# Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation

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## Abstract

**:**

## 1. Introduction

- 1
- Construct the POD projection basis matrix and define the interpolation points using DEIM.
- 2
- Generate the training dataset.
- •
- Generate the input data.
- •
- Solve the poroelasticity problem with the partially explicit discretization at each time step.

- 3
- Train the Deep Neural Network to obtain values of the implicit flow part at the interpolation points at some time steps.

- 1
- The Deep Neural Network obtains the implicit part of the pressure at the interpolation points at some time moments.
- 2
- The POD projection basis matrix restores the complete implicit parts of the flow at some time moments.
- 3
- Linear interpolation over time.
- 4
- For each time step,
- •
- Compute the explicit flow part using the learned and interpolated implicit one.
- •
- Solve the displacement using the implicit and explicit parts of the flow.

## 2. Problem Formulation

## 3. Approximation

- Pressures $({p}_{1}^{n+1},{p}_{2}^{n+1})\in ({W}_{1}\times {W}_{2})$ such that$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {d}_{1}\left(\frac{{u}^{n}-{u}^{n-1}}{\tau},{w}_{1}\right)+{c}_{1}\left(\frac{{p}_{1}^{n+1}-{p}_{1}^{n}}{\tau},{w}_{1}\right)+{b}_{1}({p}_{1}^{n+1},{w}_{1})+{q}_{12}({p}_{1}^{n+1}-{p}_{2}^{n+1},{w}_{1})=0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {d}_{2}\left(\frac{{u}^{n}-{u}^{n-1}}{\tau},{w}_{2}\right)+{c}_{2}\left(\frac{{p}_{2}^{n+1}-{p}_{2}^{n}}{\tau},{w}_{2}\right)+{b}_{2}({p}_{2}^{n},{w}_{2})-{q}_{21}({p}_{1}^{n+1}-{p}_{2}^{n+1},{w}_{2})=l\left({w}_{2}\right),\hfill \end{array}$$
- Displacements ${u}^{n+1}$ such that$$a({u}^{n+1},v)+{g}_{1}({p}_{1}^{n+1},v)+{g}_{2}({p}_{2}^{n+1},v)=0,$$

- Pressures $({p}_{1}^{n+1},{p}_{2}^{n+1})$ such that$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {B}_{1}{p}_{1}^{n+1}+{C}_{1}\frac{{p}_{1}^{n+1}-{p}_{1}^{n}}{\tau}+{D}_{1}\frac{{u}^{n}-{u}^{n-1}}{\tau}+{Q}_{12}({p}_{1}^{n+1}-{p}_{2}^{n+1})=0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {B}_{2}{p}_{2}^{n}+{C}_{2}\frac{{p}_{2}^{n+1}-{p}_{2}^{n}}{\tau}+{D}_{2}\frac{{u}^{n}-{u}^{n-1}}{\tau}-{Q}_{21}({p}_{1}^{n+1}-{p}_{2}^{n+1})=L.\hfill \end{array}$$
- Displacements ${u}^{n+1}$ such that$$A{u}^{n+1}+{G}_{1}{p}_{1}^{n+1}+{G}_{2}{p}_{2}^{n+1}=0.$$

## 4. Discrete Empirical Interpolation Method with Proper Orthogonal Decomposition

- Computing the Proper Orthogonal Decomposition (POD) basis functions;
- Determining the interpolation nodes using the DEIM algorithm.

## 5. Machine Learning Approach

- The first output layer: ${\mathcal{N}}^{1}\left({x}^{0}\right)={Y}^{1}$, i.e., ${Y}^{1}={\sigma}^{1}({W}^{1}x+{b}^{1})$;
- For i’s layer: ${Y}^{i}={\sigma}^{i}({W}^{i}{x}^{i-1}+{b}^{i})$, where i = 1,2,…, L.

- Activation function: ReLU (Rectified Linear Unit) activation function for all layers (first input and hidden layers), no activation function at the last output layer;
- DNN structure: 2 hidden layers, each layer comprises 12 neurons;
- Kernel initializer: normal for input and output layers and he_normal for all hidden layers;
- Training optimizer: Adam.

## 6. Numerical Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Flow charts of the offline and online stages of partial learning using partially explicit discretization with limited observation.

**Figure 4.**Elasticity parameter E (

**left**) and heterogeneous permeabilities ${k}_{1}$ (

**center**) and ${k}_{2}$ (

**right**).

**Figure 5.**Distribution of pressure for the first continuum at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 6.**Distribution of pressure for the second continuum at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 7.**Distribution of displacement in ${x}_{1}$ direction at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 8.**Distribution of displacement in ${x}_{2}$ direction at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 9.**Relative ${L}_{2}$ errors in % for the reference mesh (6561 vertices) with other meshes (121, 441, and 1681 vertices) for the first and second continuum and displacement (from

**left**to

**right**).

**Figure 10.**Distributions of the stress at the final time. The coupled explicit–implicit solution, the split explicit–implicit solution, and the proposed approach’s solution (from

**left**to

**right**).

**Figure 11.**Distributions of the strain at the final time. The coupled explicit–implicit solution, the split explicit–implicit solution, and the proposed approach’s solution(from

**left**to

**right**).

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**MDPI and ACS Style**

Tyrylgin, A.; Stepanov, S.; Ammosov, D.; Grigorev, A.; Vasilyeva, M.
Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation. *Mathematics* **2022**, *10*, 2629.
https://doi.org/10.3390/math10152629

**AMA Style**

Tyrylgin A, Stepanov S, Ammosov D, Grigorev A, Vasilyeva M.
Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation. *Mathematics*. 2022; 10(15):2629.
https://doi.org/10.3390/math10152629

**Chicago/Turabian Style**

Tyrylgin, Aleksei, Sergei Stepanov, Dmitry Ammosov, Aleksandr Grigorev, and Maria Vasilyeva.
2022. "Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation" *Mathematics* 10, no. 15: 2629.
https://doi.org/10.3390/math10152629