Solving Fredholm Integral Equations of the First Kind Using Neural Networks
Fredholm integral equations of the first kind appear in various fields such as signal processing, physics, and inverse problems. Solving these equations can be challenging due to their ill-posed nature, meaning small perturbations in the input can cause large deviations in the output. In this post, we explore how neural networks can be applied to solve Fredholm integral equations, focusing on both the theoretical and practical aspects of the problem.
Mathematical Background
A Fredholm integral equation of the first kind has the following general form:
where:
- g(x) is the known function (given by data),
- f(t) is the unknown function (which we want to recover),
- K(x,t) is the kernel function, and
- a and b define the limits of integration.
The challenge lies in recovering f(t) from the integral equation, given the known function g(x) and the kernel K(x,t). This is an inverse problem, as we are trying to deduce the unknown function f(t) from the given data g(x).
