论文标题
物理学为带有振荡差异操作员的椭圆方程式的神经网络告知
Physics informed neural networks for elliptic equations with oscillatory differential operators
论文作者
论文摘要
物理知情的神经网络（PINN）用于微分方程的解决方案方法最近在各种科学计算应用中表现出了成功。但是，当使用PINN求解具有多尺度功能的方程时，有几位作者报告了困难。本工作的目的是说明和解释在特定情况下使用标准PINN用于差异操作员中存在振荡系数的特定情况。我们表明，如果椭圆运算符$ a^ε（x）$的系数为1 $ a（x/ε）$，用于1-过程强制胁迫函数$ a（\ cdot）$，则与神经切线内核（ntk）损失功能相关的神经切线（NTK）的Frobenius Norm frobenius Norm af loss offerage norm frolbenius norm aflobenius。这意味着，随着问题中量表的分离增加，基于梯度下降的方法训练神经网络以实现对PDE的溶液的准确近似变得越来越困难。数值示例说明了优化问题的刚度。
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 $a^ε(x)$ is of the form $a(x/ε)$ 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 $1/ε^2$. 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.
