  Multiscale elliptic equations with scale separation are often approximated by
the corresponding homogenized equations with slowly varying homogenized
coefficients (the G-limit). The traditional homogenization techniques typically
rely on the periodicity of the multiscale coefficients, thus finding the
G-limits often requires sophisticated techniques in more general settings even
when multiscale coefficient is known, if possible. Alternatively, we propose a
simple approach to estimate the G-limits from (noisy-free or noisy) multiscale
solution data, either from the existing forward multiscale solvers or sensor
measurements. By casting this problem into an inverse problem, our approach
adopts physics-informed neural networks (PINNs) algorithm to estimate the
G-limits from the multiscale solution data by leveraging a priori knowledge of
the underlying homogenized equations. Unlike the existing approaches, our
approach does not rely on the periodicity assumption or the known multiscale
coefficient during the learning stage, allowing us to estimate homogenized
coefficients in more general settings beyond the periodic setting. We
demonstrate that the proposed approach can deliver reasonable and accurate
approximations to the G-limits as well as homogenized solutions through several
benchmark problems.
