One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
Department of Mathematics（数学系）；Hunan University（湖南大学）；Academy for Multidisciplinary Studies（跨学科研究学院）；Beijing National Center for Applied Mathematics（北京应用数学中心）；Capital Normal University（首都师范大学）
Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations
Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing（数学与计算方法协调国家科学计算实验室）；Systems and Computational Engineering Program, COPPE/UFRJ, Federal University of Rio de Janeiro（系统与计算工程系，COPPE/UFRJ，里约热内卢联邦大学）
Meta-Auto-Decoder for Solving Parametric Partial Differential Equations
Peking University（北京大学）；Huawei Technologies Co. Ltd（华为技术有限公司）；Beijing International Center for Mathematical Research, Peking University（北京国际数学研究中心（北京大学））；Center for Machine Learning Research, Peking University（北京大学机器学习研究中心）
Cluster Attention Neural Operators for Solving Parametric Partial Differential Equations
University of Chinese Academy of Sciences（中国科学院大学）；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）；International School for Advanced Studies (SISSA)（国际高等研究学院（SISSA））；Zhongyuan University of Technology（中原工学院）
Adaptation of uncertainty-penalized Bayesian information criterion for parametric partial differential equation discovery
Graduate School of Information Science and Technology, Osaka University（大阪大学信息科学与技术研究生院）；SANKEN (The Institute of Scientific and Industrial Research), Osaka University（大阪大学产业科学研究所（SANKEN））
Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations
Carnegie Mellon University（卡内基梅隆大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Ningbo Institute of Digital Twin, Eastern Institute of Technology（东方理工大学宁波数字孪生研究院）；Hefei University of Technology（合肥工业大学）；RealAI（瑞莱智慧）；Tsinghua University（清华大学）
Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations
University of Stuttgart（斯图加特大学）；Stuttgart Center for Simulation Science (SimTech)（斯图加特模拟科学中心）；International Max Planck Research School for Intelligent Systems (IMPRS-IS)（国际马克斯·普朗克智能系统研究院）
