Abstract:

In this paper, we present a novel deep learning-based approach for solving singularly perturbed delay partial differential equations. The method combines a physics-informed neural network with a deep reinforcement learning-guided mesh refinement strategy to capture sharp boundary layers and time delay effects. The neural network is a feedforward model trained using exact solution data, while the reinforcement learning component adaptively refines the spatial mesh in regions of rapid variation. The delay term is incorporated directly into the training process. Numerical experiments demonstrate that the method maintains high accuracy and stability. Performance is evaluated using maximum absolute error, mean squared error, and the discrete square-integral error norm. Visualization of predicted solutions confirms the model’s ability to approximate the exact solution even in the presence of strong singular perturbations and delay. In this approach, we achieve low maximum absolute error and mean square error while maintaining computational efficiency, highlighting its potential for solving complex time-dependent partial differential equations with multiple scales..