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 (noise-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.

具有尺度分离的多尺度椭圆方程通常由具有缓慢变化的均质系数（G 极限）的相应均质方程来近似。传统的均匀化技术通常依赖于多尺度系数的周期性，因此即使在多尺度系数已知的情况下，如果可能的话，在更一般的设置中找到 G 极限通常也需要复杂的技术。或者，我们提出了一种简单的方法来估计来自（无噪声或有噪声的）多尺度解数据的 G 限制，无论是来自现有的正向多尺度求解器还是传感器测量。通过将此问题转换为逆问题，我们的方法采用物理信息神经网络 (PINNs) 算法，通过利用底层同质化方程的先验知识，从多尺度解数据中估计 G 极限。与现有方法不同，我们的方法在学习阶段不依赖于周期性假设或已知的多尺度系数，这使我们能够在超出周期性设置的更一般的设置中估计同质化系数。我们证明了所提出的方法可以通过几个基准问题提供对 G 极限的合理和准确的近似以及同质化的解决方案。允许我们在周期性设置之外的更一般的设置中估计同质化系数。我们证明了所提出的方法可以通过几个基准问题提供对 G 极限的合理和准确的近似以及同质化的解决方案。允许我们在周期性设置之外的更一般的设置中估计同质化系数。我们证明了所提出的方法可以通过几个基准问题提供对 G 极限的合理和准确的近似以及同质化的解决方案。