We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral Neural Operators (FREDINOs), for FIEs and prove that they are universal approximators of linear and non-linear integral operators and corresponding solution operators. We furthermore prove that the learned operators are guaranteed to be contractive, thereby strictly satisfying the mathematical property required for the convergence of the fixed point scheme. Finally, we also demonstrate how FREDINOs can be used to learn the solution operator of non-linear elliptic PDEs, via a Boundary Integral Equation (BIE) formulation. We assess the proposed methodology numerically, via several benchmark problems: linear and non-linear FIEs in arbitrary dimensions, as well as a non-linear elliptic PDE in 2D. Built on tailored mathematical/numerical analysis theory, FREDINOs offer high-accuracy approximations and interpretable schemes, making them well suited for scientific machine learning/numerical analysis computations.
A new iterated Tikhonov regularization method for Fredholm integral equation of first kind
Hebei Normal University（河北师范大学）；Hebei Key Laboratory of Computational Mathematics and Applications（河北省计算数学与应用重点实验室）；Hebei Research Center of the Basic Discipline Pure Mathematics（河北省基础学科纯数学研究中心）
Monte-Carlo method for multiple parametric integrals calculation and solving of linear integral Fredholm equations of a second kind, with confidence regions in uniform norm
Adaptive Multi-Grade Deep Learning for Highly Oscillatory Fredholm Integral Equations of the Second Kind
School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机科学与工程学院）；Department of Mathematics and Statistics, Old Dominion University（老道明大学数学与统计系）
Oscillation Preserving Galerkin Methods for Fredholm Integral Equations of the Second Kind with Oscillatory Kernels
College of Science, National University of Defense Technology（国防科技大学理学院）；Department of Mathematics, Syracuse University（雪城大学数学系）；Guangdong Province Key Lab of Computational Science, School of Mathematics and Computational Science, Sun Yat-Sen University（中山大学数学与计算科学学院）
Fredholm Neural Networks for inverse problems in elliptic PDEs
Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Statistics and Stochastic Modelling and Applications Laboratory, Athens University of Economics and Business（雅典经济与商业大学统计学与随机建模与应用实验室）
De la Vallée Poussin type approximation for solving some Fredholm integral equations
University of Basilicata（巴西利卡塔大学）；Institute for Applied Mathematics ”Mauro Picone”, National Research Council of Italy（意大利国家研究委员会应用数学“毛罗·皮切内”研究所）；Department of Mathematics and Applications ”Renato Caccioppoli”, University of Naples Federico II（那不勒斯费德里科二世大学雷纳托·卡乔普利数学与应用系）
提出基于Jacobi零点的de la Vallée Poussin型多项式逼近求解第二类Fredholm积分方程，该方法具有一致有界Lebesgue常数、近最佳一致逼近和抑制Gibbs现象的优势，并证明在加权一致空间中的稳定性和收敛性。
Periodic B-spline-Heaviside collocation for Fredholm integral equations with piecewise Hölder data
Institute of Mathematics and Computer Science, Moldova State University（摩尔多瓦国立大学数学与计算机科学研究所）；Department of Computer Science, Moldova State University（摩尔多瓦国立大学计算机科学系）
A Rational Discrete Collocation Method for Second Kind Fredholm Equations
University of Basilicata（巴西利卡塔大学）；Institute for Applied Mathematics “Mauro Picone”, National Research Council of Italy（意大利国家研究委员会应用数学“毛罗·皮乔内”研究所）；University of Naples Federico II（那不勒斯费德里科二世大学）
