标题： 一种求解涉及边界层的双参数奇异摄动问题的参数驱动物理信息神经网络框架 显示英文标题

标题： A Parameter-Driven Physics-Informed Neural Network Framework for Solving Two-Parameter Singular Perturbation Problems Involving Boundary Layers

摘要： 本文的目标是使用改进的物理信息神经网络 (PINN) 方法求解一维和二维双参数奇异摄动问题 (SPP)。 这类问题在工程和科学领域中具有重要意义，例如在控制理论、流体和气体动力学、金融建模等领域。 这类问题的解通常包含边界层和/或内部层，这使得它们难以处理。 文献已证实，标准 PINN 精度较低，无法有效处理此类问题。 最近，Cao 等人\cite{cao2023physics} 提出了一种新的参数渐近 PINN（PA-PINN），用于求解单参数奇异摄动对流主导问题。 结果表明，PA-PINN 在精度、收敛性和稳定性方面均优于标准 PINN 和 gPINN。 本文将首次验证 PA-PINN 对于解决双参数 SPP 的鲁棒性。 显示英文摘要

摘要： 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 sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \cite{cao2023physics} proposed a new parameter asymptotic PINNs (PA-PINNs) to solve one-parameter singularly perturbed convection-dominated problems. It was observed that PA-PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA-PINNs will be validated for solving two-parameter SPPs.