Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by industrial simulations that involve complex geometries. In the spirit of advancing neural PDE solvers to real industrial applications, we present Transolver++, a highly parallel and efficient neural solver that can accurately solve PDEs on million-scale geometries. Building upon previous advancements in solving PDEs by learning physical states via Transolver, Transolver++ is further equipped with an extremely optimized parallelism framework and a local adaptive mechanism to efficiently capture eidetic physical states from massive mesh points, successfully tackling the thorny challenges in computation and physics learning when scaling up input mesh size. Transolver++ increases the single-GPU input capacity to million-scale points for the first time and is capable of continuously scaling input size in linear complexity by increasing GPUs. Experimentally, Transolver++ yields 13% relative promotion across six standard PDE benchmarks and achieves over 20% performance gain in million-scale high-fidelity industrial simulations, whose sizes are 100$\times$ larger than previous benchmarks, covering car and 3D aircraft designs.
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.
Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Beihang University（北京航空航天大学）；The Hong Kong University of Science and Technology（香港科技大学）
BENO: Boundary-embedded Neural Operators for Elliptic PDEs
National Engineering Research Center for Software Engineering, Peking University（北京大学软件工程国家工程研究中心）；Department of Engineering, Westlake University（西湖大学工学院）；Department of Astronautics and Aeronautics, Stanford University（斯坦福大学航空航天系）
HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries
School of Mathematics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学数学学院）；Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT（工业和信息化部南京航空航天大学数学建模与高性能计算重点实验室（NUAA））；School of Mathematics and Information Sciences, Yantai University（烟台大学数学与信息科学学院）
NPSolver: Neural Poisson Solver with Iterative Physics Supervision
Gaoling School of Artificial Intelligence（高岭人工智能学院）；Renmin University of China（中国人民大学）；School of Mechanics and Engineering Science（力学与工程科学学院）；Peking University（北京大学）；AI for Science Institute（AI for Science研究院）；University of Chinese Academy of Sciences（中国科学院大学）
RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network
College of Aerospace Science and Engineering, National University of Defense Technology（国防科技大学航天科学与工程学院）；National Innovation Institute of Defense Technology, Chinese Academy of Military Science（中国人民解放军军事科学院国防科技创新研究院）
Efficient High-Accuracy PDEs Solver with the Linear Attention Neural Operator
School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences（中国科学院大学先进交叉学科学院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学科学国家重点实验室）；School of Mathematics and Information Science, Zhongyuan University of Technology（中原工学院数学与信息科学学院）
A hybrid numerical methodology coupling Reduced Order Modeling and Graph Neural Networks for non-parametric geometries: applications to structural dynamics problems
Université Paris-Saclay（巴黎萨克雷大学）；CentraleSupélec（中央理工-高等电力学院）；ENS Paris-Saclay（巴黎萨克雷高等师范学校）；CNRS（法国国家科学研究中心）；LMPS - Laboratoire de Mécanique Paris-Saclay（巴黎萨克雷力学实验室）；Safran Tech（赛峰技术公司）
DAMTP University of Cambridge（剑桥大学数学部）；Institute for Foundations of Data Science（数据科学基础研究所）；Yale University（耶鲁大学）；Yau Mathematical Sciences Center Tsinghua University（清华大学姚贝泰数学科学中心）
CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries
Tsinghua University（清华大学）；Massachusetts Institute of Technology（麻省理工学院）；KU Leuven（荷语鲁汶大学）
Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptation
Tsinghua University（清华大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）；CNPC Engineering Technology RD Company Limited（中国石油集团工程技术研究院有限公司）；Queensland University of Technology（昆士兰科技大学）
