  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.
