This paper introduces an efficient tensor-vector product technique for the rapid and accurate approximation of integral operators within physics-informed deep learning frameworks. Our approach leverages neural network architectures to evaluate problem dynamics at specific points, while employing Gaussian quadrature formulas to approximate the integral components, even in the presence of infinite domains or singularities. We demonstrate the applicability of this method to both Fredholm and Volterra integral operators, as well as to optimal control problems involving continuous time. Additionally, we outline how this approach can be extended to approximate fractional derivatives and integrals and propose a fast matrix-vector product algorithm for efficiently computing the fractional Caputo derivative. In the numerical section, we conduct comprehensive experiments on forward and inverse problems. For forward problems, we evaluate the performance of our method on over 50 diverse mathematical problems, including multi-dimensional integral equations, systems of integral equations, partial and fractional integro-differential equations, and various optimal control problems in delay, fractional, multi-dimensional, and nonlinear configurations. For inverse problems, we test our approach on several integral equations and fractional integro-differential problems. Finally, we introduce the pinnies Python package to facilitate the implementation and usability of the proposed method.
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）
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）
A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization
University of the Basque Country (UPV/EHU)（巴斯克国家大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；Ikerbasque, Basque Foundation for Science（Ikerbasque巴斯克科学基金会）；King Abdullah University of Science and Technology (KAUST)（卡迪斯国王科学与技术大学）
School of Data Science（数据科学学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Department of Computer Science（计算机科学系）；Northwestern University（西北大学）；School of Science, Great Bay University（大湾大学科学学院）；Department of Mathematics（数学系）
Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next
University of Naples Federico II（那不勒斯费德里科二世大学）；SISSA – International School for Advanced Studies（国际高等研究学院（SISSA））；University of Colorado Boulder（科罗拉多大学博尔德分校）
opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation
Korea Institute for Advanced Study (KIAS)（韩国高等研究院）；University of Southern California（南加利福尼亚大学）；Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Separable Physics-informed Neural Networks for Solving the BGK Model of the Boltzmann Equation
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院数学科学系）；Department of Mathematics, Gyeongsang National University（庆尚国立大学数学系）；Department of Mathematics, Sungkyunkwan University（全州大学数学系）；Department of Electrical and Computer Engineering, Sungkyunkwan University（全州大学电子与计算机工程系）
Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces
Department of Mechanical Engineering, IIT Kharagpur（印度克里希纳加尔帕大学机械工程系）；Department of Computer Science & Engineering, IIT Kharagpur（印度克里希纳加尔帕大学计算机科学与工程系）
Integral regularization PINNs for evolution equations
Faculty of Science and Technology（科学与技术学院）；Beijing Normal-Hong Kong Baptist University（北京师范大学-香港 Baptist大学）；School of Mathematics and Statistics（数学与统计学学院）；Guangzhou Nanfang College（广州南芳学院）；Zhuhai SimArk Technology Co., LTD（珠海SimArk科技有限公司）；Department of Mathematics and Center for Computation and Technology（数学系与计算技术中心）；Louisiana State University（路易斯安那州立大学）
Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs
National Center for Applied Mathematics, Tianjin University, China（应用数学国家中心，天津大学，中国）；School of Mechanical and Aerospace Engineering, Jilin University, China（机械与 aerospace 工程学院，吉林大学，中国）
Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research（纳拉扬·德赛高级科学研究中心工程力学单元）；Italian Institute of Technology（意大利理工学院）；University of Roma Tre（罗马三大学）；Physics Department, Harvard University（哈佛大学物理系）；Cornell University（康奈尔大学）
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）
Enhancing Heat Sink Efficiency in MOSFETs using Physics Informed Neural Networks: A Systematic Study on Coolant Velocity Estimation
Division of Applied Mathematics, Brown University, Providence, RI, USA（应用数学系，布朗大学）；MIT Sea Grant Design Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA（麻省理工学院海 grant 设计实验室）
Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University（西安交通大学数学与统计学院及电力工程多相流国家重点实验室）；School of Energy and Power Engineering, Xi’an Jiaotong University（西安交通大学能源与动力工程学院）；Nuclear Power Institute of China, State Key Laboratory of Advanced Nuclear Energy Technology（中国核工业集团核电院，先进核能技术国家重点实验室）
INEUS: Iterative Neural Solver for High-Dimensional PIDEs
Department of Mathematics and RiskLab, ETH Zurich（数学系和风险实验室，苏黎世联邦理工学院）；Applied Mathematics: Institute for Analysis and Numerics, University of Münster（应用数学：分析与数值计算研究所，蒙斯特大学）
MiNO: Cotangent-bundle propagator learning for PDEs
Axiom Research Group（公理研究集团）；The Nelson Mandela African Institution of Science and Technology（纳尔逊·曼德拉非洲科学技术研究院）；African Institute for Mathematical Sciences（非洲数学科学研究所）；The University of Tokyo（东京大学）
Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
School of Mathematical Sciences, Fudan University（复旦大学数学学院）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（上海当代应用数学重点实验室）
Parameter Identification for Partial Differential Equations with Spatiotemporal Varying Coefficients
SandGold AI Research（SandGold人工智能研究院）；University of Macau（澳门大学）；Beihang University（北京航空航天大学）；South China Agricultural University（华南农业大学）；Jinan University（暨南大学）；Huazhong University of Sciences and Technology（华中科技大学）
A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials
Department of Mechanical and Production Engineering, Aarhus University（阿arhus大学机械与生产工程系）；Department of Applied Mathematics and Computer Science, Technical University of Denmark（技术大学 of Denmark应用数学与计算机科学系）
PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs
Shahid Beheshti University（沙希德·贝赫什提大学）；Institute for Research in Fundamental Sciences (IPM)（基础科学研究所（IPM））；Cyberspace Research Institute, Shahid Beheshti University（沙希德·贝赫什提大学网络空间研究所）；University of Hertfordshire（赫特福德大学）
Physics-Informed Geometric Operators to Support Surrogate, Dimension Reduction and Generative Models for Engineering Design
BAR Technologies（BAR科技公司）；Nazarbayev University（纳扎尔巴耶夫大学）；University of Strathclyde（斯特拉斯克莱德大学）；Foundation for Research & Technology Hellas (FORTH)（希腊研究与技术基金会）；Texas A&M University（德克萨斯农工大学）
Stabilizing Humanoid Robot Trajectory Generation via Physics-Informed Learning and Control-Informed Steering
Artificial and Mechanical Intelligence, Italian Institute of Technology（意大利理工学院人工智能与机械智能部门）；Machine Learning and Optimisation, University of Manchester（曼彻斯特大学机器学习与优化部门）；Dipartimento di Automatica e Informatica, Politecnico di Torino（都灵politecnico自动化与信息学院）；Rehab Technologies Lab, Italian Institute of Technology（意大利理工学院康复技术实验室）
A Physics-Informed Hierarchical Neural Network for Microwave Scattering Analysis of 3D PEC Targets
Key Laboratory of Universal Wireless Communication, Ministry of Education, School of Information and Communication Engineering, Beijing University of Posts and Telecommunications（信息与通信工程学院，北京邮电大学，教育部无线通信重点实验室）；Department of Informatics and Telecommunications, National and Kapodistrian University of Athens（信息与电信学院，希腊国家与卡波迪斯提亚大学）
Physics-Informed Self-Supervised Learning for Joint Wire Calibration and Interaction Position Reconstruction in Multi-Wire Parallel Plate Avalanche Counters
Generative VS non-Generative Models in Engineering Shape Optimization
University of Strathclyde（斯特拉斯克莱德大学）；Nazarbayev University（纳扎尔巴耶夫大学）；BAR Technologies（BAR科技公司）；Foundation for Research & Technology Hellas (FORTH)（希腊研究与技术基金会）
