用PINN修正边界层难题的数值解，提升精度与稳定性。
Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation
- 将PINN用于修正有限差分法产生的振荡解
- 通过输入变换使解在边界层更精确，误差降低40%以上
- 适合从事微分方程数值求解或神经网络物理建模的研究者
奇摄动问题通常具有陡峭的边界层，数值求解难度大。传统方法如有限差分法（FDM）需精细网格才能获得稳定准确解。由于物理信息神经网络（PINNs）在多个领域微分方程求解中表现优异，我们考察其在奇摄动问题中的应用。以对流-扩散方程为例，研究了两种使用PINNs的策略：一是修正由FDM得到的振荡离散解，二是修正无摄动问题的简化解。两种方法均结合输入变换以提升精度，并借助神经正切核理论分析输入变换的作用机制。
原文摘要 · Abstract (English)
Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection-diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.
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