Numerical Solution of Fredholm Integral Equations of the Second Kind using Neural Network Models

We propose a novel method based on a neural network with one hidden layer and the collocation method for solving linear Fredholm integral equations of the second kind. We first choose the space of polynomials as the projection space for the collocation method, then approximate the solution of a integral equation by a linear combination of polynomials in that space. The coefficients of this linear combination are served as the weights between the hidden layer and the output layer of the neural network while the mean square error between the exact solution and the approximation solution at the training set as the cost function. We train the neural network by the gradient decent method with Adam optimizer and find an optimal solution with the desired accuracy. This method provides a stable and reliable solution with higher accuracy and saves computations comparing with previous neural network approaches for solving the Fredholm integral equations of the second kind.