EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations
Institute for Software and Systems Engineering, Clausthal University of Technology（软件与系统工程研究所，克劳斯塔尔大学）；Institute of Applied Mechanics, Clausthal University of Technology（应用力学研究所，克劳斯塔尔大学）
Hamidreza Eivazi, Jendrik-Alexander Tröger, Stefan Wittek, Stefan Hartmann, Andreas Rausch
Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, such as uncertainty quantification, remeshing applications, and topology optimization. This limitation has motivated the development of data-driven surrogate models, where microscale computations are substituted by black-box mappings between macroscale quantities. While these approaches offer significant speedups, they typically struggle to incorporate microscale physical constraints, such as the balance of linear momentum. In this contribution, we propose the Equilibrium Neural Operator (EquiNO), a physics-informed PDE surrogate in which equilibrium is hard-enforced by construction. EquiNO achieves this by projecting the solution onto a set of divergence-free basis functions obtained via proper orthogonal decomposition (POD), thereby ensuring satisfaction of equilibrium without relying on penalty terms or multi-objective loss functions. We compare EquiNO with variational physics-informed neural and operator networks that enforce physical constraints only weakly through the loss function, as well as with purely data-driven operator-learning baselines. Our framework, applicable to multiscale FE$^{\,2}$ computations, introduces a finite element-operator learning (FE-OL) approach that integrates the finite element (FE) method with operator learning (OL). We apply the proposed methodology to quasi-static problems in solid mechanics and demonstrate that FE-OL yields accurate solutions even when trained on restricted datasets. The results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers a robust and physically consistent alternative to existing data-driven surrogate models.
Robust Parameter and State Estimation in Multiscale Neuronal Systems Using Physics-Informed Neural Networks
Department of Mathematics, University of Iowa（数学系，爱荷华大学）；Department of Mathematics, Volen National Center for Complex Systems, Brandeis University（数学系，布兰迪大学复杂系统国家中心）
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Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, 560012, India（计算与数据科学系，印度科学研究院，班加罗尔，560012，印度）；Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK（数学系，曼彻斯特大学，牛津路，曼彻斯特 M13 9PL，英国）
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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）
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High Technology High School（高科技高中）；Institute for Gravitation and the Cosmos（引力与宇宙研究所）；The Pennsylvania State University（宾夕法尼亚州立大学）；Department of Physics（物理系）；Princeton University（普林斯顿大学）
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Department Modeling and Artificial Intelligence（建模与人工智能系）；Fraunhofer Institute for Integrated Systems and Device Technology IISB（弗劳恩霍夫集成系统与器件技术研究所IISB）
School of Aeronautics and Astronautics, Shanghai Jiao Tong University（上海交通大学航空航天学院）；Department of Aeronautics, Imperial College London（伦敦帝国理工学院航空系）；School of Energy and Power Engineering, Xi’an Jiao Tong University（西安交通大学能源与动力工程学院）
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Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona（巴塞罗那大学宇宙科学研究所）；Departament de Física Quàntica i Astrofísica, Universitat de Barcelona（巴塞罗那大学量子物理与天体物理学系）；ICREA；School of Mathematics, University of Minnesota（明尼苏达大学数学学院）
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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交叉科学与应用数学研究所）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
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University of Utah（犹他大学）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；University of Utah Health（犹他大学健康中心）；Brigham and Women’s Hospital（布莱根妇女医院）；Harvard Medical School（哈佛医学院）；Veterans Affairs Salt Lake City Health Care System（退伍军人事务部盐湖城医疗系统）；Beth Israel Deaconess Medical Center（贝斯以色列女执事医疗中心）；University of Pittsburgh（匹兹堡大学）
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Department of Mechanical Engineering, The Hong Kong Polytechnic University（香港理工大学机械工程系）；Department of Civil and Systems Engineering, Johns Hopkins University（约翰霍普金斯大学土木与系统工程系）；Innovation Research Institute（创新研究院）
Uni-Flow: a unified autoregressive-diffusion model for complex multiscale flows
Centre for Computational Science, University College London, London, UK；Department of Earth Science；Engineering, Imperial College London, London, UK；Department of Chemical Engineering, University College London, London, UK；Department of Physics, Eindhoven University of Technology, Eindhoven, Netherlands；School of Civil \& Environmental Engineering, Queensland University of Technology, Brisbane, Australia；Australian Centre for Water；Environmental Biotechnology, The University of Queensland, Brisbane, Australia；Institute for Mechanics, Computational Mechanics Group, Technical University of Darmstadt, Germany；Centre for Advanced Research Computing, University College London, London, UK
Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology
Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Cardiovascular Institute, Stanford University（斯坦福大学心血管研究所）；Pediatric Cardiology, Stanford University（斯坦福大学儿科心脏病研究所）；Department of Bioengineering, Stanford University（斯坦福大学生物工程系）
A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations
School of Mathematical Sciences, University of Electronic Science（数学科学学院，电子科技大学）；Institute of Applied Physics（应用物理研究所）；Institute of Fundamental and Frontier Sciences, University of Electronic Science（基础与前沿科学研究所，电子科技大学）
Neural operator accelerated atomistic to continuum concurrent multiscale simulations of viscoelasticity
National Center for Computational Sciences, Oak Ridge National Laboratory（橡树岭国家实验室计算科学国家中心）；Department of Engineering, University of Cambridge（剑桥大学工程系）
Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；School of Mathematics, Jilin University（吉林大学数学学院）；Key Laboratory of Digital Technology in Medical Diagnostics of Zhejiang（浙江省数字医疗诊断技术重点实验室）；School of Physics, Nankai University（南开大学物理学院）
