Physics informed neural network (PINN) based solution methods for differential equations have recently shown success in a variety of scientific computing applications. Several authors have reported difficulties, however, when using PINNs to solve equations with multiscale features. The objective of the present work is to illustrate and explain the difficulty of using standard PINNs for the particular case of divergence-form elliptic partial differential equations (PDEs) with oscillatory coefficients present in the differential operator. We show that if the coefficient in the elliptic operator $a^{\epsilon}(x)$ is of the form $a(x/\epsilon)$ for a 1-periodic coercive function $a(\cdot)$, then the Frobenius norm of the neural tangent kernel (NTK) matrix associated to the loss function grows as $1/\epsilon^2$. This implies that as the separation of scales in the problem increases, training the neural network with gradient descent based methods to achieve an accurate approximation of the solution to the PDE becomes increasingly difficult. Numerical examples illustrate the stiffness of the optimization problem.

翻译：基于物理信息神经网络（PINN）的微分方程求解方法已在众多科学计算应用中展现出成功。然而，多位研究者指出，当使用PINN求解含有多尺度特征的方程时存在困难。本文旨在阐明并解释标准PINN在求解特定类型椭圆型偏微分方程（PDE）时面临的挑战，这类方程的微分算子中包含振荡系数。研究表明，若椭圆算子中的系数$a^{\epsilon}(x)$形如$a(x/\epsilon)$（其中$a(\cdot)$为1-周期胁迫函数），则损失函数对应的神经正切核（NTK）矩阵的Frobenius范数将随$1/\epsilon^2$增长。这意味着随着问题中尺度分离程度的增加，通过基于梯度下降的方法训练神经网络以精确逼近PDE解的难度将显著提升。数值算例进一步验证了该优化问题的刚性特征。