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Multiscale elliptic equations with scale separation are often approximated bythe corresponding homogenized equations with slowly varying homogenizedcoefficients (the G-limit). The traditional homogenization techniques typicallyrely on the periodicity of the multiscale coefficients, thus finding theG-limits often requires sophisticated techniques in more general settings evenwhen multiscale coefficient is known, if possible. Alternatively, we propose asimple approach to estimate the G-limits from (noisy-free or noisy) multiscalesolution data, either from the existing forward multiscale solvers or sensormeasurements. By casting this problem into an inverse problem, our approachadopts physics-informed neural networks (PINNs) algorithm to estimate theG-limits from the multiscale solution data by leveraging a priori knowledge ofthe underlying homogenized equations. Unlike the existing approaches, ourapproach does not rely on the periodicity assumption or the known multiscalecoefficient during the learning stage, allowing us to estimate homogenizedcoefficients in more general settings beyond the periodic setting. Wedemonstrate that the proposed approach can deliver reasonable and accurateapproximations to the G-limits as well as homogenized solutions through severalbenchmark problems.

Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations