**2. Problem Settings**

In this section, we will present some basic ideas involving the use of the generalized multiscale finite element method (GMsFEM) for parameter-dependent problems. Let *D* be a bounded domain in R<sup>2</sup> and Ω be the parameter space in R *N*. We consider the following parameter-dependent elliptic problem:

$$-\nabla \cdot \left( \kappa(\mathbf{x}, \mathbf{s}) \nabla u(\mathbf{x}, \mathbf{s}) \right) = f(\mathbf{x}, \mathbf{s}), (\mathbf{x}, \mathbf{s}) \in D \times \Omega,\tag{1}$$

$$u(\mathbf{x}, \mathbf{s}) = \mathbf{0}, (\mathbf{x}, \mathbf{s}) \in \partial D \times \Omega,\tag{2}$$

where *<sup>κ</sup>*(*<sup>x</sup>*,*<sup>s</sup>*) is a heterogeneous coefficient depending on both the spatial variable *x* and the parameter *s*, and *f* ∈ *L*<sup>2</sup>(*D*) is a given source. We remark that the differential operators in Equation (1) are defined with respect to the spatial variable *x*. This is the case for the rest of the paper.

#### *2.1. The Coarsening of the Parameter Space. The Main Idea*

The parameter space Ω is assumed to be of very high dimension (i.e., large *N*) and consists of very large number of realizations. For a given realization, the idea is to find its representation in the coarse space and use the coarse space to perform the computation. We will use the deep cluster learning algorithm to perform the coarsening. Due to the heterogeneous properties of the proposed problem, fine mesh is used; this will bring difficulties in coarsening the parameter space and in computation of the solution. We hence perform the parameter coarsening locally in the space *D*, i.e., we also coarsen the spatial domain. To coarsen the spatial domain, we use coarse grids and consider the GMsFEM.

In Figure 1, we present an illustration of the proposed coarsening technique. On the left figure, the coarse grid blocks in the space are shown. Each coarse grid has a different cluster in the uncertainty space Ω, which corresponds to the coarsening of the uncertainty space. The main objective in multiscale methods is efficiently finding the clustering of the uncertainty space, which is our main goal.

**Figure 1.** Illustration of coarsening of space and uncertainties. Different clusters for different coarse blocks. On the left plot, two coarse blocks are shown. On the right plot, clusters are illustrated.

#### *2.2. Space Coarsening—Generalized Multiscale Finite Element Method*

It is computationally expensive to capture heterogeneous properties using very fine mesh. For this reason, we use GMsFEM to coarsen the spatial representation of the solution. The coarsening of the parameter space will be performed in each local spatial neighborhood. We will achieve this goal by the GMsFEM, which will briefly be discussed. Consider the second order elliptic equation *Lu* = *f* in *D* with proper boundary conditions; denote the the elliptic operator as:

$$L(\boldsymbol{u}) = -\frac{\partial}{\partial \mathbf{x}\_i}(k\_{i\bar{j}}(\mathbf{x})\frac{\partial}{\partial \mathbf{x}\_{\bar{j}}}\boldsymbol{u}).\tag{3}$$

Let the spatial domain *D* be partitioned by a coarse grid T *H*; this does not resolve the multiscale features. Let us denote *K* as one cell in T *H* and refine *K* to obtain the fine grid partition T *h* (blue box in Figure 2). We assume the fine grid is a conforming refinement of the coarse grid. See Figure 2 for details.

**Figure 2.** Domain Partition T *H*

For the *i*-th coarse grid node, let *ωi* be the set of all coarse elements having the vertex *i* (green region in Figure 2). We will solve local problem in each coarse neighborhood to obtain set of multiscale basis functions {*φ<sup>ω</sup><sup>i</sup> i*} and seek solution in the form:

$$
\mu = \sum\_{i} \sum\_{j} c\_{ij} \phi\_{j}^{\omega\_{i}} \, \prime \, \tag{4}
$$

.

where *φ<sup>ω</sup><sup>i</sup> j* is the offline basis function in the *i*-th coarse neighborhood *ωi* and *j* denotes the *j*-th basis function. Before we construct the offline basis, we first need to derive the snapshot basis.
