**4. Numerical Experiments**

In this section, we will demonstrate a series of experiments. We are going to apply our method on problems with high contrast including moving background and moving channels. he experiments are related to subsurface simulations. The moving background and moving channels simulate some important characteristics in the field. We numerically generate heterogeneous fields which contain moving channels and varying well rates. In Section 4.1, we first demonstrate a set of simulated heterogeneous oil fields to be used to train and solve the PDE modeling the reservoirs simulation; the deep learning model settings are also detailed in this section. In Section 4.2, we simulate some other more complicated heterogeneous fields and conduct the experiments to demonstrate the power of clustering algorithm. This experiments can show that our method is robust to handle complicated problems. In the last section, we will solve the PDE using the proposed method based on the heterogeneous field proposed in Section 4.1 and compute the relative error to demonstrate the accuracy of our method.

#### *4.1. High Contrast Heterogeneous Fields with Moving Channels*

We consider solving Equations (1)–(2) for a heterogeneous field with moving channels and changing background. Let us denote the heterogeneous field as *<sup>κ</sup>*(*x*), where *x* ∈ [0, 1]2, then *κ*(*x*) = 1000 if *x* is in some channels which will be illustrated later and otherwise,

> *κ*(*x*)= *<sup>e</sup>η*·*sin*(<sup>7</sup>*πx*)*sin*(<sup>8</sup>*<sup>π</sup>y*)+*sin*(<sup>10</sup>*πx*)*sin*(<sup>12</sup>*<sup>π</sup>y*),

where *η* follows discrete uniform distribution in [0, 1]. The channels are moving and we include cases of the intersection of two channels and formation and dissipation of the channels in the fields. These simulate the realistic petroleum oil fields. In Figure 7, we demonstrate 20 heterogeneous fields.

**Figure 7.** Heterogeneous fields, the yellow strips are the channels.

It can be observed from the images that, vertical channel (at around *x* = 30) (not always) intersects with horizontal channels (at around *y* = 40); and the channel at *x* = 75, *y* = 25 demonstrates the case of generation and degeneration of a channel.

We train the network using 600 samples using the Adam gradient descent. We find that the cluster assignment of 600 realizations in uncertain space is stable(fixed) when the gradient descent epoch reaches a certain number, so we set the stopping criteria to be: the assignment does not change for 100 iteration epochs; and the maximum number of iteration epochs is set to be 1000. We also find that the coefficients in Equation (16) can affect the training result. We set *λ*1 = *λ*2 = *λ*3 = 1.

It should be noted that we train the network locally in each coarse neighborhood. The fine mesh element has size 1/100 and 5 fine elements are merged into one coarse element.

The dimension reduction network *F* contains 4 fully connected layers to reduce the size of local coarse elements from 100 to 60, 40, 30, 20 gradually. The K-means clustering is conducted in space *<sup>F</sup>*(*x*) of dimension 20; the reconstruction net *G* is designed symmetrically with the reduction network *F*. The adversary net is fully convolutional. All convolution layers except the last layer have kernels of size 3 by 3 with stride 1; we use 1 by 1 convolution in the last layer to reduce the number of channels to 1. The number of channels is doubled if the spatial dimension is reduced and half-ed if the spatial dimension is increased. Max pooling of size 2 by 2 is used to reduce the spatial dimension in the encoder; and to increase the dimension in the decoder, we perform the nearest neighbor resize followed by convolution [52].
