Related papers: Fredholm integral equations for function approxima…
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…
Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel…
In this paper we investigate the connection between supervised learning and linear inverse problems. We first show that a linear inverse problem can be view as a function approximation problem in a reproducing kernel Hilbert space (RKHS)…
In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…
We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as…
We present a fundamentally new regularization method for the solution of the Fredholm integral equation of the first kind, in which we incorporate solutions corresponding to a range of Tikhonov regularizers into the end result. This method…
Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values…
The paper deals with numerical solution of the Fredholm integral equation associated with the classical problem of extrapolating bandlimited functions known on $(-1,1)$ to the entire real line. The approach presented can be characterized as…
We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In…
We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely,…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…
This paper proposes a new family of algorithms for training neural networks (NNs). These are based on recent developments in the field of non-convex optimization, going under the general name of successive convex approximation (SCA)…
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower…
Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering…
In this Master Thesis, we study the approximation capabilities of Neural Networks in the context of numerical resolution of elliptic PDEs and Approximation Theory. First of all, in Chapter 1, we introduce the mathematical definition of…
High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty…
Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and…
Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose…
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…