Numerical solution of nonlinear integral equations with multi-layer neural networks

In this thesis, we present an approach to solve systems of nonlinear integral equations of Fredholm and Volterra types using a multilayer neural network (MLNN) trained through unconstrained optimization. The learning process involves minimizing the integral equation error function using the Nelder-Mead simplex method (NMSM), enabling efficient convergence in both Fredholm and Volterra equations. Our method ensures high accuracy despite potentially time-consuming computations for complex problems, maintaining convergence and approximation precision. Furthermore, the proposed approach demonstrates robustness against high-dimensional and variant equations. Numerical experiments employing the NMSM showcase its superior performance compared to existing methods. This research offers a convenient way for solving a system of nonlinear integral equations efficiently and accurately, with broad applicability across diverse domains.