A shallow physics-informed neural network for solving partial differential equations on surfaces
National Central University（中央大学）；National Yang Ming Chiao Tung University（国立阳明交通大学）；National Center for Theoretical Sciences（理论科学研究中心）；National Taiwan University（台湾大学）
Wei-Fan Hu, Yi-Jun Shih, Te-Sheng Lin, Ming-Chih Lai
In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We perform a series of performance study for the present methodology by solving Laplace-Beltrami equation and surface diffusion equation on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. Then we extend the present methodology to solve the advection-diffusion equation on an evolving surface with given velocity. To track the surface, we additionally introduce a prescribed hidden layer to enforce the topological structure of the surface and use the network to learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate the surfactant transport on the droplet surface under shear flow and obtain some physically plausible results.
A physics informed neural network approach to simulating ice dynamics governed by the shallow ice approximation
Department of Mathematics and the Institute for Scientific Computing and Applied Mathematics, Indiana University（数学系和科学计算与应用数学研究所，印第安纳大学）；Department of Mathematics and Cognitive Science Program, Indiana University（数学系和认知科学项目，印第安纳大学）
Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm
organization= Department of Mathematics, Institut Teknologi Sepuluh Nopember , city= Surabaya , postcode= 60111 , country= Indonesia；organization= Department of Mathematics, Khalifa University of Science \& Technology , city= Abu Dhabi , postcode= PO Box 127788 , country= United Arab Emirates；organization= Department of Mathematics, Faculty of Mathematics
Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
Tsinghua University（清华大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；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）；Leibniz University Hannover（汉诺威莱布尼茨大学）
An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator
The University of Arizona（亚利桑那大学）；Massachusetts Institute of Technology（麻省理工学院）
CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations
Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）；Beijing Normal University（北京师范大学）；Guangzhou Nanfang College（广州南方学院）；Louisiana State University（路易斯安那州立大学）；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）
Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance
Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA, USA（宾夕法尼亚州立大学土木与环境工程系，大学公园，PA，美国）；Institute for Computational and Data Sciences, Pennsylvania State University, University Park, PA, USA（宾夕法尼亚州立大学计算与数据科学研究所，大学公园，PA，美国）
A Theory-guided Weighted $L^2$ Loss for solving the BGK model via Physics-informed neural networks
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Center for Mathematical Machine Learning and its Applications (CM2LA), Department of Mathematics, Pohang University of Science and Technology（浦项科技大学数学系数学机器学习及其应用中心(CM2LA)）；Department of Mathematics, Gyeongsang National University（庆尚国立大学数学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
Quantum-Classical Physics-Informed Neural Networks for Solving Reservoir Seepage Equations
School of Petroleum Engineering, Yangtze University（长江大学石油工程学院）；College of Future Technology, Yangtze University（长江大学未来技术学院）；State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University)（长江大学低碳催化与二氧化碳利用国家重点实验室）；King Abdullah University of Technology, Physical Science and Engineering Division（卡布斯大学物理科学与工程学院）；Western Research Institute, Yangtze University（长江大学西部研究院）
Solving a class of stochastic optimal control problems by physics-informed neural networks
School of Mathematics and Statistics（数学与统计学学院）；MOE Key Laboratory for Complexity Science in Aerospace（复杂科学在航空航天中的关键实验室）；State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics（流体动力与机械系统国家重点实验室，力学系）
Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations
Department of Computer and Data Sciences, Shahid Beheshti University（计算机与数据科学系，谢赫·贝赫什提大学）；Department of Cognitive Modeling, Shahid Beheshti University（认知建模系，谢赫·贝赫什提大学）
Solving Euler equations with Multiple Discontinuities via Separation-Transfer Physics-Informed Neural Networks
Graduate School, China Academy of Engineering Physics（中国工程物理研究院研究生院）；China Academy of Engineering Physics（中国工程物理研究院）；Zhejiang Institute of Modern Physics（浙江现代物理研究所）；School of Physics, Zhejiang University（浙江大学物理学院）；Beijing Computational Science Research Center（北京计算科学研究院）
Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems
College of Mathematics and Statistics, Northwest Normal University（西北师范大学数学与统计学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；Department of Information Technology, Uppsala University（乌普萨拉大学信息科技系）
QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs
Computational Science and Engineering Department, Informatics Institute Istanbul Technical University（信息学院理工学院计算科学与工程系）；Scientific Computing, Rutherford Appleton Laboratory Science and Technology Facilities Council (STFC)（科学与技术设施委员会（STFC）科学计算部）
