Multiscale problems can usually be approximated through numerical homogenization by an equation with some effective parameters that can capture the macroscopic behavior of the original system on the coarse grid to speed up the simulation. However, this approach usually assumes scale separation and that the heterogeneity of the solution can be approximated by the solution average in each coarse block. For complex multiscale problems, the computed single effective properties/continuum might be inadequate. In this paper, we propose a novel learning-based multi-continuum model to enrich the homogenized equation and improve the accuracy of the single continuum model for multiscale problems with some given data. Without loss of generalization, we consider a two-continuum case. The first flow equation keeps the information of the original homogenized equation with an additional interaction term. The second continuum is newly introduced, and the effective permeability in the second flow equation is determined by a neural network. The interaction term between the two continua aligns with that used in the Dual-porosity model but with a learnable coefficient determined by another neural network. The new model with neural network terms is then optimized using trusted data. We discuss both direct back-propagation and the adjoint method for the PDE-constraint optimization problem. Our proposed learning-based multi-continuum model can resolve multiple interacted media within each coarse grid block and describe the mass transfer among them, and it has been demonstrated to significantly improve the simulation results through numerical experiments involving both linear and nonlinear flow equations.

翻译：多尺度问题通常可通过数值均匀化方法，利用某些能捕捉原始系统宏观行为的有效参数，在粗网格上以方程近似，从而加速模拟。然而，该方法通常假设尺度分离，且解的异质性可通过各粗网格块内的解平均值近似。对于复杂多尺度问题，计算得到的单一有效属性/连续介质可能不足。本文提出一种新颖的基于学习的多连续介质模型，以丰富均匀化方程并提升单一连续介质模型在给定数据下的多尺度问题求解精度。不失一般性，我们考虑双连续介质情形。第一个流动方程保留原始均匀化方程信息，并增加一个相互作用项。第二个连续介质为新引入，其流动方程中的有效渗透率由神经网络确定。两个连续介质间的相互作用项与双孔隙模型所用形式一致，但其系数可通过另一神经网络学习得到。随后利用可信数据对含神经网络项的新模型进行优化。我们讨论了PDE约束优化问题的直接反向传播与伴随方法。所提出的基于学习的多连续介质模型能在每个粗网格块内解析多个相互作用的介质并描述其间的质量传递，数值实验在线性与非线性流动方程中均证明该模型能显著改善模拟结果。