Physics informed neural networks for elliptic equations with oscillatory differential operators

We consider standard physics informed neural network solution methods for elliptic partial differential equations with oscillatory coefficients. We show that if the coefficient in the elliptic operator contains frequencies on the order of $1/\epsilon$, then the Frobenius norm of the neural tangent kernel matrix associated to the loss function grows as $1/\epsilon^2$. Numerical examples illustrate the stiffness of the optimization problem.