Multistep Asymptotic Pre-Training Strategy Based on Pinns for Solving Steep Boundary Singular Perturbation Problems

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. To manage singularly perturbed parabolic problems with a strong gradient in the spatio-temporal domain, a novel time and parameter multi-step asymptotic pre-training technique based on PINNs is proposed. 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 with out 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 PINNs (PTPINNs) by solving singular convection-diffusion-reaction equations and magnetohydrodynamic equations. The results show that the proposed strategy outperforms PINNs and PTPINNs 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.