A neural network approach for solving Fredholm integral equations of the second kind

Deep neural network processing of DEER data

“…At a more general level, neural network “surrogate” solutions to Fredholm equations are well researched in their own right ( 51 ), with rigorous accuracy bounds available ( 52 , 53 ). In 2013, Jafarian and Nia ( 54 ) proposed a two-layer feedback network built around a Taylor expansion of the solution; Effati and Buzhabadi ( 55 ) published a feedforward network proposition. Both groups considered a generic Fredholm equation without any specific physical model or context.…”

“…At that time, neither group had the computing power to train a network of sufficient width and depth to perform the tasks encountered in this work. However, both groups observed that, for such problems as they could handle, neural networks provided very accurate solutions ( 54 , 55 ). Promising neural network results also exist for two-dimensional (2D) integral equations ( 56 , 57 ), meaning that processing triple electron resonance spectroscopy ( 58 ) data with neural networks may also be possible.…”

A Novel Numerical Method of Two-Dimensional Fredholm Integral Equations of the Second Kind

“…There exist many methods dealing with one-dimensional integral equation [1,2,4,[6][7][8]. However, high dimensional problem is still a challenge, a few numerical approaches dealing with high dimensional problems [9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Surrogate Modelling of Solutions of Integral Equations by Neural Networks

“…(see, e.g., [3,4,5] [6,Chapter 11]. As an alternative to the traditional approach, surrogate models formed by neural networks with perceptrons [7] and Gaussian radial-basis units [8] were proposed and explored experimentally. These experiments were motivated by higher flexibility of neural networks than flexibility of polynomials.…”

“…In some cases, especially in the case of functions of large numbers of variables, neural networks achieve better approximation rates than linear models with much smaller model complexity [9,10]. Motivated by the experimental exploration [8,7] of surrogate models of solutions of Fredholm equations by neural networks, Gnecco et al [11] initiated a theoretical analysis of this modelling. In [11], they estimated approximation errors in supremum norm for surrogate solutions by networks with kernel units induced by the kernels of the equations.…”