Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA（应用数学系，布朗大学）；School of Engineering, Brown University, Providence, RI 02912, USA（工程学院，布朗大学）
Siavash Khodakarami, Vivek Oommen, Nazanin Ahmadi Daryakenari, Maxim Beekenkamp, George Em Karniadakis
Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit spectral bias, whereby low-frequency components of the solution are learned significantly faster than high-frequency modes. While spectral bias is often treated as an intrinsic representational limitation of neural architectures, its interaction with optimization dynamics and physics-based loss formulations remains poorly understood. In this work, we provide a systematic investigation of spectral bias in physics-informed and operator learning frameworks, with emphasis on the coupled roles of network architecture, activation functions, loss design, and optimization strategy. We quantify spectral bias through frequency-resolved error metrics, Barron-norm diagnostics, and higher-order statistical moments, enabling a unified analysis across elliptic, hyperbolic, and dispersive PDEs. Through diverse benchmark problems, including the Korteweg-de Vries, wave and steady-state diffusion-reaction equations, turbulent flow reconstruction, and earthquake dynamics, we demonstrate that spectral bias is not simply representational but fundamentally dynamical. In particular, second-order optimization methods substantially alter the spectral learning order, enabling earlier and more accurate recovery of high-frequency modes for all PDE types. For neural operators, we further show that spectral bias is dependent on the neural operator architecture and can also be effectively mitigated through spectral-aware loss formulations without increasing the inference cost.
An explainable operator approximation framework under the guideline of Green's function
State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University（湍流与复杂系统国家重点实验室，工程学院，北京大学）；Ningbo Institute of Digital Twin, Eastern Institute of Technology（宁波数字孪生研究所，东部技术研究所）；Zhejiang Key Laboratory of Industrial Intelligence and Digital Twin, Eastern Institute of Technology（浙江省工业智能与数字孪生重点实验室，东部技术研究所）
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）
Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networks
The University of Tokyo, Japan（东京大学）；Axiom Research Group（Axiom研究组）；Department of Applied Mathematics and Computational Science, NM-AIST, Tanzania（应用数学与计算科学系，NM-AIST，坦桑尼亚）；African Institute for Mathematical Sciences (AIMS), Research and Innovation Centre, Rwanda（非洲数学科学研究所（AIMS），研究与创新中心，卢旺达）
On the training of physics-informed neural operators for solving parametric partial differential equations
College of Civil Engineering, Tongji University（同济大学土木工程学院）；Department of Engineering Science, University of Oxford（牛津大学工程科学系）；Institute for Foundations of Data Science, Yale University（耶鲁大学数据科学基础研究所）
Variational Quantum Physics-Informed Neural Networks for Hydrological PDE-Constrained Learning with Inherent Uncertainty Quantification
Faculty of Computer Science and Multimedia, Lincoln University College, Petaling Jaya, Selangor, Malaysia（计算机科学与多媒体学院，林肯大学学院，Petaling Jaya，Selangor，马来西亚）
Scalable Quantum Error Mitigation with Physically Informed Graph Neural Networks
National Supercomputing Center in Zhengzhou（郑州国家超级计算中心）；School of Computer and Artificial Intelligence（计算机与人工智能学院）；School of Communication and Artificial Intelligence（通信与人工智能学院）；School of Integrated Circuits（集成电路学院）；Nanjing Institute of Technology（南京理工大学）
Scalable Quantum Error Mitigation with Neighbor-Informed Learning
Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong（香港中文大学机械与自动化工程系）；College of Computer and Data Science, Fuzhou University（福州大学计算机与数据科学学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学姚期 Banking 数学科学中心）；Department of Mathematics, Tsinghua University（清华大学数学系）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（燕琦湖北京数学科学与应用研究院）
Biologically Informed Deep Neural Networks for Multi-Omic Integration, Pathway Activity Inference and Risk Stratification in Cancer
King’s College London（伦敦国王学院）；Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR)（科技研究局信息与通信研究所）；Shenzhen University（深圳大学）；University of Manchester（曼彻斯特大学）
Data Bias Mitigation under Coverage Constraints & The Price of Fairness
Khoury College of Computer Sciences, Northeastern University（东北大学库里计算机科学学院）；Cheriton School of Computer Science, University of Waterloo（滑铁卢大学切里顿计算机科学学院）
GraphFire-X: Physics-Informed Graph Attention Networks and Structural Gradient Boosting for Building-Scale Wildfire Preparedness at the Wildland-Urban Interface
Urban Reslience.AI Lab, Zachry Department of Civil and Environmental Engineering, Texas A&M University（Urban Reslience.AI实验室，Zachry土木与环境工程系，德克萨斯A&M大学）；Department of Computer Science and Engineering, Texas A&M University（计算机科学与工程系，德克萨斯A&M大学）
