Solving Differential Equations by Artificial Neural Networks and Domain Decomposition

In this paper, parallel neural networks are proposed to solve various kinds of differential equations using domain decomposition techniques. First, trigonometric neural networks are designed based on the truncated Fourier series. Second, a group of these networks is calculated to estimate the initial approximation in each decomposed domain. Third, special modifier networks for decomposed domains and boundary networks for the related boundaries are determined. We successfully achieved the solution by considering an iterative method for learning modifier and boundary networks. Convergent properties for this method are proved. Neural network solutions are given and compared with some other numerical methods. Simulation results confirmed this hybrid approach's efficiency, validity, and accuracy.