A Long Short-Term Memory Based Neural Network Approach for Numerical Solutions of Integral Equations

In this paper, neural networks with long short-term memories have been used to solve integral equations. The integral equations have been discretized using a Monte-Carlo method. While neural networks have been widely used for various applications, their application to integral equations, especially in conjunction with LSTMs, is relatively unexplored which is filled by this paper. By discretizing the stochastic integral equations into a system of linear algebraic equations, the proposed method provides a practical solution that can be easily implemented and computationally efficient. This addresses a practical gap in solving integral equations, as traditional methods can often be complex and time-consuming. The use of Monte-Carlo methods allows for a probabilistic approach, which can handle the inherent randomness in some integral equations. A numerical example of the Fredholm Integral equation has been provided to show how adaptable our suggested systems are. Adam optimization has been used to create a neural network model based on long short-term memory in order to find solutions. Loss function graphs, actual and predicted function value graphs, and other visualizations have also been described to provide a better grasp of the behavior of the solutions based on our suggested methodology.