The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the Automatic Differentiation (AD). This study proposes the hybrid finite difference with the physics-informed neural network (HFD-PINN) to fully use the domain knowledge. The main idea is to use the finite difference method (FDM) locally instead of AD in the framework of PINN. In particular, we use AD at complex boundaries and the FDM in other domains. The hybrid learning model shows promising results in experiments. To use the FDM locally in the complex boundary domain and avoid the generation of background mesh, we propose the HFD-PINN-sdf method, which locally uses the finite difference scheme at random points. In addition, the signed distance function is used to avoid the difference scheme from crossing the domain boundary. In this paper, we demonstrate the performance of our proposed methods and compare the results with the different number of collocation points for the Poisson equation, Burgers equation. We also chose several different finite difference schemes, including the compact finite difference method (CDM) and crank-nicolson method (CNM), to verify the robustness of HFD-PINN. We take the heat conduction problem and the heat transfer problem on the irregular domain as examples to demonstrate the efficacy of our framework. In summary, HFD-PINN, especially HFD-PINN-sdf, are more instructive and efficient, significantly when solving PDEs in complex geometries.
Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs
School of Petroleum Engineering, Yangtze University（扬州大学石油工程学院）；School of Computer Science, Yangtze University（扬州大学计算机科学学院）；State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University)（扬州大学低碳催化与二氧化碳利用国家重点实验室）；Western Research Institute, Yangtze University（扬州大学西部研究院）
PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations
Department of Statistics and Data Science, Yale University（统计与数据科学系，耶鲁大学）；Department of Applied Mathematics and Statistics, Johns Hopkins University（应用数学与统计学系，约翰霍普金斯大学）；Department of Applied Computational Mathematics and Statistics, University of Notre Dame（应用计算数学与统计学系，圣母大学）；Department of Mathematics, University of California Los Angeles（数学系，加州大学洛杉矶分校）；Department of Chemical and Environmental Engineering, Yale University（化学与环境工程系，耶鲁大学）
Parameterized Physics-informed Neural Networks for Parameterized PDEs
Yonsei University（延世大学）；Arizona State University（亚利桑那州立大学）；LG CNS；Texas A&M University-Commerce（德州农工大学Commerce分校）；Oregon State University（俄勒冈州立大学）；KAIST（韩国科学技术院）
Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion
College of Engineering, Peking University（北京大学工学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEs
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China（上海交通大学航空宇航科学与技术学院）；School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China（浙江大学航空宇航科学与技术学院）；Key Laboratory for Satellite Digitalization Technology, Innovation Academy for Microsatellites of CAS, Shanghai, China（中国科学院卫星数字化技术重点实验室）
Reconstructing Relativistic Magnetohydrodynamics with Physics-Informed Neural Networks
Harvard College（哈佛大学）；Harvard John A. Paulson School of Engineering and Applied Sciences（哈佛约翰·A·保尔森工程与应用科学学院）；Institute for Advanced Study（高等研究院）；OpenAI
Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs
Brown University（布朗大学）；NVIDIA（英伟达）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Julia Computing（朱莉娅计算公司）；Pacific Northwest National Lab（太平洋西北国家实验室）；Massachusetts Institute of Technology（麻省理工学院）
正问题和反问题的不确定性量化是物理学和生物医学学科的核心挑战。我们解决这一挑战的问题，模拟地下水流在汉福德网站的随机计算模型与观测数据相结合，使用物理信息的GAN模型。汉福德站点的地理范围、空间异质性和多个相关长度尺度要求训练计算密集型GAN模型到数千个维度。我们开发了一个层次化的计划，利用域并行，映射鉴别器和发电机到多个GPU，并采用有效的通信方案，以确保训练的稳定性和收敛性。我们开发了一个高度优化的方案，在Summit超级计算机上扩展到27，500个NVIDIA Volta GPU和4584个节点，扩展效率为93.1%，实现了1228 PF/s和1207 PF/s的峰值和持续半精度速率。
Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs
Dept. of Aerospace Eng. Sciences University of Colorado Boulder（航空航天工程科学系，科罗拉多大学博尔德分校）；Center for Systems and Control Delft University of Technology（系统与控制中心，代尔夫特理工大学）
