# Groundwater Contaminant Transport Solved by Monte Carlo Methods Accelerated by Deep Learning Meta-Model

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## Abstract

**:**

## 1. Introduction

## 2. Groundwater Contaminant Transport Model

## 3. Deep Learning Meta-Model

#### 3.1. Graph Convolutional Neural Network Meta-Model

#### 3.2. ChebNet GCNN

#### 3.3. Architecture of Meta-Model

#### 3.4. Assessment of Meta-Model

## 4. Multilevel Monte Carlo Method

#### 4.1. Optimal Number of Samples

## 5. Monte Carlo Methods with a Meta-Model

#### 5.1. MC-M

#### 5.2. MLMC-M

## 6. Results

#### 6.1. Analysis of Meta-Models

#### 6.2. Comparison of MC-M and MLMC-M

#### 6.3. Multilevel Case

- 1LMC and 1LMC-M: standard MC of models on 18,397 mesh elements and its extension by meta-level;
- 2LMC and 2LMC-M: 2-level MLMC on models with 18,397 and 2714 mesh elements, and its extension by meta-level trained on the model of 2714 mesh elements;
- 3LMC and 3LMC-M: 3 level MLMC with models on 18,397, 2714, and 474 mesh elements and its extension by meta-level trained on the model of 474 mesh elements

#### 6.4. Approximation of Probability Density Function

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DGR | deep geological repository |

DNN | deep neural networks |

GCNN | graph convolutional neural network |

MC | Monte Carlo method |

MEM | Maximum entropy method |

MLMC | multilevel Monte Carlo method |

NRMSE | normalized mean squared error |

probability density function |

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**Figure 1.**Contaminant transport problem. A high concentration (X_conc) of a contaminant is spreading from the DGR to the surface. Its course is influenced by the depicted hydraulic conductivity (conductivity) of the rock environment.

**Figure 2.**An illustrative example of a random field on an unstructured mesh with the corresponding graph representation.

**Figure 3.**Diagram of the architecture of the used deep learning meta-model. The ChebNet graph convolutional neural network is followed by the global summation pool and the deep feed-forward neural network with hidden layers of 64 and 32 neurons. The convolutional kernel has $Ch$ channels.

**Figure 4.**Cost (blue) and moments error (red) of Monte Carlo methods on six different model mesh sizes. MC (square)—standard Monte Carlo method; MC-M (circle)—standard Monte Carlo using meta-model samples; MLMC-M (triangle)—MC extended by a meta-level. $J(\widehat{\mathbf{\mu}},\tilde{\mathbf{\mu}})$ denote error between MC moments ($\widehat{\mathbf{\mu}}$) and MC-M or MLMC-M moments ($\tilde{\mathbf{\mu}}$).

**Figure 5.**Computational costs of different Monte Carlo methods and number of moments R. XLMC denotes MLMC of X levels, whereas XLMC-M has an additional meta-level.

**Figure 6.**Variances of moments across levels for 3LMC (circle) and 3LMC-M (triangle). For legibility, the values are slightly shifted on the x-axis. The leftmost triangles correspond to the meta-level values. The numbers added represent ${C}_{l}\left[sec\right]$ of 3LMC-M.

**Figure 7.**PDFs approximated by moments estimated via 3LMC (blue) and 3LMC-M (red dashed). Both compared to the reference MC (black dotted) by the Kullback–Leibler divergence D.

Mesh Size | Accuracy of Meta-Model (${\mathit{J}}_{\mathcal{L}}$) | |
---|---|---|

Deep Meta-Model | Shallow Meta-Model | |

53 | $0.779$ | $0.945$ |

115 | $0.799$ | $0.949$ |

474 | $0.763$ | $0.885$ |

2714 | $0.773$ | $0.896$ |

10,481 | $0.799$ | $0.898$ |

18,397 | $0.792$ | $0.954$ |

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**MDPI and ACS Style**

Špetlík, M.; Březina, J. Groundwater Contaminant Transport Solved by Monte Carlo Methods Accelerated by Deep Learning Meta-Model. *Appl. Sci.* **2022**, *12*, 7382.
https://doi.org/10.3390/app12157382

**AMA Style**

Špetlík M, Březina J. Groundwater Contaminant Transport Solved by Monte Carlo Methods Accelerated by Deep Learning Meta-Model. *Applied Sciences*. 2022; 12(15):7382.
https://doi.org/10.3390/app12157382

**Chicago/Turabian Style**

Špetlík, Martin, and Jan Březina. 2022. "Groundwater Contaminant Transport Solved by Monte Carlo Methods Accelerated by Deep Learning Meta-Model" *Applied Sciences* 12, no. 15: 7382.
https://doi.org/10.3390/app12157382