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.

中文翻译：

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基于PINN的多步渐进预训练策略求解陡边界奇异摄动问题


奇异摄动问题的特点是存在狭窄的边界层，由于复杂性和高成本，对传统数值方法提出了挑战。当代深度学习物理信息神经网络（PINN）在学习初始条件时存在准确性问题，无法捕获尖锐的梯度行为，并且对快速振荡的解决方案提供不充分的近似。引入了一种名为 PATPINN 的新技术，利用基于 PINN 的独特时间和参数多步渐近预训练方法，有效解决时空域中具有显着梯度的奇扰动抛物线问题。所提出的技术可以帮助模型学习系统动态行为并提高初始条件的准确性。它还使 PINN 能够在无需事先了解边界层位置的情况下捕获解中的突然变化，从而增强其逼近振荡解的能力。这种创新方法不需要超参数微调，并为处理进化奇异扰动问题提供了可靠的深度学习方法。通过求解奇异对流-扩散-反应方程和磁流体动力学方程，将所提出的方法与 PINN 和预训练 PINN (PTPINN) 进行比较。结果表明，所提出的策略在捕获边界层梯度、提高逼近精度和加速训练过程方面优于PINNs和PTPINN，此外还显着提高了PINNs逼近初始条件的精度。