Numerical method for solving Fredholm and Volterra integral equations using artificial neural networks

Many problems in mathematics, mechanics, physics and other engineering disciplines lead to equations in which the unknown function appears under the integral sign. Integral equations are useful mathematical tools in many fields, so they are studied in many different aspects, such as the existence of solutions, approximation of solutions, calculation of correction or incorrigibility, correction of solutions, etc. Many articles mention the so-called PINN (physics-informed neural networks, which can be translated as physically conditioned neural networks), which have found application for solving differential equations, both ordinary and partial derivatives, as well as systems of differential equations. PINNs are also used to solve integral equations, but publications usually provide methods for solving a certain class of equations, for example, the Fredholm equation of the 2nd kind or the Volterra equation of the 2nd kind. This article will describe a general method for solving continuous integral equations using neural networks that generalizes them to both Fredholm and Volterra integral equations. The essence of the method is that the desired function is approximated by a neural network, which is essentially a huge function with a large number of adjustable parameters, which are selected from the condition of minimal squared residual, for which the parameters of the neural network are adjusted using the L-BFGS optimization algorithm. The results of the ANN method are compared with the exact solution for several typical integral equations.