Adaptive Physics-Informed Neural Networks for Singularly Perturbed Convection-Diffusion Problems

This paper presents an adaptive physics-informed neural network (PINN) framework for the numerical solution of one-dimensional singularly perturbed differential equations. Such problems are characterized by the presence of small perturbation parameters multiplying the highest-order derivatives, which typically generate sharp boundary or interior layers and lead to severe numerical difficulties for standard discretization methods. The proposed approach integrates a residual-based adaptive sampling strategy with a dynamically refined neural network training process, allowing the method to focus computational effort in regions of rapid solution variation. The governing differential equation and associated boundary conditions are incorporated directly into the loss function, ensuring consistency with the underlying physics. To enhance stability and accuracy in the layer regions, the training data are progressively enriched using an error indicator derived from the local PDE residual. Numerical experiments on representative singularly perturbed convection--diffusion and reaction--diffusion problems demonstrate that the adaptive PINN significantly improves pointwise accuracy compared to standard PINNs, particularly in boundary-layer regions, while maintaining computational efficiency. The results confirm that adaptive sampling combined with physics-informed learning provides a robust and flexible tool for solving one-dimensional singularly perturbed problems without requiring a priori knowledge of layer locations or specialized meshes.