Integral Operator

Integral operators, which map functions to other functions through integration, are central to solving many scientific problems, particularly in modeling systems with nonlocal interactions. Current research focuses on developing efficient numerical methods for approximating these operators, including physics-informed neural networks, kernel-based spectral methods, and neural integral equation approaches, often leveraging techniques like Gaussian quadrature and self-attention mechanisms. These advancements are improving the accuracy and scalability of solving integral equations and related problems in diverse fields such as partial differential equations, optimal control, and data analysis from high-dimensional noisy datasets, leading to more powerful tools for scientific modeling and computation.

Papers

September 3, 2024

PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems
Alireza Afzal Aghaei, Mahdi Movahedian Moghaddam, Kourosh Parand
High Efficiency Physic Informed Neural Network Physic Informed Deep Learning Fractional Derivative Matrix Vector Product Integral Operator

May 20, 2024

Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators
Xiucai Ding, Rong Ma
Kernel Hilbert Space Single Cell Kernel Extension Spectral Embedding Single Cell Multi Omics Integral Operator High Dimensional Noisy

January 27, 2024

Integral Operator Approaches for Scattered Data Fitting on Spheres
Shao-Bo Lin
Spherical Surface Tikhonov Regularization Spectral Filtering Integral Operator

December 9, 2023

Spectral methods for Neural Integral Equations
Emanuele Zappala
Deep Learning Model Spectral Method Neural Integral Equation Boundary Integral Integral Operator

December 5, 2023

GIT-Net: Generalized Integral Transform for Operator Learning
Chao Wang, Alexandre Hoang Thiery
Neural Operator Partial Differential Equation Operator Learning GIT Net Differential Operator Integral Operator Integral Transforms

October 13, 2023

Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations
Hardeep Bassi, Yuanran Zhu, Senwei Liang, Jia Yin, Cian C. Reeves, Vojtech Vlcek, Chao Yang
LeArning Abstract Application Proficiency Recurrent Neural Network Integral Operator

September 30, 2022

Neural Integral Equations
Emanuele Zappala, Antonio Henrique de Oliveira Fonseca, Josue Ortega Caro, Andrew Henry Moberly, Michael James Higley, Jessica Cardin, David van Dijk
Spatiotemporal Dynamical System Neural Integral Equation Boundary Integral Integral Operator

May 1, 2022

On the speed of uniform convergence in Mercer's theorem
Rustem Takhanov
Speed Effect Theorem Driven Question Positive Definite Kernel Weak Convergence Integral Operator Separable Kernel

February 28, 2022

Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach
Xiucai Ding, Rong Ma
Kernel Hilbert Space Low Dimensional Manifold Dimensional Embeddings Low Dimensional Structure Integral Operator High Dimensional Noise High Dimensional Noisy

January 28, 2022

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations
Jin Young Shin, Jae Yong Lee, Hyung Ju Hwang
Partial Differential Equation Fourier Neural Operator Solution Operator Integral Operator

Integral Operator

Papers

PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems

Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators

Integral Operator Approaches for Scattered Data Fitting on Spheres

Spectral methods for Neural Integral Equations

GIT-Net: Generalized Integral Transform for Operator Learning

Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations

Neural Integral Equations

On the speed of uniform convergence in Mercer's theorem

Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations