Physics informed neural networks for elliptic equations with oscillatory differential operators

Published: 18 Oct 2023, Last Modified: 18 Oct 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: 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.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: We moved some of the figures from the numerical experiments in sections 4.3 and 4.4 to the appendix, and we added a table summarizing the key points of these subsections.
Supplementary Material: zip
Assigned Action Editor: ~Atsushi_Nitanda1
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 1373