Solving Inhomogeneous Constant Coefficient Ordinary Differential Equations Using Artificial Neural Networks

Ordinary differential equations are fundamental tools for modeling dynamic systems in science, engineering, and applied mathematics. Solving these equations accurately and efficiently is crucial, particularly in cases where analytical solutions are challenging or impossible to obtain. This paper presents a method for solving inhomogeneous linear ordinary differential equations using an artificial neural network. The network is composed of a single input layer with one neuron, one hidden layer with three neurons, and a single output layer with one neuron. A multiple regression model is employed to determine the weights from the input layer to the hidden layer, while radial basis functions are used to compute the weights from the hidden layer to the output layer. The bias values are chosen within the range of −1 to 1 to optimize learning behavior. A trial solution is constructed as a sum of two parts. One part satisfies the initial condition, and the other part is the output of the network to approximate the function. The neural network is trained to minimize the mean squared error of the residuals obtained by doing the substitution of the trial solution into the given ordinary differential equation. The methodology is tested on first-order and second-order ordinary differential equations to evaluate its accuracy, stability and how its capability can be generalized. The results show that the method can approximate the exact solutions of these ordinary differential equations with reasonable accuracy.