Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain "stiff" problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method.
Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones
Lishui University（丽水学院）；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）；University of Chinese Academy of Sciences（中国科学院大学）
Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations
College of Engineering and Computer Science, VinUniversity（越南 Vin 大学工程与计算机科学学院）；Center for AI Research, VinUniversity（越南 Vin 大学人工智能研究中心）；Department of Computer Science and Engineering, University of North Texas（北德克萨斯大学计算机科学与工程系）；Department of Mathematics and Informatics, Hanoi University of Science and Technology（河内科学技术大学数学与信息学系）；Group of Materials and Structures, School of Mechanical Engineering, Hanoi University of Science and Technology（河内科学技术大学机械工程学院材料与结构组）
Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings
School of Information and Software Engineering, University of Electronic Science and Technology of China（信息与软件工程学院，电子科学与技术大学）；Computer Science and Engineering, University of Electronic Science and Technology of China（计算机科学与工程，电子科学与技术大学）；Institute of Software Chinese Academy of Sciences, Beijing, China（软件研究所，中国科学院北京）；School of Robotics and Advanced Manufacture, Harbin Institute of Technology, Shenzhen, China（机器人与先进制造学院，哈尔滨工业大学深圳）
LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems
Institute for Advanced Research, Great Bay University（先进研究院，大湾大学）；College of Computer Science and Technology, Dongguan University of Technology（计算机科学与技术学院，东莞理工大学）；Department of Mathematics, Northwestern University（数学系，西北大学）
An efficient wavelet-based physics-informed neural network for multiscale problems
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）
Learnable Viscosity Modulation in Physics-Informed Neural Networks for Incompressible Flow Reconstruction
Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子学与生物光子学重点实验室，长春理工大学物理学院）
Learning characteristic parameters and dynamics of centrifugal pumps under multiphase flow using physics-informed neural networks
Faculdade de Engenharia Mecânica (FEM), Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学机械工程学院）；Centro de Estudos de Energia e Petróleo (CEPETRO), Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学能源与石油研究中心）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；School of Engineering, Brown University（布朗大学工程学院）
Learning Fluid-Structure Interaction with Physics-Informed Machine Learning and Immersed Boundary Methods
Computational Science and Engineering Department, Informatics Institute Istanbul Technical University（信息学院伊斯坦布尔技术大学计算科学与工程系）；Scientific Computing, Rutherford Appleton Laboratory Science and Technology Facilities Council (STFC)（科学与技术设施理事会（STFC）科学计算部）
PhysAttNet: Enhancing Predictive Performance in Industrial and Astrophysical Time Series via Physics-Informed Attention
Faculty of Physics and Astronomy, Ruhr University Bochum（鲁尔大学波鸿分校物理与天文学学院）；Virtual Machining Group, TU Dortmund（多特蒙德工业大学虚拟加工组）；Faculty of Physics（物理学院）
Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems
Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子与生物光子重点实验室，物理学院，长春理工大学）
Recovering Sharp Conductivity Features in the Finite-Data Calderón Problem with Physics-Informed Neural Networks
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）
Accelerating physics-informed neural networks for full waveform inversion using a hybrid quantum-classical finite-basis architecture
Department of Geophysics, Colorado School of Mines（地质学系，科罗拉多矿业学院）；Department of Energy Science and Engineering, Stanford University（能源科学与工程系，斯坦福大学）；Department of Petroleum Engineering, Colorado School of Mines（石油工程系，科罗拉多矿业学院）
Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefields
Basque Center for Applied Mathematics（巴斯克应用数学中心）；University of the Basque Country（巴斯克大学）；Ikerbasque, Basque Foundation for Science（伊克巴斯克基金会（巴斯克科学基金会））；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems
Department of Computer Science, Norwegian University of Science and Technology（挪威科学与技术大学计算机科学系）；Department of Mathematics and Cybernetics, SINTEF Digital（SINTEF数字研究院数学与控制论系）；Aneo AI Research（Aneo人工智能研究）
Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations
School of Automation, Central South University（中南大学自动化学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；China Institute of Water Resources and Hydropower Research（中国水利水电科学研究院）
Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems
Department of Computer Technologies, Antalya Bilim University（安塔利亚比林大学计算机技术系）；Institute of Applied Mathematics, Middle East Technical University（中东技术大学应用数学研究所）；Department of Mathematics, Indian Institute of Technology Guwahati（印度理工学院古瓦哈提数学系）
Discontinuity-aware physics-informed neural network for phase-field method in three-phase flow with phase change
School of Interdisciplinary Science, Beijing Institute of Technology（北京理工大学交叉科学学院）；School of Mechatronical Engineering, Beijing Institute of Technology（北京理工大学机电学院）；IFP Energies nouvelles（法国石油研究院）；School of Aerospace Engineering, Beijing Institute of Technology（北京理工大学航天航空学院）；Beijing Institute of Technology (Zhuhai)（北京理工大学（珠海））
Physics informed wavelet Fourier representation for multiscale fluid dynamics
College of Shipbuilding Engineering, Harbin Engineering University（哈尔滨工程大学船舶工程学院）；Qingdao Innovation and Development Base, Harbin Engineering University（哈尔滨工程大学青岛创新发展基地）；National Key Laboratory of Hydrodynamics, Harbin Engineering University（哈尔滨工程大学水动力学国家重点实验室）；Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology（南方科技大学力学与航空航天工程系）；Centre for Interdisciplinary Research in Fluids, Institute of Mechanics, Chinese Academy of Sciences（中国科学院力学研究所流体交叉研究中心）；Nanhai Institute of Harbin Engineering University, Harbin Engineering University（哈尔滨工程大学南海研究院）；Department of Modern Mechanics, University of Science and Technology of China（中国科学技术大学近代力学系）
A Gaussian-based spatially weighted loss formulation for Physics-Informed Neural Networks with Shocks
University Salhi Ahmed（Salhi Ahmed大学）；University Mohamed Khider Biskra（Mohamed Khider Biskra大学）；Universidad Politécnica de Madrid（马德里理工大学）；University Center of El Bayadh Nour El Bachir（El Bayadh Nour El Bachir大学中心）
School of Interdisciplinary Science, Beijing Institute of Technology（北京理工大学交叉科学学院）；School of Mechatronical Engineering, Beijing Institute of Technology（北京理工大学机电工程学院）；Beijing Institute of Technology (Zhuhai)（北京理工大学（珠海））
Order of Magnitude Analysis and Data-Based Physics-Informed Symbolic Regression for Turbulent Pipe Flow
Fluid Dynamics and Spray Laboratory, Mechanical Engineering Department, Özyegin University（Özyegin大学机械工程系流体动力学与喷雾实验室）；Cloud Computing Research Laboratory, Computer Science Department, Özyegin University（Özyegin大学计算机系云计算研究实验室）；Institute for Thermal Energy Technology and Safety, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院热能与安全技术研究所）
PT-PINNs: A Parametric Engineering Turbulence Solver based on Physics-Informed Neural Networks
Zhejiang University（浙江大学）；Shanghai Institute for Advanced Study of Zhejiang University（浙江大学上海高等研究院）；State Key Laboratory of Clean Energy Utilization（清洁能源利用国家重点实验室）
