Physics-informed neural networks in iterative form of nonlinear equations for numerical algorithms and simulations of delay differential equations

Abstract

This paper proposes a new high-precision and efficient algorithm for solving delay differential equations using a physics-informed neural network. We utilize initial conditions and two types of neural network methods, namely the Extreme Learning Machine and the Multilayer Perceptron, to construct trial functions that accurately satisfy the initial conditions. These trial functions are then used to discretize the delay differential equations. In contrast to the original physics-informed neural network, we employ an iterative approach by transforming the form of the loss function into an algebraic system generated at configuration points. The algebraic system is iteratively computed to obtain the optimal parameters, which correspond to the optimal solution of the equation. Finally, we validate the effectiveness of our method through six numerical examples, including complex delay differential systems, demonstrating that our approach yields high-precision and efficient numerical results.