Learning-based Multi-continuum Model for Multiscale Flow Problems

TL;DR

This work addresses the limitations of numerical homogenization for complex multiscale flow by proposing a learning-based two-continuum framework. A first continuum retains the homogenized dynamics with an added interaction to a newly introduced second continuum, whose permeability and the inter-continuum transfer are learned via neural networks, with gradients computed by both direct backpropagation and the adjoint method. The approach is validated on linear and nonlinear flow problems, showing substantial accuracy gains over standard homogenization and revealing robust learned corrections that capture mass transfer between continua. The method offers a data-driven path to enhance reduced-order models for fractured and heterogeneous media with potential for broader multiscale PDE applications.

Abstract

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.

Figures (11)

• Figure 1: The schematic illustration of our proposed learning-based multi-continuum model. The black color indicates the forward process, and the green color indicates the back-propagation to optimize network parameters.
• Figure 2: The configuration of the permeability field and the source term of the linear flow equation. (a) The permeability field on the fine grid. (b) The source term. (c) The value $\kappa^\star_{11}$ of the homogenized permeability. (d) The value $\kappa^\star_{22}$ of the homogenized permeability.
• Figure 3: The training trajectories of the loss function. The gradients of netwrok parameters are calculated by (a) direct BP and (b) the adjoint method.
• Figure 4: The FEM solutions of the linear flow equation, the homogenized equation and our proposed learning-based multi-continuum equation. (a)(d) The reference solutions at $t=0.1$ and 1. (b)(e) The homogenized solutions at $t=0.1$ and 1. (c)(f) The solutions of our proposed multi-continuum equation at $t=0.1$ and 1.
• Figure 5: The learned permeability field $\kappa_2$ and transfer coefficient $\sigma$. (a-b) The diagonal elements $\kappa_2^{11}$ and $\kappa_2^{22}$ in domain $\Omega$. (c) The learned transfer coefficient $\sigma$ in domain $\Omega$.