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.

翻译：此外,我们建议一种简单的方法,从现有的前方多尺度解决器或传感器测量器或传感器测量器中估算G(无噪音或噪音)的多尺度解决方案数据。通过将这一问题推入一个反向问题,我们的方法是采用物理知情神经网络(PINNs)算法,利用多尺度解决方案数据的预知性知识,从多尺度系数中估算G(G)界限,从而从多尺度系数中估算G界限。与现有的方法不同,我们的方法并不依赖周期假设或已知的多尺度系数,从而使我们能够在学习阶段从现有的前方多尺度解决器或传感器测量器中估算同质系数。我们的方法通过将这一问题推入一个反向问题,采用物理知情神经网络(PINNs)算法,利用多尺度解决方案数据的先入为主,从多尺度数据中估算G界限。我们的方法与现有方法不同,在学习阶段不依赖周期假设或已知的多尺度系数,从而使我们能够在超出定期设定的更普遍情况下估算同质系数。我们证明,拟议的方法能够通过几个基准问题,向G界限提供合理和准确的近似近似的同质解决方案,作为同质解决方案。