# Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations

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## Abstract

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## 1. Introduction

## 2. Problem Settings

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

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

#### 2.2.1. Snapshot Space

#### 2.2.2. Offline Spaces

#### 2.3. The Idea of the Proposed Method

- (Training) For a given input local neighborhood ${\omega}_{j}$, we train the cluster (which will be detailed in next section) of the parameter space $\mathrm{\Omega}$ and get the clusters ${S}_{1}^{j},\dots ,{S}_{n}^{j}$, where n is the number of clusters and is uniform for all j. Please note that we may have different cluster assignments in different local neighborhoods.
- (Training) For each local neighborhood ${\omega}_{j}$ and cluster ${S}_{i}^{j}$, define the average ${\overline{\kappa}}_{ij}$ and compute generalized multiscale basis for ${\overline{\kappa}}_{ij}$.
- (Testing) Given a new $\kappa (x,s)$ and for each local neighborhood ${\omega}_{j}$, fit $\kappa (x,s)$ into a ${S}_{i}^{j}$ by the trained network (step 1) and use the pre-computed GMsFEM basis (step 2) to find the solution.

## 3. Deep Learning

#### 3.1. Clustering Net

- Initialize the networks and clustering the output basis function.
- Compute the loss function L (defined later) and run optimization.
- Cluster the latent space by K-means algorithm (reduced dimension space, which is a middle layer of the cluster network); the latent space data are computed using the previous optimized parameters; the assignment will be denoted as A.
- Basis functions whose corresponding inputs are in the same cluster (basing on assignment A) will be grouped together. No training or fitting-in involved in this step.
- Repeat step 2 to step 4 until the stopping criteria is met.

#### 3.2. Loss Functions

- Clustering loss $C({\theta}_{F},{\theta}_{G})$: this is the mean standard deviation of all clusters of the learned basis and $\theta $ is the parameters we need to optimize. It should be noted that the loss here is computed using the learned basis instead of the input of the network. This loss controls the clustering process, i.e., the smaller the loss, the better the clustering in the sense of clustering the multiscale basis. Let us denote ${\kappa}_{ij}$ as jth realization in ith cluster; $G\left(F\left({\kappa}_{ij}\right)\right)\in {\mathbb{R}}^{d}$ will then be jth learned basis in cluster i and let ${\theta}_{G}$ and ${\theta}_{F}$ be the parameters associated with G and F, the loss is then defined as follow,$$\begin{array}{c}\hfill C({\theta}_{F},{\theta}_{G})=\frac{1}{\left|A\right|}\sum _{i}^{\left|A\right|}\sum _{j}^{{A}_{i}}\frac{1}{{A}_{i}}{\parallel G\left(F\left({\kappa}_{ij}\right)\right)-{\overline{\varphi}}_{i}\parallel}_{2}^{2},\end{array}$$
- Reconstruction loss $R({\theta}_{F},{\theta}_{G})$: this is the mean square error of multiscale basis $Y\in {\mathbb{R}}^{m,d}$, where m is the number of samples. This loss controls the construction process, i.e., if the loss is small, the learned basis are close to the real multiscale basis. This loss will supervise the learning of the cluster. It is defined as follow:$$\begin{array}{c}\hfill R({\theta}_{F},{\theta}_{G})=\frac{1}{m}\sum _{i}^{m}{\parallel G\left(F\left({\kappa}_{i}\right)\right)-{\varphi}_{i}\parallel}_{2}^{2},\end{array}$$

#### 3.3. Adversary Network Severing as an Additional Loss

## 4. Numerical Experiments

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

#### 4.2. Results

#### 4.2.1. Cluster Assignment in a Local Coarse Element

#### 4.2.2. Relation of Error and the Number of Clusters

#### 4.2.3. Comparison of Cluster-Based Method with Tradition Method

#### 4.3. Effect of the Adversary Net

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**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.

**Figure 8.**Cluster results of 28 samples, images shown are heterogeneous fields, the number on top of each image is the cluster assignment ID number.

**Figure 9.**Cluster results of 20 samples, images shown are heterogeneous fields, the number on top of each image is the cluster assignment ID number.

**Figure 10.**The ${l}_{2}$ error when the number of clusters changes, colors represent the number of GMsFEM basis.

**Figure 11.**The ${l}_{2}$ error cluster solution (11 clusters) vs. solution by real $\kappa (x,\widehat{s})$. Color represents number of basis.

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**MDPI and ACS Style**

Zhang, Z.; Chung, E.T.; Efendiev, Y.; Leung, W.T.
Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations. *Mathematics* **2020**, *8*, 720.
https://doi.org/10.3390/math8050720

**AMA Style**

Zhang Z, Chung ET, Efendiev Y, Leung WT.
Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations. *Mathematics*. 2020; 8(5):720.
https://doi.org/10.3390/math8050720

**Chicago/Turabian Style**

Zhang, Zecheng, Eric T. Chung, Yalchin Efendiev, and Wing Tat Leung.
2020. "Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations" *Mathematics* 8, no. 5: 720.
https://doi.org/10.3390/math8050720