Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all potential scenarios through data-driven methodologies alone. Moreover, there are legitimate concerns regarding the generalization and reliability of such approaches, as they often overlook inherent physical constraints. In response to these challenges, this study introduces a novel machine-learning architecture that is highly generalizable and adheres to conservation laws and physical symmetries, thereby ensuring greater reliability. The foundation of this architecture is graph neural networks (GNNs), which are adept at accommodating a variety of shapes and forms. Additionally, we explore the parallels between GNNs and traditional numerical solvers, facilitating a seamless integration of conservative principles and symmetries into machine learning models. Our findings from experiments demonstrate that the model's inclusion of physical laws significantly enhances its generalizability, i.e., no significant accuracy degradation for unseen spatial domains while other models degrade. The code is available at https://github.com/yellowshippo/fluxgnn-icml2024.
Adaptive Correction for Ensuring Conservation Laws in Neural Operators
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK（应用数学与理论物理系，剑桥大学）；Department of Mathematical Sciences, University of Bath, UK（数学科学系，巴斯大学）
How simple can you go? An off-the-shelf transformer approach to molecular dynamics
BIFOLD – Berlin Institute for the Foundations of Learning and Data Machine Learning Group, Technische Universität Berlin（柏林学习与数据基础研究所）；Technische Universität Berlin（柏林技术大学）；Google DeepMind（谷歌DeepMind）；Department of Artificial Intelligence, Korea University（人工智能系，韩国大学）；Max-Planck Institute for Informatics（马克斯·普朗克信息研究所）
Learning Under Laws: A Constraint-Projected Neural PDE Solver that Eliminates Hallucinations
Department of Physics, The Catholic University of America（天主教美国大学物理系）；Center for Research and Exploration in Space Science and Technology, NASA, Goddard Space Flight Center（空间科学与技术研究中心）
GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws
Modelling Engineering Risk and Complexity, Scuola Superiore Meridionale（建模工程风险与复杂性，南方高级学校）；Dept. of Electrical Engineering and Information Technology, University of Naples Federico II（电气工程与信息技术系，那不勒斯费德里科二世大学）；Institute of Science and Technology for Energy and Sustainable Mobility, Consiglio Nazionale delle Ricerche（能源与可持续交通科学与技术研究所，意大利国家研究委员会）
Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；MIT BCS（麻省理工学院脑与认知科学系）；Center for Brains, Minds and Machines（脑、心智与机器中心）；Tsinghua University（清华大学）；Shanghai Qi Zhi Institute（上海期智研究院）；UMass Amherst（马萨诸塞大学阿默斯特分校）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
Kinetic-based regularization: Learning spatial derivatives and PDE applications
Engineering Mechanics Unit（工程力学单元）；Jawaharlal Nehru Centre for Advanced Scientific Research（贾瓦哈拉尔·奈尔中心先进科学研究所）；Fondazione Istituto Italiano di Tecnologia（意大利技术研究院）
