  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.
