Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation

Physics-informed neural networks (PINNs) are a recent trend in scientific machine learning research and modeling of differential equations. Despite progress in PINN research, large gradients and highly nonlinear patterns remain challenging to model. Thin boundary layer problems are prominent examples of large gradients that commonly arise in transport problems. In this study, boundary-layer PINN (BL-PINN) is proposed to enable a solution to thin boundary layers by considering them as a singular perturbation problem. Inspired by the classical perturbation theory and asymptotic expansions, BL-PINN is designed to replicate the procedure in singular perturbation theory. Namely, different parallel PINN networks are defined to represent different orders of approximation to the boundary layer problem in the inner and outer regions. In different benchmark problems (forward and inverse), BL-PINN shows superior performance compared to the traditional PINN approach and is able to produce accurate results, whereas the classical PINN approach could not provide meaningful solutions. BL-PINN also demonstrates significantly better results compared to other extensions of PINN such as the extended PINN (XPINN) approach. The natural incorporation of the perturbation parameter in BL-PINN provides the opportunity to evaluate parametric solutions without the need for retraining. BL-PINN demonstrates an example of how classical mathematical theory could be used to guide the design of deep neural networks for solving challenging problems.