Interest is rising in Physics-Informed Neural Networks (PINNs) as a mesh-free alternative to traditional numerical solvers for partial differential equations (PDEs). However, PINNs often struggle to learn high-frequency and multi-scale target solutions. To tackle this problem, we first study a strong Boundary Condition (BC) version of PINNs for Dirichlet BCs and observe a consistent decline in relative error compared to the standard PINNs. We then perform a theoretical analysis based on the Fourier transform and convolution theorem. We find that strong BC PINNs can better learn the amplitudes of high-frequency components of the target solutions. However, constructing the architecture for strong BC PINNs is difficult for many BCs and domain geometries. Enlightened by our theoretical analysis, we propose Fourier PINNs -- a simple, general, yet powerful method that augments PINNs with pre-specified, dense Fourier bases. Our proposed architecture likewise learns high-frequency components better but places no restrictions on the particular BCs or problem domains. We develop an adaptive learning and basis selection algorithm via alternating neural net basis optimization, Fourier and neural net basis coefficient estimation, and coefficient truncation. This scheme can flexibly identify the significant frequencies while weakening the nominal frequencies to better capture the target solution's power spectrum. We show the advantage of our approach through a set of systematic experiments.
Physics-Informed Neural Networks with Fourier Features and Attention-Driven Decoding
High Technology High School（高科技高中）；Institute for Gravitation and the Cosmos（引力与宇宙研究所）；The Pennsylvania State University（宾夕法尼亚州立大学）；Department of Physics（物理系）；Princeton University（普林斯顿大学）
Multiscale lubrication simulation based on fourier feature networks with trainable frequency
School of Mechanical Engineering, Shanghai Jiao Tong University（上海交通大学机械工程学院）；Shanghai marine diesel research institute（上海船用柴油机研究所）；National key laboratory of marine engine science and technology（船用发动机科学技术国家重点实验室）
针对传统 PINN 因谱偏差难以处理粗糙表面高频信号的问题，提出采用可训练 Fourier 特征网络的多尺度润滑神经网络，在多种形貌上达到与 FEM 高度一致，并优于固定频率网络。
Fourier Feature Pyramids for Physics-Informed Neural Networks
Department of Computing and Mathematical Sciences, California Institute of Technology（计算与数学科学部，加州理工学院）；Cahill Center for Astronomy and Astrophysics, California Institute of Technology（天文与天体物理中心，加州理工学院）；Google DeepMind（谷歌DeepMind）
Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddings
Department Modeling and Artificial Intelligence（建模与人工智能系）；Fraunhofer Institute for Integrated Systems and Device Technology IISB（弗劳恩霍夫集成系统与器件技术研究所IISB）
Operator-informed initialization for Fourier features physics-informed neural networks
National Center for Artificial Intelligence（国家人工智能中心）；University of Pennsylvania（宾夕法尼亚大学）；Pontificia Universidad Católica de Chile（智利天主教大学）；Millennium Institute for Intelligent Healthcare Engineering, iHEALTH（千禧智能医疗工程研究所）
Multi-scale Dynamic Wake Modeling and Prediction of Floating Offshore Wind Turbines via Physics-Informed Neural Networks and Fourier Neural Operators
College of Renewable Energy, Hohai University, Changzhou, 213200, China（能源学院，河海大学，常州，213200，中国）；College of Water Conservancy and Hydropower Engineering（水利水电工程学院）；National Technology Innovation Center for Wind Power, Hohai University, Changzhou, 213200, China（风能技术创新中心，河海大学，常州，213200，中国）
A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity
The University of New South Wales（新南威尔士大学）；Massachusetts Institute of Technology（麻省理工学院）；University of British Columbia（不列颠哥伦比亚大学）；Sharif University of Technology（谢里夫理工大学）
Physics-Informed Neural Network with Squeeze-Excitation-like Attention
Key Laboratory of Quark and Lepton Physics (MOE) & Institute of Particle Physics, Central China Normal University（华中师范大学夸克与轻子物理教育部重点实验室及粒子物理研究所）；Artificial Intelligence and Computational Physics Research Center, Central China Normal University（华中师范大学人工智能与计算物理研究中心）；Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) & Institute of Modern Physics, Fudan University（复旦大学核物理与离子束应用教育部重点实验室及现代物理研究所）；Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University（国家自然科学基金委员会-复旦大学上海理论核物理研究中心）；Northwest Institute of Nuclear Technology（西北核技术研究所）
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）
B-PL-PINN: Stabilizing PINN Training with Bayesian Pseudo Labeling
Institute of Human-Centred Computing, Graz University of Technology, Graz, Austria（以人为中心计算研究所，格拉茨技术大学，格拉茨，奥地利）；Know Center Research GmbH, Graz, Austria（格拉茨know研究中心有限责任公司，格拉茨，奥地利）；Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz, Austria（信号处理与语音通信实验室，格拉茨技术大学，格拉茨，奥地利）
Enhancing Physics-Informed Neural Networks Through Feature Engineering
Department of Mathematics, Stanford University（数学系，斯坦福大学）；Department of Management Science & Engineering, Stanford University（管理科学与工程系，斯坦福大学）；ICME, Stanford University（ICME，斯坦福大学）
Hybrid Quantum-Classical PINNs for Scientific Computing: A Multi-GPU Open-Source Framework
Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Haifa, Israel（技术学院机械工程系，以色列技术学院，海法，以色列）；Andrew and Erna Viterbi Faculty of Electrical & Computer Engineering, Technion Israel Institute of Technology, Haifa, Israel（安德鲁和伊尔纳·维特比电气与计算机工程学院，技术学院，海法，以色列）；Helen Diller Quantum Center, Technion Israel Institute of Technology, Haifa, Israel（海伦·迪尔量子中心，技术学院，海法，以色列）；Faculty of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA（电气与计算机工程学院，康奈尔大学，纽约州伊萨克，美国）
Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds
Department of Bio-Functions and Systems Science（生物功能与系统科学系）；Graduate School of Bio-Applications and Systems Engineering（生物应用与系统工程研究生学校）；Department of Mathematics（数学系）；Faculty of Mathematics and Natural Sciences（数学与自然科学学院）；Universitas Padjadjaran（帕迪贾兰大学）；Department of Informatics（信息系）；Faculty of Science and Informatics（科学与信息学院）；Jenderal of Achmad Yani University（阿赫马德·亚尼大学）
