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，利用包含有效参数的方程来近似描述，该方程能在粗网格上捕捉原始系统的宏观行为以加速模拟。然而，该方法通常假设尺度分离，且解的异质性可通过各粗块上的解平均值来近似。对于复杂多尺度问题，计算得到单一有效属性/连续介质可能不够充分。本文针对给定数据下的多尺度问题，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高单连续介质模型的精度。不失一般性，我们考虑双连续介质情形。第一流动方程保留了原始均匀化方程的信息，并附加一个相互作用项。第二连续介质为新引入项，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度模型中的形式一致，但其可学习系数由另一神经网络决定。随后，利用可信数据对包含神经网络项的新模型进行优化。我们讨论了直接反向传播方法和伴随方法两种PDE约束优化问题的求解策略。本文提出的基于学习的多连续介质模型，能够解析每个粗网格块内的多种相互作用介质并描述其间的质量传递。通过线性和非线性流动方程的数值实验，该模型被证明能显著提升模拟结果。