engineering, Northwestern Polytechnical University（西北工业大学航空工程系）；MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University（西北工业大学工信部复杂系统动力学与控制重点实验室）
