Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI
Graduate School of Social and Data Sciences（社会科学与数据科学研究生院）；Hitotsubashi University（立命馆大学）；Department of Electrical and Electronic Engineering（电子工程系）；Nihon University（日立大学）
Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data-driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data-driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.
Resolving positive semi-definiteness in physics-informed kernels for scientific machine learning
Institute of Theoretical & Computational Physics, Graz University of Technology（格拉茨工业大学理论与计算物理研究所）；Courant Institute, New York University（纽约大学柯朗数学科学研究所）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；School of Mechanical Engineering, Purdue University（普渡大学机械工程学院）
Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles
Departamento de Ciencia de la Computación, Pontificia Universidad Católica de Chile（计算机科学系，天主教智利大学）；Department of Computer Science, University of Tübingen（计算机科学系，图宾根大学）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（工程与应用科学学院，哈佛大学）
Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；University of North Carolina, Charlotte（北卡罗来纳大学夏洛特分校）；Delft University of Technology（代尔夫特理工大学）；University of Washington, Applied Mathematics（华盛顿大学应用数学系）；Brown University, Applied Mathematics（布朗大学应用数学系）
RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations
Department of Civil and Environmental Engineering, Hong Kong Polytechnic University（香港理工大学土木与环境工程系）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）
Leveraging Influence Functions for Resampling Data in Physics-Informed Neural Networks
Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute（人工智能系，弗劳恩霍夫海因里希·赫兹研究所）；Department of Electrical Engineering and Computer Science, Technische Universität Berlin（电气工程与计算机科学系，柏林技术大学）；BIFOLD -- Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；Centre of eXplainable Artificial Intelligence, Technological University Dublin（可解释人工智能中心，都柏林技术大学）
Physics-Informed Tree Search for High-Dimensional Computational Design
Department of Mechanical and Industrial Engineering, University of Illinois, Chicago（机械与工业工程系，伊利诺伊大学芝加哥分校）；Center for Nanoscale Materials, Argonne National Laboratory（纳米材料中心，阿贡国家实验室）；Mathematics and Computer Science Division, Argonne National Laboratory（数学与计算机科学部，阿贡国家实验室）
Gryphon Center for Artificial Intelligence and Theoretical Sciences（格里芬人工智能与理论科学中心）；Georgia Institute of Technology（佐治亚理工学院）；University of Pennsylvania（宾夕法尼亚大学）
Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators
Brown University（布朗大学）；Huazhong University of Science and Technology（华中科技大学）；Institute of Interdisciplinary Research for Mathematics and Applied Science, School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院数学与应用科学交叉研究院）
Design of Turing Systems with Physics-Informed Neural Networks
Nanyang Technological University（南洋理工大学）；Institute of Bioengineering and Bioimaging, A*STAR（新加坡科技研究局生物工程与生物成像研究所）；Bioinformatics Institute, A*STAR（新加坡科技研究局生物信息学研究所）；Institute of High Performance Computing（高性能计算研究所）；Centre for Frontier AI Research（前沿人工智能研究中心）
Beyond Residuals: Energy based solutions of partial differential equations using scientific machine learning
Bauhaus University Weimar（魏玛包豪斯大学）；Institute of Structural Mechanics, Bauhaus University Weimar（魏玛包豪斯大学结构力学研究所）；Fudan University（复旦大学）；Institute of Computational Mechanics and Artificial Intelligence, Fudan University（复旦大学计算力学与人工智能研究所）；Tsinghua University（清华大学）；Department of Engineering Mechanics, Tsinghua University（清华大学工程力学系）
Neuroevolution of Physics-Informed Neural Nets: Benchmark Problems and Comparative Results
Center for Frontier AI Research（前沿人工智能研究中心）；Agency for Science, Technology, and Research（新加坡科技研究局）；Institute for High Performance Computing（高性能计算研究所）；Nanyang Technological University（南洋理工大学）；Singapore Institute of Manufacturing Technology（新加坡制造技术研究所）
Bayesian Reasoning for Physics Informed Neural Networks
Institute for Theoretical Physics, University of Wroc aw（沃拉夫大学理论物理研究所）；Institute of Low Temperature and Structure Research（低温与结构研究所）；Polish Academy of Sciences（波兰科学院）
