Multistep asymptotic pre-training strategy based on PINNs for solving steep boundary singular perturbation problems

Abstract

The singularly perturbed problem is characterized by the presence of narrow boundary layers, which poses challenges for traditional numerical methods due to complexity and high costs. The contemporary deep learning physics-informed neural networks (PINNs) suffer from accuracy issues while learning initial conditions, fail to capture the sharp gradient behaviors, and provide inadequate approximations to rapidly oscillating solutions. A novel technique named PATPINN is introduced to effectively address singularly perturbed parabolic problems with significant gradients in the spatio-temporal domain, utilizing a unique time and parameter multi-step asymptotic pre-training approach based on PINNs. The presented technique can assist the model in learning the system dynamic behavior and improve the accuracy of the initial conditions. It also enables PINNs to capture abrupt changes in the solution without prior knowledge of the boundary layer position, boosting its ability to approximate oscillatory solutions. This innovative approach does not require hyperparameter fine-tuning and provides a dependable deep learning approach for handling evolutionary singular perturbation problems. The proposed method is compared to PINNs and pre-training PINN (PTPINN) by solving singular convection–diffusion–reaction equations and magnetohydrodynamic equations. The results show that the proposed strategy outperforms PINNs and PTPINN in capturing the boundary layer gradient, improving the approximation accuracy and accelerating the training process, in addition to significantly improving the accuracy of PINNs in approximating the initial conditions.