1. Introduction
Computer simulations play an essential role in nuclear reactor safety analysis, design, and licensing. While these simulations can model real physical phenomena, they are approximations with inherent uncertainties from various sources. Quantifying these uncertainties is an important step in the simulation model validation process because the assessment of model accuracy requires a reliable measure of uncertainty for model predictions. Uncertainty quantification (UQ) is the process of quantifying the uncertainties in the model outcomes (Quantities-of-Interest, or QoIs) by propagating the uncertainties from the input parameters through the computer model. In the nuclear energy community, UQ holds a particularly pivotal role, compared to other field, as it aids in minimizing over-conservatism in systems with potentially severe consequences. Key activities in nuclear power plant development, such as nuclear reactor design, safety analysis, and licensing, all rely on computer codes whose credibility has been established through a rigorous verification, validation and uncertainty quantification (VVUQ) process.
Inverse Uncertainty Quantification (IUQ) is the process of inversely quantifying the input uncertainties based on experimental data. IUQ seeks statistical descriptions of the uncertain input parameters that are consistent with the experimental data [
1]. The uncertainty information of the input parameters can thus be used for further VVUQ tasks. IUQ has emerged as a particularly prominent segment of UQ in recent years, which can be attributed to the increasing requirements from using computer codes for nuclear reactor simulations, as well as the continuous expansion in computational power, coupled with advancements in machine learning and sophisticated statistical methodologies.
A comprehensive framework for computer model calibration under the Bayesian formulation was initially introduced by Kennedy and O’Hagan [
2]. Subsequently, this methodology has been adapted for use in various domains. While the technique offers a broad framework for implementing Bayesian calibration in computational models, it requires customization for specific fields. Because the method is dependent on the available measurement data and the parameter space, which are very different across different domains. IUQ is a natural extension to Bayesian calibration because instead of calibrating the input parameters with a point estimate, it primarily focuses on the distribution information of input parameters. In the nuclear TH field, IUQ has primarily focused on the parametric uncertainty from closure relations in TH codes.
Parametric uncertainty can come from the empirical equations used in two-fluid models or other fluid dynamics models, like equations of state and constitutive equations (also known as closure laws). These model parameters are not always known precisely. For large-scale modeling, two-phase transport phenomena are usually described with a multi-fluid continuum formulation and a system of mass, momentum, and energy conservation equations derived for each phase. On smaller scales, complex interactions like forces between phases, evaporation at walls, and others, are typically approximated using various empirical or semi-empirical closure models. The precision and reliability of model predictions largely depend on how well these closure models are calibrated, a process that is traditionally carried out step-by-step. Initially, empirical or semi-empirical submodels undergo calibration and validation using separate-effect test (SET) data, after which the entire model is validated against integral-effect test (IET) data [
3]. SETs, commonly executed at a scale smaller than real reactors, primarily concentrate on specific experimental QoIs, minimizing the interference with other phenomena [
4]. After thorough analysis of these tests, thermal-hydraulic experts determine the closure relationships. During this procedure, closure models are developed to minimize added computational load. However, these models might possess considerable uncertainties from knowledge gaps, scalability concerns, and oversimplifications.
In recent years, many IUQ methodologies have been developed in the nuclear community and several international projects have also been performed. For example, (1) the PREMIUM (Post-BEMUSE Reflood Models Input Uncertainty Methods) [
5] benchmark, which focused on core reflood problems to quantify and validate input uncertainties in system TH models, (2) the SAPIUM (Systematic APproach for Input Uncertainty quantification Methodology) [
6] project that aimed to develop a systematic approach for input UQ methodology in nuclear TH codes, and (3) the ATRIUM (Application Tests for Realization of Inverse Uncertainty quantification and validation Methodologies in thermal-hydraulics) [
7] project that was initiated in 2022 to perform practical IUQ exercises of a demonstration of the SAPIUM approach. Wu et al. [
1] conducted a comprehensive survey where twelve IUQ methods for nuclear TH applications are reviewed, compared, and evaluated. More recently, Liu et al. [
8] developed a SAM-ML framework for calibration of closure laws in the SAM system code. A nonlinear extension of the CIRCE method was introduced in [
4], employing Bayesian inference for IUQ in closure relations of TH codes. This was further expanded to account for multiple experimental groups in [
9]. Furthermore, Xie et al. [
10] combined IUQ and quantitative validation via Bayesian hypothesis testing to improve the predictive capability of computer simulations. Besides the applications in nuclear energy, the Bayesian approach for IUQ has also been applied to many other fields such as biotechnology [
11], geophysics [
12], additive manufacturing [
13], computational fluid dynamics [
14,
15], etc.
Many recent advancements in IUQ are attributed to the rapid evolution of ML/AI technologies. These technologies have not only enhanced the accuracy and reliability of simulation models across diverse industries but have also proven effective in addressing numerous complex challenges [
16]. They have been successfully applied in various fields such as healthcare [
17,
18,
19], agriculture [
20], transportation [
21,
22,
23], clinical studies [
24,
25], medical imaging [
26,
27], civil engineering [
28], industrial engineering [
29,
30], etc. The effectiveness of ML/AI methodologies in these domains demonstrates their versatility and offers valuable insights for our IUQ research in the nuclear energy sector.
In the field of Bayesian IUQ studies for nuclear TH, the research focus has primarily been on using single-level Bayesian inference to investigate the posterior distributions of parameters. These approaches have the advantage of being applicable with relatively small datasets. Nevertheless, the posterior distributions derived from these calculations are typically specific to the chosen experimental scenarios. They may change if different datasets are used. In practice, researchers may be more interested in inferring the population distribution of the parameters given complete experimental datasets. The hierarchical Bayesian models can help in applications where observations are organized into distinct groups. In many nuclear TH applications, the physical model parameters (PMPs) may differ across groups, caused by the group effect introduced by experimental data. For example, many PMPs use different constitutive equations at different experimental conditions (boundary conditions, geometries, flow regimes, etc.). These different constitutive equations impose “group” characteristics—the PMPs within a group show similar behaviors because their constitutive equations are be derived from similar separate-effect test (SET), while the PMPs at very different experimental conditions and constitutive equations are likely to have different level of uncertainty and accuracy. Within each group, the “individual” characteristics of PMPs are due to the parametric uncertainties and measurement error in deriving each constitutive equations. Thus, employing single-level Bayesian inference may introduce errors by ignoring the group-level characteristics in the input parameters. Hierarchical models allow for the creation and identification of “hyperparameters” to make sure that both “group” characteristics and “individual” characteristics of PMPs are considered.
The idea of the hierarchical Bayesian model is not new but has not been extensively explored in the nuclear energy community. The CIRCE method [
4] employed a hierarchical structure for a nuclear TH application, where a two-level structure is used and each group consists of a single data point. Robertson et al. [
31] used a similar structure to aid the calibration of a fuel performance model. However, the power and flexibility of the hierarchical Bayesian model is not fully explored. Wang et al. [
32,
33] proposed a flexible hierarchical Bayesian model where data observations can be grouped according to their experimental boundary conditions, and conducted a comprehensive comparison between the hierarchical model and the non-hierarchical model. The hierarchical model structure is demonstrated to be less prone to overfitting caused by model discrepancies or outliers, and has the potential to be used for larger sets of experimental data [
32].
However, a practical challenge in the hierarchical Bayesian model is the computation cost. As the number of groups increases, the number of parameters to be quantified also increases. Established Markov Chain Monte Carlo (MCMC) methods scale poorly with data size and parameter space, and become prohibitive when the dimension of parameters is very high. The variational inference (VI) method, also called variational Bayesian, provides a more scalable alternative to MCMC sampling and has been widely used in many applications such as Bayesian neural networks [
34], where a large number of parameters need to be estimated. In the field of nuclear engineering, a variational Bayesian Monte Carlo (VBMC) method has been utilized for the IUQ of a doped UO
2 fission gas release model [
35]. The results showed that VBMC has similar accuracy and superior efficiency compared to traditional MCMC sampling methods.
In this paper, we propose to use VI in the hierarchical Bayesian model to improve the scalability of the Bayesian IUQ framework for nuclear TH applications. We will firstly describe the essential steps in the IUQ framework, then introduce the hierarchical Bayesian model, VI, and explain how they are integrated in the Bayesian IUQ framework. The framework is then applied to a demonstrative example using manufactured data, and later a real-world application using the PMPs in a nuclear TH simulation code and BFBT (BWR Full-size Fine-mesh Bundle Test) benchmark experimental data [
36]. Both examples involve more than 300 parameters and VI is used for parameter estimation in the hierarchical model. The resulting posterior distributions of PMPs are compared with the results from No-U-Turn sampler (NUTS) sampling methods, and the efficiency benefit using VI is demonstrated. The proposed method shows a promising framework for reducing the IUQ computation burden and scaling IUQ to large datasets.
4. Conclusions and Future Work
In this study, we have developed and demonstrated an extension to the existing Bayesian IUQ framework that employs a hierarchical Bayesian model and VI to quantify uncertainties of PMPs more efficiently than most of the previous studies using MCMC sampling. The proposed approach offers a scalable, efficient, and accurate means to obtain the posteriors of PMPs. We provided a comprehensive comparison between the proposed method and a hierarchical model that utilizes NUTS sampling in a previous study [
32]. Through a synthetic data experiment and a case study on IUQ of TRACE based on the BFBT benchmark data, we demonstrated that our VI-based method delivers significant computational advantages without sacrificing the quality of the posterior estimates. Similarly to the NUTS algorithm, the VI-based method requires no manual tuning, thereby extending its utility across diverse applications with minimal adjustments.
We note that, while VI and NUTS generated different posterior distributions for certain hyper-parameters, their predictive performance, as determined by PPC and FUQ, showed similar results. This suggests that the hierarchical Bayesian IUQ framework is robust to the choice of the inference algorithms. The difference could be attributed to the correlations between the parameters, because ADVI assumes that all parameters are independent during the optimization process. This can potentially limit the applicability of ADVI in scenarios where capturing such correlations is desired. But in applications where we are mainly interested in the final distributions of PMPs, such simplification is acceptable. The correlations also lead to the “identifiability” issue in IUQ, which remains an active research area.
For future research, the framework can be extended to incorporate more sophisticated surrogate models for more complex problems, such as those with strong non-linearity. Additionally, the currently example does not consider model discrepancy, while model discrepancy can play an important role in IUQ [
1]. Exploring the potential benefits of the hierarchical model when model discrepancy exists can also be an interesting area.