PINNIES: An efficient physics-informed neural network framework for integral operator problems
Description
This paper introduces an efficient tensor-vector product technique for the fast and accurate approximation of integral operators within physics-informed deep learning frameworks. 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. For forward problems, we evaluate the performance of our method on over 50 diverse mathematical problems, including multi-dimensional integral equations, systems of integral equations, partial and fractional integro-differential equations, and various optimal control problems in delay, fractional, multi-dimensional, and nonlinear configurations. For inverse problems, we test our approach on several integral equations and fractional integro-differential problems. Finally, we introduce the pinnies Python package to facilitate the implementation and usability of the proposed method.