PINNIES: An efficient physics-informed neural network framework for integral operator problems

Our approach leverages Kolmogorov-Arnold networks to evaluate problem dynamics at specific points, while employing Gaussian quadrature formulas to approximate the integral components, even in the presence of semi-infinite domains or singularities.

We demonstrate the applicability of the proposed method to both Fredholm and Volterra integral operators, as well as to optimal control problems involving continuous time. Additionally, we outline how this approach can be extended to approximate fractional derivatives and integrals and propose a fast matrix-vector product algorithm for efficiently computing the fractional Caputo derivative. In the numerical section, we conduct comprehensive experiments on forward and inverse problems.