A parameter-driven physics-informed neural network framework for solving two-parameter singular perturbation problems involving boundary layers

In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and the sciences, as they arise in control theory, fluid and gas dynamics, financial modeling, and related areas. These problems often exhibit boundary and/or interior layers, making them particularly challenging to solve. Prior studies have shown that standard PINNs suffer from low accuracy and struggle to effectively address such issues. A recently proposed enhancement, known as the parameter asymptotic PINNs (PA-PINNs), has demonstrated improved performance over standard PINNs and Gradient enhanced PINNs (gPINNs) in handling one-parameter singularly perturbed convection-dominated problems, offering better accuracy, convergence, and stability. In this article, we extend and evaluate the robustness of PA-PINNs for the first time in solving two-parameter SPPs. Furthermore, we derive bounds on the generalization error, linking training residuals to quadrature errors, thereby providing a theoretical foundation for the reliability of PINNs in solving multi-scale two-parameter SPPs. We also present a comprehensive comparison with standard finite difference and finite element methods, analyzing accuracy and computational efficiency across various neural network architectures.