# Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference

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## Abstract

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## 1. Introduction

_{2}fission gas release model [35]. The results showed that VBMC has similar accuracy and superior efficiency compared to traditional MCMC sampling methods.

## 2. Materials and Methods

#### 2.1. Bayesian IUQ Framework Overview

- Problem Definition. Identify the problem being studied and choose relevant experimental data and simulation codes. In this work, we utilize the BFBT (BWR Full-size Fine-mesh Bundle Test) benchmark data [36] and build the corresponding models using TRACE. Details of the experimental data and the simulation codes will be introduced in the following session.
- Sensitivity Analysis. In this step, we aim to identify the key and influential input parameters. For the chosen TH codes, these input parameters are typically multipliers for the coefficients in closure equations, such as single-phase/two-phase heat transfer coefficients, interfacial drag coefficients, etc. This identification can be accomplished through two successive SA steps. Initially, a relatively straightforward perturbation method is applied to all parameters to identify those that are active in the model. Following this, a more precise SA method, known as Sobol indices, is employed to determine the influential parameters within a constrained variable space. In the field of nuclear TH, a variety of SA techniques have been effectively employed. These include Subset Simulation, Line Sampling [44], Sobol indices [45,46], Pearson correlation coefficient, elementary effects method [47], adjoint method [48] etc.
- Surrogate Model. Surrogate models are approximations of the input/output relation of the original computer model. They are developed using a limited set of full model simulations (known as the training set) combined with a learning algorithm. Typical MCMC sampling algorithms takes at least thousands of samples, thus if the simulation codes are computationally expensive, it would be impossible to to conduct the computation efficiently. In this scenario, we can use a surrogate model, also referred to as emulators or metamodels, to replace the computationally prohibitive simulation code. Many learning algorithms are available and have been successfully applied to TH applications, such as, Polynomial Regression (PR) [43], Gaussian Process (GP) [41,49], Artificial Neural Networks [50,51,52], etc.
- Hierarchical Bayesian Model. Once the problem is clearly defined and the uncertain inputs are identified, a hierarchical Bayesian model is formulated accordingly. The hierarchical structure should be defined based on the group effects in the experimental data. This step also involves establishing the likelihood function and formulating the posterior distributions.
- Posterior Exploration. In this step, inference algorithms such as MCMC and VI are employed compute the approximate posterior distributions of the parameters.
- Posterior Predictive Check (PPC). PPC is the process of comparing the observed experimental data to the posterior predictions of the model. The core concept is that if the posterior parameter distributions are good approximations of the “true” underlying distributions, then the predicted data from the model should actually “look like” real observed data. If the patterns in the predicted data do not mirror the patterns in the observed data, then we are motivated to invent models that can produce the QoIs [53]. PPC provides a great way to confirm the the obtained posteriors.
- Forward UQ (FUQ). In applications where we are interested in the uncertainty ranges of QoIs, the derived posterior distributions are propagated through the simulation model. This FUQ process is expected to produce more accurate model prediction uncertainties by using the PMP uncertainties quantified from IUQ.

#### 2.2. TRACE PMPs and BFBT Benchmark Data

#### 2.3. Hierarchical Bayesian Model

- Draw global variables $\mathit{b}\sim p\left(\mathit{b}\right)$ and ${\mathbf{\Sigma}}_{\mathit{\theta}}\sim p\left({\mathbf{\Sigma}}_{\mathit{\theta}}\right)$.
- Draw group-specific variables ${\theta}_{i}$ according to $p\left({\theta}_{i}\right|{\mathbf{\Sigma}}_{\mathit{\theta}}),i=1,2,\dots M.$
- Draw observed data point ${y}_{i}\sim p\left({y}_{i}\right|\mathit{\theta},\mathit{b}),i=1,2,\dots N$.

#### 2.4. Markov Chain Monte Carlo

#### 2.5. Variational Inference

## 3. Results

#### 3.1. Synthetic Data Example

#### 3.1.1. Problem Definition

#### 3.1.2. Results and Discussions

- Draw samples of global variables, (${\mu}_{{\theta}_{1}},{\sigma}_{{\theta}_{1}},{\mu}_{{\theta}_{2}},{\sigma}_{{\theta}_{2}},{\mu}_{{\theta}_{3}},{\sigma}_{{\theta}_{3}}$), from their prior distributions. We use wide uniform distributions as priors to reflect our ignorance of knowledge.
- For $i=1,2,3$ and $n=1,2,\dots N$, draw samples of group-specific parameters ${\theta}_{i}^{n}\sim \mathcal{N}({\mu}_{{\theta}_{i}},{\sigma}_{{\theta}_{i}})$.

#### 3.2. Nuclear Thermal-Hydraulics Application

#### 3.2.1. Problem Definition

#### 3.2.2. Sensitivity Analysis and Surrogate Model

#### 3.2.3. Results

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ADVI | Automatic Differentiation Variational Inference |

BFBT | BWR Full-size Fine-mesh Bundle Test |

ELBO | Evidence Lower Bound |

FUQ | Forward Uncertainty Quantification |

GP | Gaussian Process |

IET | Integral-effect test |

IUQ | Inverse Uncertainty Quantification |

KDE | Kernel Density Estimate |

LHS | Latin Hypercube Sampling |

MAE | Mean Absolute Error |

MCMC | Markov Chain Monte Carlo |

NUTS | No-U-Turn-Sampler |

PMP | Physical Model Parameters |

PPC | Posterior Predictive Check |

PR | Polynomial Regression |

QoI | Quantity-of-Interest |

SET | Separate-Effect Test |

TH | Thermal-Hydraulics |

UQ | Uncertainty Quantification |

VBMC | Variational Bayesian Monte Carlo |

VF | Void Fraction |

VI | Variational Inference |

VVUQ | Verification, Validation, and Uncertainty Quantification |

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**Figure 1.**Key Elements of the IUQ framework [32].

**Figure 2.**Experimental facility for void fraction measurement in the BFBT benchmark [36].

**Figure 3.**Comparison of TRACE-predicted and experimentally-measured void fractions, assembly 4 in the BFBT benchmark.

**Figure 7.**KDE plots of the posterior distributions sampled by NUTS and ADVI algorithms and comparison with true values.

**Figure 11.**Comparison of the posterior distributions of total variance ${\sigma}_{t}$ by NUTS and ADVI methods.

**Figure 12.**Correlations between two pairs of hyper-parameters, (${\mu}_{{P}_{1008}}$, ${\mu}_{{P}_{1012}}$) and (${\mu}_{{P}_{1022}}$, ${\mu}_{{P}_{1028}}$), using bivariate KDE plots.

**Figure 14.**Comparison of simulation results among (1) original model; (2) updated model using posteriors obtained by ADVI; and (3) updated model using posteriors obtained by NUTS. The error bars represent the 2.5–97.5 percentile of the model QoIs obtained from the FUQ step, using the posterior distributions of the PMPs that have been quantified during IUQ.

ADVI | NUTS | |
---|---|---|

Number of fitting/sampling steps required | 120,000 | 6000 |

Computational time | 12 s | 50 s |

Parameter | Definition |
---|---|

${P}_{1008}$ | Single phase liquid to wall heat transfer coefficient |

${P}_{1012}$ | Subcooled boiling heat transfer coefficient |

${P}_{1022}$ | Wall drag coefficient |

${P}_{1028}$ | Interfacial drag (bubbly/slug Rod Bundle–Bestion) coefficient |

Parameters | Distributions | Dist. Parameter 1 | Dist. Parameter 2 |
---|---|---|---|

${\mu}_{{P}_{1008}},{\mu}_{{P}_{1012}},{\mu}_{{P}_{1022}},{\mu}_{{P}_{1028}}$ | Uniform | $a=0$ | $b=3$ |

${\sigma}_{{P}_{1008}},{\sigma}_{{P}_{1012}},{\sigma}_{{P}_{1022}},{\sigma}_{{P}_{1028}}$ | Uniform | $a=0$ | $b=1$ |

${\sigma}_{t}$ | Normal | $\mu =0$ | $\sigma =1$ |

Parameters | NUTS | ADVI | ||
---|---|---|---|---|

${P}_{1008}$ | $\mu =1.63$ | $\sigma =0.66$ | $\mu =1.34$ | $\sigma =0.49$ |

${P}_{1012}$ | $\mu =1.32$ | $\sigma =0.18$ | $\mu =1.45$ | $\sigma =0.21$ |

${P}_{1022}$ | $\mu =0.89$ | $\sigma =0.12$ | $\mu =0.89$ | $\sigma =0.33$ |

${P}_{1028}$ | $\mu =1.26$ | $\sigma =0.15$ | $\mu =1.32$ | $\sigma =0.22$ |

ADVI | NUTS | |
---|---|---|

Number of fitting/sampling steps required | 300,000 | 100,000 |

Computational time | 58 s | 1520 s |

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**MDPI and ACS Style**

Wang, C.; Wu, X.; Xie, Z.; Kozlowski, T.
Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference. *Energies* **2023**, *16*, 7664.
https://doi.org/10.3390/en16227664

**AMA Style**

Wang C, Wu X, Xie Z, Kozlowski T.
Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference. *Energies*. 2023; 16(22):7664.
https://doi.org/10.3390/en16227664

**Chicago/Turabian Style**

Wang, Chen, Xu Wu, Ziyu Xie, and Tomasz Kozlowski.
2023. "Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference" *Energies* 16, no. 22: 7664.
https://doi.org/10.3390/en16227664